Spectral Applications of Vertex-Clique Incidence Matrices Associated with a Graph

Abstract

Using the notions of clique partitions and edge clique covers of graphs, we consider the corresponding incidence structures. This connection furnishes lower bounds on the negative eigenvalues and their multiplicities associated with the adjacency matrix, bounds on the incidence energy, and on the signless Laplacian energy for graphs. For the more general and well-studied set $S\left(G\right)$ of all real symmetric matrices associated with a graph G, we apply an extended version of an incidence matrix tied to an edge clique cover to establish several classes of graphs that allow two distinct eigenvalues.

1. Introduction

Let $G=(V,E)$ be a simple undirected graph with n vertices and m edges. A clique in G is a subset $C\subseteq V$ such that all vertices in C are pairwise adjacent. An edge clique cover F of G is a set of cliques $F=\{C_1,C_2,\dots ,C_k\}$ that together contain each edge of G at least once. The smallest size of an edge clique cover of G is called the edge clique cover number of G and is denoted by cc$\left(G\right)$. An edge clique cover of G with size $cc\left(G\right)$ is called a minimum edge clique cover of G. A special case of an edge clique cover in which every edge belongs to exactly one clique is called a clique partition of G. The size of the smallest clique partition of G is called the clique partition number of G, and is denoted by $cp\left(G\right)$. A clique partition of G with size $cp\left(G\right)$ is referred to as a minimum clique partition of G. It is clear that both $cc\left(G\right)$ and $cp\left(G\right)$ exist as E forms a clique partition (and hence an edge clique cover) of G. Further, note that any minimum clique partition does not contain any cliques of order one, and, by convention, the clique partition number of the empty graph is defined to be zero. Information concerning clique partitions and edge clique covers of a graph can be found in the works [1,2,3,4].

Before defining the various matrices associated with a graph, we make note of standard matrix notations: $I_n$ denotes the $n\times n$ identity matrix; O denotes the zero matrix (size determined by context); J denotes the all ones matrix (size determined by context); and $𝟙$ denotes the all ones vector (size determined by context).

Given a graph G with $V=\{1,2,\dots ,n\}$ and $E=\{e_1,e_2,\dots ,e_m\}$, the (vertex-edge) incidence matrix M of G is the $n\times m$ matrix defined as follows: the rows and the columns of M are indexed by V and E, respectively, and the $(i,j)$-entry of M is 0 if $i\notin e_j$ and 1 otherwise. Similarly, the adjacency matrix $\mathcal{A}=\mathcal{A}\left(G\right)=\left(a_{ij}\right)$ is a $(0,1)$-matrix of G such that $a_{ij}=1$ if $ij\in E\left(G\right)$ and 0 otherwise. It is well known that [5]

$$M{M}^T=Q\left(G\right),\mathrm{and}{M}^{T}M=\mathcal{A}\left(L_G\right)+2I_m,$$

(1)

where $D\left(G\right)=\left(d_{ij}\right)$ is the diagonal matrix of vertex degrees ($d_{i,i}=deg\left(i\right):=d_i$, $i=1,2,\dots ,n$) and the matrix $Q\left(G\right)=D\left(G\right)+\mathcal{A}\left(G\right)$ is known as the signless Laplacian matrix of the graph G; the line graph, $L_G$, of the graph G is the graph whose vertex set is in one-to-one correspondence with the set of edges of G, where two vertices of $L_G$ are adjacent if and only if the corresponding edges in G have a vertex in common [6]. Finally, the equations in (1) imply an important spectral relation between the signless Laplacian matrix $Q\left(G\right)$ and $\mathcal{A}\left(L_G\right)$, see Lemma 6.

As we are also interested in studying more general symmetric matrices associated with a graph on n vertices, we let $S\left(G\right)$ denote the collection of real symmetric matrices $A=\left(a_{ij}\right)$ such that for $i\ne j$, $a_{ij}\ne 0$ if and only if $ij\in E\left(G\right)$. The main diagonal entries of any such A in $S\left(G\right)$ are not constrained. Observe that for any graph G, both $Q\left(G\right)$ and $\mathcal{A}\left(G\right)$ belong to $S\left(G\right)$.

We denote the spectrum of A, i.e., the multiset of eigenvalues of A, by $\mathrm{Spec}\left(A\right)$. In particular, $\mathrm{Spec}\left(A\right)=\{{\lambda }_1^{\left[m_1\right]},{\lambda }_2^{\left[m_2\right]},\dots ,{\lambda }_q^{\left[m_q\right]}\},$ where the distinct eigenvalues of A are given by ${\lambda }_1<{\lambda }_2<\cdots <{\lambda }_q$ with corresponding multiplicities of these eigenvalues are $m_1,m_2,\dots ,m_q$, respectively. Further, we consider the ordered multiplicity list of A as the sequence $m\left(A\right)=(m_1,m_2,\dots ,m_q)$. For brevity, a simple eigenvalue ${\lambda }_k^{\left[1\right]}$ is simply denoted by ${\lambda }_k$.

With respect to the set $S\left(G\right)$, the parameter $q\left(G\right)$ is defined by $q\left(G\right)=min\left\{q\right(A):A\in S(G\left)\right\},$ where $q\left(A\right)$ is the number of distinct eigenvalues of A (see [7,8]). The number $q\left(G\right)$ is known as the minimum number of distinct eigenvalues of the graph G. The class of matrices $S\left(G\right)$ has been of recent interest (see [9,10,11] and the references therein), and there has been considerable development on the inverse eigenvalue problem for graphs (see [12]) which continues to receive considerable and deserved attention, as it remains one of the most interesting unresolved issues in combinatorial matrix theory.

Using the notions of clique partitions and edge clique covers of a graph we generalize the conventional vertex-edge incidence matrix M by considering an incidence matrix called the vertex-clique incidence matrix of a graph. Eigenvalues of graphs and clique partitions have arisen previously, see, for example, the works [13,14], and for other types of graph decompositions see [15]. Suppose $F=\{C_1,C_2,\dots ,C_k\}$ is an edge clique cover of a graph G with $V=\{1,2,\dots ,n\}$. The vertex-clique incidence matrix $M_F$ of G associated with the edge clique cover F is defined as follows: the $(i,j)$-entry of $M_F$ is real and nonzero if and only if the vertex i belongs to the clique $C_j\in F$. In the particular case when F is actually a clique partition, the vertex-clique incidence matrix is denoted by ${\mathcal{M}}_F$, and the $(i,j)$-entry of ${\mathcal{M}}_F$ is equal to one if and only if the vertex i belongs to the clique $C_j\in F$. We observe that for any graph G, the vertex-clique incidence matrix corresponding to a clique partition F preserves several main properties of its vertex-edge incidence matrix. For instance, in Section 3, ${\mathcal{M}}_F{\mathcal{M}}_F^T={\mathcal{D}}_F+\mathcal{A}$, where ${\mathcal{D}}_F=diag(t_1^F,t_2^F,\dots ,t_n^F)$, where $t_i^F$ is the number of cliques in F containing the vertex i (this parameter is discussed in more detail in Section 3). Note that for each i, $t_i^F\le d_i$. This fact enables us to determine lower bounds for the negative eigenvalues of the graph.

This paper is organized as follows. In Section 2, we provide the necessary notions, notations, and known results that are needed in the sections containing our main observations. In Section 3, using the notion of a clique partition F of a graph G, we define a signless Laplacian matrix of the graph G associated with the clique partition F. A graph $P_G$ is introduced as a generalization for the line graph of G. In Section 3.1, applying this theory of a vertex-clique incidence matrix, we produce lower bounds for the negative eigenvalues of the graph. Moreover, we present lower bounds for the negative inertia ${\nu }^{-}\left(G\right)$ of a graph G in terms of its order n and the rank of its vertex-clique incidence matrix. We also provide a sufficient condition under which the well-known inequality ${\nu }^{-}\left(G\right)\le n-\alpha \left(G\right)$ holds with equality, where $\alpha \left(G\right)$ is the independence number of G. In Section 3.2, we introduce graph energies associated with a clique partition F of the graph G and study several associated properties. Moreover, upper bounds for the energies of the graph G and its clique partition graph and line graph are determined. In Section 4, studies on the vertex-clique incidence matrix of a graph associated with an edge clique cover lead to a derivation of some new classes of graphs with $q\left(G\right)=2$ (see also Section 4.1).

2. Notations and Preliminaries

In this section, we provide known notions, notations, and results that are used later in this work.

We begin by introducing the notion of the eigenvalues of a graph. The eigenvalues ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ of the adjacency matrix $\mathcal{A}\left(G\right)$ (or shortened to $\mathcal{A}$ when reference to the graph G is clear from context) of the graph G are also called the eigenvalues of G. The number of positive (negative) eigenvalues in the spectrum of the graph G is called the positive (negative) inertia of the graph G, and is denoted by ${\nu }^{+}\left(G\right)$ (${\nu }^{-}\left(G\right)$). The energy of the graph G is defined as

$$\mathcal{E}\left(G\right)=\sum _{i=1}^n\left|{\lambda }_i\right|.$$

(2)

Further details on various properties of graph energy can be found in [16,17,18,19,20]. Suppose $q_1,q_2,\dots ,q_n$ be the eigenvalues of the matrix $Q\left(G\right)$. Then, the signless Laplacian energy of the graph G is defined as

$$L{E}^{+}=L{E}^{+}\left(G\right)=\sum _{i=1}^n|q_i-\frac{2m}{n}|.$$

(3)

More information on properties of the signless Laplacian energy can be found in [21], and the energy of a line graph and its relations with other graph energies are studied in [22,23].

A subgraph H of a graph G is a graph whose vertex set and edge set are subsets of those of G. If H is a subgraph of G, then G is said to be a supergraph of H. The subgraph of G obtained by deleting either a vertex v of G or an edge e of G is denoted by $G-v$ and $G-e$, respectively. Suppose H is a graph on n vertices. Then, we let $K_n\backslash H$ denote the graph obtained from the complete graph, $K_n$, by removing the edges from H (this graph is also known as the complement of the graph H). An independent set in the graph G is a set of vertices in G, no two of which are adjacent. The independence number $\alpha \left(G\right)$ of G is the number of vertices in the largest independent set of G. A matching in a graph G, is a collection of independent edges from G (i.e., no two edges in a matching share a common vertex from G). Additionally, a matching is referred to as perfect if each vertex from G is incident with one edge from the matching.

An $n\times n$ real symmetric matrix B is a positive semi-definite matrix if all of its eigenvalues are nonnegative. In this case, we denote $B\ge 0$. For real symmetric matrices B and C, if $B-C\ge 0$, then we write $B\ge C$.

Lemma 1 

([24]). Let A and B be real symmetric matrices of order n, and assume that $A\le B$. Then, for all $i=1,2,\dots ,n$, ${\lambda }_i\left(A\right)\le {\lambda }_i\left(B\right),$ where ${\lambda }_i\left(M\right)$ is the ith largest eigenvalue of a square matrix M.

The following result was obtained in [5].

Lemma 2 

([5]). If B and C are matrices such that $BC$ and $CB$ are both defined, then $BC$ and $CB$ have the same nonzero eigenvalues with the same multiplicity.

The Schur product of two matrices $A=\left(a_{ij}\right)$ and $B=\left(b_{ij}\right)$ of the same size is defined to be $A\circ B=\left(a_{ij}{b}_{ij}\right)$. An $n\times n$ symmetric matrix A is said to have the Strong Spectral Property (or A has the SSP for short) if the only symmetric matrix X satisfying $A\circ X=O$, $I\circ X=O$ and $[A,X]=AX-XA=O$ is $X=O$ (see [25]). The following result is given in Theorem. 10 [25].

Lemma 3 

([25]). If $A\in S\left(G\right)$ has the SSP, then every supergraph of G with the same vertex set has a matrix realization that has the same spectrum as A and has the SSP.

Given two graphs G and H, the join of G and H, denoted by $G\vee H$, is the graph obtained from $G\cup H$, by adding all possible edges between G and H. Suppose G is a graph with $q\left(G\right)=2$. Then, among all matrix realizations A in $S\left(G\right)$ with two distinct eigenvalues, we define the multiplicity bi-partition $[n-k,k]$ associated with A if the two eigenvalues of A have respective multiplicities $n-k$ and k. Further, we define the minimal multiplicity bi-partition $MB\left(G\right)$ to be the least integer $k\le \lfloor \frac{n}{2}\rfloor$ such that G achieves the multiplicity bi-partition $[n-k,k]$. We close this section with two useful results concerning specific classes of graphs realizing two distinct eigenvalues with respect to the set $S\left(G\right)$.

Lemma 4 

([26,27]). Let G be a connected graph on n vertices. Then,

(1) 

$MB\left(G\right)=1$ if and only if G is the complete graph, $K_n$.

(2) 

$MB\left(G\right)=2$ if and only if $G=(K_{p_1}\cup K_{q_1})\vee (K_{p_2}\cup K_{q_2})\vee \cdots \vee (K_{p_k}\cup K_{q_k})$ for non-negative integers $p_1,\dots ,p_k,q_1,\dots ,q_k$ with $k>1$, and G is not isomorphic to either one of a complete graph or $G=(K_{p_1}\cup K_{q_1})\vee K_1$.

Lemma 5 

([28]). If G is a connected graph of order $n\in \{l,l+1,l+2\}$ and $n_1,\dots ,n_l\in \mathbb{N}$, then $q(G\vee {\cup }_{j\in \left[l\right]}{K}_{n_j})=2$, where $\left[l\right]:=\{1,2,\dots ,l\}$.

3. Matrices Associated with a Clique Partition

In this section, we consider the vertex-clique incidence matrix associated with a clique partition of a graph G. Recall from the introduction that for a given clique partition $F=\{C_1,C_2,\dots ,C_k\}$ of G, the matrix ${\mathcal{M}}_F$ has $(i,j)$-entry is equal to one if and only if the vertex i belongs to the clique $C_j\in F$ (see also [13,14]). Observe that when $F=E$, ${\mathcal{M}}_F$ is simply the conventional incidence matrix of the graph G. For each vertex $i\in \left[n\right]$ of the graph G, we define the parameter $t_i^F=t_i^F\left(G\right)$ to be the number of cliques in F containing the vertex i, that is, $t_i^F=\left|\{j\in \left[k\right]:C_j\in F,i\in C_j\}\right|.$ We call $t_i^F\left(G\right)$the clique-degree of the vertex i in graph G associated with F, and, without loss of generality, after a re-labelling of the vertices if necessary, we assume that $t_1^F\ge t_2^F\ge \dots \ge t_n^F$ (see also [13]). Given clique partition $F=\{C_1,C_2,\dots ,C_k\}$ of G, we consider different possible classes of graphs as follows:

(i)

The graph G is t clique-regular if $t_1^F=\cdots =t_n^F=t$, for some positive integer t;

(ii)

The graph G is s clique-uniform if $|C_1|=\cdots =|C_k|=s$, for some positive integer s;

(iii)

The graph G is $(s,t)$ regular if $t_1^F=\cdots =t_n^F=t$ and $|C_1|=\cdots =|C_k|=s$, for positive integers $s,t$. Any graph is 2 clique-uniform and any d-regular graph is also d clique-regular using the trivial clique partition $F=E$.

Let ${\mathcal{D}}_F$ be the $n\times n$ diagonal matrix with row and column indexed by the vertex set V with $(i,i)$-entry equal to $t_i^F$, that is, ${\mathcal{D}}_F=diag(t_1^F,\dots ,t_n^F)$. It follows that the inner product of row i and row j (with $i\ne j$) of ${\mathcal{M}}_F$ equals the number of cliques in F containing the vertices i and j. By definition of the clique partition F, if i and j are adjacent, then this number is equal to 1 and otherwise 0. This leads to the following result:

Theorem 1. 

Let ${\mathcal{M}}_F$ be the vertex-clique incidence matrix of G associated with a given clique partition F. Then ${\mathcal{M}}_F{\mathcal{M}}_F^T={\mathcal{D}}_F+\mathcal{A},$ where ${\mathcal{D}}_F=diag(t_1^F,\dots ,t_n^F)$ and $\mathcal{A}$ is the adjacency matrix of G.

As mentioned above, in the case of $F=E$, the matrix ${\mathcal{M}}_F$ is the incidence matrix M of G and consequently, ${\mathcal{M}}_F{\mathcal{M}}_F^T=M{M}^T$ is the signless Laplacian matrix of G, where we assume, after possibly re-labelling, that the sequence of vertex degrees is ordered as $d_1\ge d_2\ge \cdots \ge d_n$. Notice that in this case, $t_i^F=d_i$ for $1\le i\le n$. Motivated by this observation, for any clique partition F we call ${\mathcal{Q}}_F={\mathcal{M}}_F{\mathcal{M}}_F^T$ the signless Laplacian matrix of the graph G associated with the clique partition F. Since we always have $D\ge {\mathcal{D}}_F$, it follows $Q=D+\mathcal{A}\ge {\mathcal{D}}_F+\mathcal{A}={\mathcal{Q}}_F\ge 0$. Now define the clique partition graph$P_G$ with k vertices, where each vertex i corresponds to each clique $C_i$ in F such that each pair of vertices of $P_G$ are adjacent if and only if the corresponding cliques in F have a vertex in common. If $F=E$, then $P_G=L_G$ the line graph of G. The inner product of two columns of ${\mathcal{M}}_F$ is nonzero if and only if the corresponding cliques have a common vertex. From the definition of a clique partition, this nonzero value must be 1. These facts immediately yield the following result:

Theorem 2. 

Let ${\mathcal{M}}_F$ be the incidence matrix of G associated with a clique partition F. Then, ${\mathcal{M}}_F^T{\mathcal{M}}_F={\mathcal{S}}_F+\mathcal{A}\left(P_G\right),$ where ${\mathcal{S}}_F=diag(s_1^F,\dots ,s_k^F)$ and $s_i^F=\left|C_i\right|$ and $\mathcal{A}\left(P_G\right)$ denote the adjacency matrix of the graph $P_G$.

For the case of $F=E$, we have ${\mathcal{M}}_F^T{\mathcal{M}}_F=M^{T}M=2I_m+\mathcal{A}\left(L_G\right)$, and $P_G=L_G$ so $s_i^F=2$ for $1\le i\le k=m$.

3.1. Applications of the Vertex-Clique Incidence Matrix to Graph Spectrum

In this section, we develop several results on the spectrum of the graph G and its clique partition graph $P_G$ by the vertex-clique incidence matrix of a graph. Considering ${\mathcal{R}}_F={\mathcal{M}}_F^T{\mathcal{M}}_F$ with Lemma 2 we conclude that the nonzero eigenvalues of matrices ${\mathcal{Q}}_F$ and ${\mathcal{R}}_F$ are the same. This fact leads to the following basic results.

Theorem 3. 

We have the following.

(i) 

If $1\le i\le min\{n,k\}$, then ${\lambda }_i\left({\mathcal{Q}}_F\right)={\lambda }_i\left({\mathcal{R}}_F\right)$.

(ii) 

If $min\{n,k\}=n$, then ${\lambda }_i\left({\mathcal{R}}_F\right)=0$ for $n+1\le i\le k$.

(iii) 

If $min\{n,k\}=k$, then ${\lambda }_i\left({\mathcal{Q}}_F\right)=0$ for $k+1\le i\le n$.

Recall that if $F=E$, then ${\mathcal{Q}}_F=Q$ and ${\mathcal{R}}_F=2I_m+\mathcal{A}\left(L_G\right)$. Combining these equations with Theorem 3 leads to the following well-known result [22,29]:

Lemma 6. 

Let G be a graph of order n with m edges. Then, $q_i\left(G\right)=2+{\lambda }_i\left(L_G\right)for1\le i\le min\{n,m\}.$ In particular if $m>n$ then ${\lambda }_i\left(L_G\right)=-2$ for $i>n$, and if $n>m$ then $q_i\left(G\right)=0$ for $i>m$.

The following result is obtained by applying Theorem 3 for a $(s,t)$ regular graph G with the clique partition F.

Theorem 4. 

Let G be a $(s,t)$ regular graph of order n with a clique partition F of size k.

(i) 

If $1\le i\le min\{n,k\}$, then ${\lambda }_i\left(G\right)-{\lambda }_i\left(P_G\right)=s-t.$

(ii) 

If $min\{n,k\}=n$, then ${\lambda }_i\left(P_G\right)=-s$ for $n+1\le i\le k$.

(iii) 

If $min\{n,k\}=k$, then ${\lambda }_i\left(G\right)=-t$ for $k+1\le i\le n$.

Proof. 

(i) By Theorem 3 (i), if $1\le i\le min\{n,k\}$, then ${\lambda }_i\left({\mathcal{Q}}_F\right)={\lambda }_i\left({\mathcal{R}}_F\right)$, that is, ${\lambda }_i({\mathcal{D}}_F+\mathcal{A}\left(G\right))={\lambda }_i({\mathcal{S}}_F+\mathcal{A}\left(P_G\right))$, that is, ${\lambda }_i(t{I}_n+\mathcal{A}\left(G\right))={\lambda }_i(s{I}_k+\mathcal{A}\left(P_G\right))$, that is, $t+{\lambda }_i\left(G\right)=s+{\lambda }_i\left(P_G\right).$

(ii)

By Theorem 3 (ii), if $min\{n,k\}=n$ then ${\lambda }_i\left({\mathcal{R}}_F\right)=0$ for $n+1\le i\le k$, that is, ${\lambda }_i(s{I}_k+\mathcal{A}\left(P_G\right))=0$ for $n+1\le i\le k$, that is, ${\lambda }_i\left(P_G\right)=-s$ for $n+1\le i\le k$.

(iii)

By Theorem 3 (iii), if $min\{n,k\}=k$ then ${\lambda }_i\left({\mathcal{Q}}_F\right)=0$ for $k+1\le i\le n$, that is, ${\lambda }_i(t{I}_n+\mathcal{A}\left(G\right))=0$ for $k+1\le i\le n$, that is, ${\lambda }_i\left(G\right)=-t$ for $k+1\le i\le n$.

□

Example 1.

(i) Considering the complete graph $K_n$ and its minimum clique partition F with only one clique, we have ${\mathcal{M}}_F={𝟙}_n$, ${\mathcal{M}}_F{\mathcal{M}}_F^T=J_n$ and ${\mathcal{M}}_F^T{\mathcal{M}}_F=\left[n\right]$. Applying Theorem 4 here we have $t_i^F=1$ for $1\le i\le n$, $k=1$ and $s_1^F=n$, that is, $K_n$ is a $(n,1)$ regular graph. From this with Theorem 4 $\left(i\right)$ we arrive at $1+{\lambda }_1\left(K_n\right)=n+{\lambda }_1\left(K_1\right),$ that is, ${\lambda }_1\left(K_n\right)=n-1$, and by Theorem 4 $\left(iii\right)$, ${\lambda }_i\left(K_n\right)=-1$ for $2\le i\le n$.

$\left(ii\right)$ Considering the clique partition $F=\left\{C_1=\{1,2,6\},C_2=\{2,3,4\},C_3=\{1,3,5\},C_4=\{4,5,6\}\right\}$ for G isomorphic to the complete tripartite graph $K_{2,2,2}$ (or $G\cong K_{2,2,2}$) in Figure 1, we have $s_i^F=3$ for $i\in \left[4\right]$ and $t_j^F=2$ for $j\in \left[6\right]$. Then, G is a $(3,2)$ regular graph. Moreover,

$${\mathcal{M}}_F=\left(\begin{array}{cccc}1& 0& 1& 0\\ 1& 1& 0& 0\\ 0& 1& 1& 0\\ 0& 1& 0& 1\\ 0& 0& 1& 1\\ 1& 0& 0& 1\end{array}\right),{\mathcal{Q}}_F=\left(\begin{array}{cccccc}2& 1& 1& 0& 1& 1\\ 1& 2& 1& 1& 0& 1\\ 1& 1& 2& 1& 1& 0\\ 0& 1& 1& 2& 1& 1\\ 1& 0& 1& 1& 2& 1\\ 1& 1& 0& 1& 1& 2\end{array}\right),{\mathcal{R}}_F=\left(\begin{array}{cccc}3& 1& 1& 1\\ 1& 3& 1& 1\\ 1& 1& 3& 1\\ 1& 1& 1& 3\end{array}\right)$$

and by Theorem 4, we have ${\lambda }_i\left(G\right)=1+{\lambda }_i\left(P_G\right)$ for $1\le i\le 4$ and ${\lambda }_i\left(G\right)=-2$ for $i=5,6$. From these facts with $P_G\cong K_4$, we arrive at $\mathrm{Spec}\left(G\right)=\{4,0,0,0,-2,-2\}$.

Figure 1. The graph $G\cong K_{2,2,2}$.

Now applying the theory of clique partitions and vertex-clique incidence matrices, we obtain a lower bound for the smallest eigenvalue of a graph. During the review of a previous version, we were made aware of the work in [15] where the first statement in the next result can also be found in Corollary 3.1 [15]. We include a proof here for completeness.

Theorem 5. 

Let G be a graph of order n and let $t_1^F$ be the largest clique-degree of G with a given clique partition F. Then

$${\lambda }_n\left(G\right)\ge -t_1^F.$$

(4)

Moreover, if equality holds in (4), then $rank\left({\mathcal{M}}_F\right)<n$ and if $rank\left({\mathcal{M}}_F\right)<n$ and G is clique-regular, then equality holds in (4).

Proof. 

Since ${\mathcal{Q}}_F={\mathcal{D}}_F+\mathcal{A}$ is a positive semi-definite matrix, we have ${\mathcal{D}}_F\ge -\mathcal{A}$ and by Lemma 1 we arrive at

$${\lambda }_i\left({\mathcal{D}}_F\right)\ge {\lambda }_i(-\mathcal{A})for1\le i\le n.$$

(5)

Considering $i=1$ we arrive at $-{\lambda }_n\left(G\right)={\lambda }_1(-\mathcal{A})\le {\lambda }_1\left({\mathcal{D}}_F\right)=t_1^F$, which gives the required result in (4).

For the second part of the proof, suppose that ${\lambda }_n\left(G\right)=-t_1^F$. Then ${\lambda }_n(t_1^{F}I+\mathcal{A})=0$. This with the relation $0\le {\mathcal{Q}}_F={\mathcal{D}}_F+\mathcal{A}\le t_1^{F}I+\mathcal{A}$, gives ${\lambda }_n\left({\mathcal{Q}}_F\right)=0$, that is, $rank\left({\mathcal{M}}_F\right)=rank\left({\mathcal{Q}}_F\right)<n$. Now we assume that $t_1^F=\cdots =t_n^F$. If $rank\left({\mathcal{M}}_F\right)<n$ then $rank\left({\mathcal{Q}}_F\right)<n$, that is, ${\lambda }_i\left({\mathcal{Q}}_F\right)=0$ for $1+k\le i\le n$, that is, $t_1^F+{\lambda }_i\left(G\right)=0$ as ${\mathcal{Q}}_F=t_1^{F}I+\mathcal{A}$, that is, ${\lambda }_n\left(G\right)=-t_1^F$ with the multiplicity at least $n-k$. □

Corollary 1. 

All regular bipartite graphs and all clique-regular graphs with $n>\left|F\right|$ satisfy the equality in (4).

Proof. 

First we assume that G is a regular bipartite graph. Since G is bipartite, we have $t_i^F=d_i$ for $i\in \left[n\right]$ and $q_n={\lambda }_n\left(Q\right)=0$. On the other hand, since G is regular, we have $t_1^F=\cdots =t_n^F$. These facts with Theorem 5 give the fact that all regular bipartite graphs satisfies the equality in (4).

Next, assume that G is a clique-regular graph with $n>k=\left|F\right|$. Since $rank\left({\mathcal{M}}_F\right)\le min\{n,k\}\le k<n$, the required result is obtained by Theorem 5. □

Theorem 5 holds for any clique partition F of G, which leads to the following. However, during the review of a previous version, we were made aware of the work [13] where a version of the next result can also be found in Corollary 3.2 [13]. We include a proof here for completeness.

Corollary 2. 

Let G be a graph of order n and let $t_1^F$ be the largest clique-degree of G with a given clique partition F. Then,

$${\lambda }_n\left(G\right)\ge -\underset{F}{min}{t}_1^F,$$

where the minimum is over all clique partitions F of G.

The following example shows that for the equality ${\lambda }_n\left(G\right)=-t_1^F$ the graph G need not be clique-regular.

Example 2. 

For the graph G given in Figure 2, we have $F=\left\{\{1,2\},\{2,3\},\{1,3,4,6,7\},\{4,5\},\{5,6\}\right\}.$ This gives $t_i^F=2$ for $i\in \left[6\right]$ and $t_7^F=1$. The graph is the line graph of the graph $H\cong K_1\vee (2K_2\cup K_1)$ of order 6 with 7 edges. Then the smallest eigenvalue of G is ${\lambda }_7\left(G\right)={\lambda }_7\left(L_H\right)=-2=-t_1^F$ while $t_1^F\ne t_7^F$.

Figure 2. The Graph G.

In the following we provide a lower bound for the negative inertia of a graph G of order n.

Theorem 6. 

Let G be a graph of order n. Then,

$${\nu }^{-}\left(G\right)\ge n-\underset{F}{min}rank\left({\mathcal{M}}_F\right),$$

(6)

where minimum is over all clique partitions F of G. Moreover, if ${min}_{F}rank\left({\mathcal{M}}_F\right)<n$, then $-t_1^F\le {\lambda }_i\left(G\right)\le -t_n^F$ for $1+\underset{F}{min}rank\left({\mathcal{M}}_F\right)\le i\le n$.

Proof. 

If $\underset{F}{min}rank\left({\mathcal{M}}_F\right)=n$, then the result in (6) is obvious. Assume that $F_1$ is a clique partition of G with $rank\left({\mathcal{M}}_{F_1}\right)=\underset{F}{min}rank\left({\mathcal{M}}_F\right)<n$. In this case, since $rank\left({\mathcal{Q}}_{F_1}\right)=rank\left({\mathcal{M}}_{F_1}\right)$ and ${\mathcal{Q}}_{F_1}$ is positive semi-definite matrix, we have ${\lambda }_i\left({\mathcal{Q}}_{F_1}\right)=0$ for $1+rank\left({\mathcal{M}}_{F_1}\right)\le i\le n$. From this and the fact that $t_n^F+{\lambda }_i\left(G\right)\le {\lambda }_i\left({\mathcal{Q}}_{F_1}\right)\le t_1^F+{\lambda }_i\left(G\right)$, we have $-t_1^F\le {\lambda }_i\left(G\right)\le -t_n^F<0$ for $1+rank\left({\mathcal{M}}_{F_1}\right)\le i\le n$, which gives the required results. □

The following result is obtained by Theorem 6 and the fact $rank\left({\mathcal{M}}_F\right)\le \left|F\right|$.

Corollary 3. 

Let G be a graph of the order n and a clique partition F such that $n>\left|F\right|$. Then,

(i) 

$-t_1^F\le {\lambda }_i\left(G\right)\le -t_n^F$ for $\left|F\right|+1\le i\le n$.

(ii) 

${\nu }^{-}\left(G\right)\ge n-\left|F\right|$.

Furthermore, if F is a minimum clique partition of G the next consequence immediately follows from Corollary 3.

Corollary 4. 

Let G be a graph of the order n and clique partition number $cp\left(G\right)$. If $cp\left(G\right)<n$, then

(i) 

$-t_1^F\le {\lambda }_i\left(G\right)\le -t_n^F$ for $cp\left(G\right)+1\le i\le n$.

(ii) 

${\nu }^{-}\left(G\right)\ge n-cp\left(G\right)$.

For any graph G of order n we have [29]

$$\alpha \left(G\right)\le min\{n-{\nu }^{-}\left(G\right),n-{\nu }^{+}\left(G\right)\},$$

(7)

where ${\nu }^{-}$ and ${\nu }^{+}$ are the negative and positive parts of the inertia, respectively of the graph G. This implies that

$${\nu }^{-}\left(G\right)\le n-\alpha \left(G\right).$$

(8)

In the following we present a sufficient condition for which equality in (8) holds.

Theorem 7. 

Let G be a graph of order n with the independence number $\alpha \left(G\right)$ and the clique partition number $cp\left(G\right)$. If F is a clique partition with $rank\left({\mathcal{M}}_F\right)=\alpha \left(G\right)$, then ${\nu }^{-}\left(G\right)=n-\alpha \left(G\right).$ In particular, if $cp\left(G\right)=\alpha \left(G\right)$, then ${\nu }^{-}\left(G\right)=n-\alpha \left(G\right).$

Proof. 

By Theorem 6 we have ${\nu }^{-}\left(G\right)\ge n-rank\left({\mathcal{M}}_F\right)=n-rank\left({\mathcal{Q}}_F\right)=\eta \left({\mathcal{Q}}_F\right).$ This fact along with (8) gives

$$\eta \left({\mathcal{Q}}_F\right)\le {\nu }^{-}\left(G\right)\le n-\alpha \left(G\right).$$

(9)

The assumption that $\mathrm{rank}\left({\mathcal{M}}_{\mathrm{F}}\right)=\alpha \left(G\right)$ is equivalent to $\eta \left({\mathcal{Q}}_F\right)=n-\alpha \left(G\right)$. This with (9) gives the first required result.

Without loss of generality, we may assume that the vertex set $\left[\alpha \right]$ is a maximum independent set in G and $C_i$ is a clique of a minimum clique partition $F_m$ containing the vertex $i\in \left[\alpha \right]$. Now in ${\mathcal{M}}_{F_m}$ we consider the submatrix induced by the rows and columns corresponding to the vertex set $\left[\alpha \right]$ and the clique set $\{C_i:i\in \left[\alpha \right]\}$, respectively. Obviously, this square principal submatrix is equivalent to the identity matrix of size $\alpha$ and hence $\mathrm{rank}\left({\mathcal{Q}}_{F_m}\right)\ge \mathrm{rank}\left(I_{\alpha }\right)=\alpha$. Since $\mathrm{rank}\left({\mathcal{Q}}_{F_m}\right)\le cp\left(G\right)$ and using the assumption $cp\left(G\right)=\alpha \left(G\right)$ we arrive at $\mathrm{rank}\left({\mathcal{Q}}_{F_m}\right)=\alpha =\mathrm{rank}\left({\mathcal{M}}_{F_m}\right)$ and, therefore, ${\nu }^{-}\left(G\right)=n-\alpha \left(G\right)$ by the first part of the theorem. □

The following result is obtained from (5). During the review of a previous version, we were made aware of the work [13] where the next result can also be found in Theorem 3.1 [13]. We include a proof here for completeness.

Theorem 8. 

Let G be a graph of order n and the negative inertia ${\nu }^{-}$. Let $t_i^F$ be the ith largest clique-degree of G with a clique partition F. Then, for $1\le i\le {\nu }^{-}$, we have

$${\lambda }_{n-i+1}\left(G\right)\ge -t_i^F.$$

(10)

Equality holds in (10) if G is a clique-regular graph with ${\nu }^{-}=n-\left|F\right|$.

Since ${\mathcal{R}}_F$ is a positive semi-definite matrix, using a similar argument as in the proof of Theorem 5, we obtain the following result.

Theorem 9. 

Let G be a graph of order n with a clique partition $F=\{C_1,\dots ,C_k\}$ and let $|C_i|=s_i^F$ for $1\le i\le k$ such that $s_1^F\ge s_2^F\ge \dots \ge s_k^F$. Then,

$${\lambda }_k\left(P_G\right)\ge -s_1^F.$$

(11)

Equality holds in (11) if G is a $s_1^F$ clique-uniform graph with $k>n$.

Proof. 

Since ${\mathcal{R}}_F={\mathcal{S}}_F+\mathcal{A}\left(P_G\right)$ is a positive semi-definite matrix, we have ${\mathcal{S}}_F\ge -\mathcal{A}\left(P_G\right)$ and by Lemma 1, it follows that

$${\lambda }_i\left({\mathcal{S}}_F\right)\ge {\lambda }_i(-\mathcal{A}\left(P_G\right))for1\le i\le k.$$

(12)

Considering $i=1$ we have $-{\lambda }_k\left(P_G\right)={\lambda }_1(-\mathcal{A}\left(P_G\right))\le {\lambda }_1\left({\mathcal{S}}_F\right)=s_1^F,$ which gives the required result in (11).

Now assume that G is a $s_1^F$ clique-uniform graph with $k>n$. By Theorem 3 (ii) with $k>n$, we arrive at ${\lambda }_i\left({\mathcal{R}}_F\right)=0$ for $n+1\le i\le k$. On the other hand, since $s_1^F=\cdots =s_k^F$ we have ${\mathcal{R}}_F=s_1^F{I}_k+\mathcal{A}\left(P_G\right)$, and consequently ${\lambda }_i\left({\mathcal{R}}_F\right)=s_1^F+{\lambda }_i\left(P_G\right)=0$. That is, ${\lambda }_i\left(P_G\right)=-s_1^F$ for $n+1\le i\le k$, that is, ${\lambda }_k\left(P_G\right)=-s_1^F$ with multiplicity at least $k-n$. □

Theorem 9 holds for any clique partition F of G, which gives the following result.

Corollary 5. 

Let G be a graph of order n with a clique partition $F=\{C_1,\dots ,C_k\}$ and let $|C_i|=s_i^F$ for $1\le i\le k$ such that $s_1^F\ge s_2^F\ge \cdots \ge s_k^F$. Then,

$${\lambda }_k\left(P_G\right)\ge -\underset{F}{min}{s}_1^F,$$

(13)

where minimum is over all clique partitions F of G.

In the case of $k>n$, we have ${\lambda }_i\left({\mathcal{R}}_F\right)=0$ for $1+n\le i\le k$ by Theorem 3. Since $s_k^F+{\lambda }_i\left(P_G\right)\le {\lambda }_i\left({\mathcal{R}}_F\right)\le s_1^F+{\lambda }_i\left(P_G\right)$, we get $-s_1^F\le {\lambda }_i\left(P_G\right)\le -s_k^F<0$. We summarize this in the next result.

Theorem 10. 

Let G be a graph of order n and a clique partition F with $\left|F\right|=k>n$. Then,

(i) 

$-s_1^F\le {\lambda }_i\left(P_G\right)\le -s_k^F$ for $1+n\le i\le k$.

(ii) 

${\nu }^{-}\left(P_G\right)\ge k-n$.

The following result follows from (12).

Theorem 11. 

Let G be a graph of order n with a clique partition $F=\{C_1,\dots ,C_k\}$ and let $|C_i|=s_i^F$ for $1\le i\le k$ such that $s_1^F\ge s_2^F\ge \cdots \ge s_k^F$. If $P_G$ is the corresponding clique partition graph of G, then for $1\le i\le {\nu }^{-}\left(P_G\right)$,

$${\lambda }_{k-i+1}\left(P_G\right)\ge -s_i^F.$$

(14)

Equality in (14) holds if G is a $s_1^F$ clique-uniform graph with ${\nu }^{-}\left(P_G\right)=k-n$.

The following concerns the signless Laplacian eigenvalues of a graph.

Theorem 12. 

Let G be a graph of order n and having a clique partition F with $\left|F\right|=k$ and assume $1\le i\le min\{n,k\}$.

(i) 

If G is a t clique-regular graph, then $q_i\left(G\right)-{\lambda }_i\left(G\right)\ge t$.

(ii) 

If G is a s clique-uniform graph, then $q_i\left(G\right)-{\lambda }_i\left(P_G\right)\ge s$.

Proof. 

From Section 3, the signless Laplacian matrix Q of G satisfies $Q\ge {\mathcal{Q}}_F$. This fact with Lemma 1 gives $q_i\left(G\right)\ge {\lambda }_i\left({\mathcal{Q}}_F\right)$, where $q_i\left(G\right)$ and ${\lambda }_i\left({\mathcal{Q}}_F\right)$ are, respectively, the ith largest signless Laplacian eigenvalue of G and the ith largest eigenvalue of matrix ${\mathcal{Q}}_F$. Using the above analysis combined with Theorem 3 and the facts ${\lambda }_i\left({\mathcal{Q}}_F\right)=t+{\lambda }_i\left(G\right)$ and ${\lambda }_i\left({\mathcal{R}}_F\right)=s+{\lambda }_i\left(P_G\right)$ implies the required results in $\left(i\right)$ and $\left(ii\right)$. □

3.2. Applications to Energy of Graphs and Matrices

In this section, using the theory of vertex-clique incidence matrices of a graph, we introduce notions of graph energies, as a generalization of the incidence energy and the signless Laplacian energy of the graph. Finally, we present upper bounds on energies of a graph, its clique partition graph and line graph.

The energy $\mathcal{E}\left(G\right)$ of the graph G defined in (2) has the equivalent expressions as follows [22]:

$$\mathcal{E}\left(G\right)=2\sum _{i=1}^{{\nu }^{+}}{\lambda }_i=2\sum _{i=1}^{{\nu }^{-}}-{\lambda }_{n-i+1}=2\underset{1\le k\le n}{max}\sum _{i=1}^k{\lambda }_i=2\underset{1\le k\le n}{max}\sum _{i=1}^k-{\lambda }_{n-i+1}$$

(15)

where ${\nu }^{+}$ and ${\nu }^{-}$ are, respectively, the positive and the negative inertia of G. Nikiforov [30,31,32] proposed a significant extension and generalization of the graph energy concept. The energy of an $r\times s$ matrix B is the summation of its singular values, that is,

$$\mathcal{E}\left(B\right)=\sum _{i=1}^s{\sigma }_i\left(B\right),$$

(16)

where ${\sigma }_i\left(B\right)$ denotes the ith singular value of B which is equal to $\sqrt{{\lambda }_i\left(B^{T}B\right)}$.

Consonni and Todeschini [33] introduced an entire class of matrix-based quantities, defined as

$$\sum _{i=1}^n|x_i-\overline{x}|,$$

(17)

where $x_1,x_2,\dots ,x_n$ are the eigenvalues of the respective matrix, and $\overline{x}$ is their arithmetic mean.

According to (16) and (17), two types of energies can then be defined for any matrix B. The incidence energy $IE\left(G\right)$ of a graph G is defined to be the energy of the incidence matrix of G of the type (16), i.e.,

$$IE\left(G\right)=\mathcal{E}\left(M\right)=\sum _{i=1}^m{\sigma }_i\left(M\right)=\sum _{i=1}^m\sqrt{{\lambda }_i\left(M^{T}M\right)}=\sum _{i=1}^n\sqrt{{\lambda }_i\left(M{M}^T\right)}=\sum _{i=1}^n\sqrt{q_i}.$$

Similarly, the vertex-clique incidence energy $I{E}_F\left(G\right)$ of G associated with the clique partition F is defined as the energy of the vertex-clique incidence matrix ${\mathcal{M}}_F$, i.e.,

$$\begin{array}{ccc}\hfill I{E}_F\left(G\right)=\mathcal{E}\left({\mathcal{M}}_F\right)& =& \sum _{i=1}^k{\sigma }_i\left({\mathcal{M}}_F\right)=\sum _{i=1}^k\sqrt{{\lambda }_i\left({\mathcal{M}}_F^T{\mathcal{M}}_F\right)}\hfill \\ & =& \sum _{i=1}^n\sqrt{{\lambda }_i\left({\mathcal{M}}_F{\mathcal{M}}_F^T\right)}=\sum _{i=1}^n\sqrt{{\lambda }_i\left({\mathcal{Q}}_F\right)}.\hfill \end{array}$$

Observe

$$Q-{\mathcal{Q}}_{\mathcal{F}}=(D+\mathcal{A})-({\mathcal{D}}_F+\mathcal{A})=D-{\mathcal{D}}_F=diag(d_1-t_1^F,d_2-t_2^F,\dots ,d_n-t_n^F)\ge 0.$$

From the above and using Lemma 1 we have $q_i={\lambda }_i\left(Q\right)\ge {\lambda }_i\left({\mathcal{Q}}_F\right)$ and, consequently, we have

$$I{E}_F\left(G\right)=\sum _{i=1}^n\sqrt{{\lambda }_i\left({\mathcal{Q}}_F\right)}\le \sum _{i=1}^n\sqrt{q_i}=IE\left(G\right)$$

with equality if and only if $F=E$.

Moreover,

$$\sum _{i=1}^n{\lambda }_i\left({\mathcal{Q}}_F\right)=\sum _{i=1}^n{t}_i^F,\sum _{i=1}^n{\lambda }_i^2\left({\mathcal{Q}}_F\right)=\sum _{i=1}^n({\left(t_i^F\right)}^2+t_i^F).$$

Applying the fact that the diagonal entries are majorized by the eigenvalues of ${\mathcal{Q}}_F$ and by a similar method given in [34] it can be shown that ${\sum }_{i=1}^n\sqrt{{\lambda }_i\left({\mathcal{Q}}_F\right)}\le {\sum }_{i=1}^n\sqrt{t_i^F}.$

Considering the energy of the matrix ${\mathcal{Q}}_F$ of the type (17) gives

$$\mathcal{E}\left({\mathcal{Q}}_F\right)=\sum _{i=1}^n\left|{\lambda }_i\left({\mathcal{Q}}_F\right)-\overline{t}\right|,$$

(18)

where $\overline{t}=\frac{{\sum }_{i=1}^n{t}_i^F}{n}$. The energy $\mathcal{E}\left({\mathcal{Q}}_F\right)$ can be viewed as a generalization of the signless Laplacian energy $L{E}^{+}\left(G\right)$ of G which is defined in [21] as follows:

$$L{E}^{+}\left(G\right)=\mathcal{E}\left(Q\right)=\sum _{i=1}^n|q_i-\frac{2m}{n}|.$$

Due to the similarity of the definitions for signless Laplacian energy $L{E}^{+}\left(G\right)$ and $\mathcal{E}\left({\mathcal{Q}}_F\right)$ it follows that in most cases, results derived about $L{E}^{+}\left(G\right)$ can be generalized to $\mathcal{E}\left({\mathcal{Q}}_F\right)$. For example, from Lemma 2.12 in [22] for $L{E}^{+}\left(G\right)$, we obtain the following:

$$\mathcal{E}\left({\mathcal{Q}}_F\right)=\underset{1\le j\le n}{max}\left\{2\sum _{i=1}^j{\lambda }_i\left({\mathcal{Q}}_F\right)-2j\overline{t}\right\}=2\sum _{i=1}^{\tau }{\lambda }_i\left({\mathcal{Q}}_F\right)-2\overline{t}\tau ,$$

(19)

where $\tau$ is the largest positive integer such that ${\lambda }_{\tau }\left({\mathcal{Q}}_F\right)>\overline{t}$.

Using a method similar to the proof of Corollary 5 in [35] for ${\mathcal{Q}}_F-\overline{t}I={\mathcal{D}}_F-\overline{t}I+\mathcal{A}$, we have $\mathcal{E}\left({\mathcal{Q}}_F\right)-\mathcal{E}\left(G\right)\le {\sum }_{i=1}^n|t_i^F-\overline{t}|.$

In the next result, we show that for a clique-regular graph G associated with a clique partition F, $\mathcal{E}\left({\mathcal{Q}}_F\right)=\mathcal{E}\left(G\right)$.

Theorem 13. 

If G is a clique-regular graph associated with a clique partition F, then $\mathcal{E}\left({\mathcal{Q}}_F\right)=\mathcal{E}\left(G\right)$.

Proof. 

Suppose that G is t clique-regular. Then,

$$\begin{array}{cc}\hfill \mathcal{E}\left({\mathcal{Q}}_F\right)=& \sum _{i=1}^n\left|{\lambda }_i\left({\mathcal{Q}}_F\right)-\frac{{\sum }_{i=1}^n{t}_i^F}{n}\right|=\sum _{i=1}^n\left|{\lambda }_i\left({\mathcal{Q}}_F\right)-t\right|\hfill \\ \hfill =& \sum _{i=1}^n\left|{\lambda }_i({\mathcal{D}}_F+\mathcal{A}\left(G\right))-t\right|=\sum _{i=1}^n\left|{\lambda }_i(tI+\mathcal{A}\left(G\right))-t\right|\hfill \\ \hfill =& \sum _{i=1}^n\left|{\lambda }_i\left(G\right)\right|=\mathcal{E}\left(G\right).\hfill \end{array}$$

□

Note that for any t clique-regular graph G, we have $I{E}_F\left(G\right)={\sum }_{i=1}^n\sqrt{{\lambda }_i\left({\mathcal{Q}}_F\right)}={\sum }_{i=1}^n\sqrt{t+{\lambda }_i}$. Next, we show that for a clique-uniform graph G we have $\mathcal{E}\left({\mathcal{R}}_F\right)=\mathcal{E}\left(P_G\right)$.

Theorem 14. 

If G is a clique-uniform graph with the clique partition graph $P_G$, then $\mathcal{E}\left({\mathcal{R}}_F\right)=\mathcal{E}\left(P_G\right)$.

Proof. 

Suppose that G is a s clique-uniform graph. Then,

$$\begin{array}{cc}\hfill \mathcal{E}\left({\mathcal{R}}_F\right)=& \sum _{i=1}^k|{\lambda }_i\left({\mathcal{R}}_F\right)-\frac{{\sum }_{i=1}^k{s}_i^F}{k}|=\sum _{i=1}^k\left|{\lambda }_i\left({\mathcal{R}}_F\right)-s\right|\hfill \\ \hfill =& \sum _{i=1}^k\left|{\lambda }_i({\mathcal{S}}_F+\mathcal{A}\left(P_G\right))-s\right|=\sum _{i=1}^k\left|{\lambda }_i(sI+\mathcal{A}\left(P_G\right))-s\right|\hfill \\ \hfill =& \sum _{i=1}^k\left|{\lambda }_i\left(P_G\right)\right|=\mathcal{E}\left(P_G\right).\hfill \end{array}$$

□

Note that for any s clique-uniform graph G with the clique partition graph $P_G$, we have

$$I{E}_F\left(G\right)=\sum _{i=1}^n\sqrt{{\lambda }_i\left({\mathcal{Q}}_F\right)}=\sum _{i=1}^k\sqrt{{\lambda }_i\left({\mathcal{R}}_F\right)}=\sum _{i=1}^k\sqrt{s+{\lambda }_i\left(P_G\right)}.$$

In [22] Theorem 3.3, a relation between the energy of the line graph $\mathcal{E}\left(L_G\right)$ and the signless Laplacian energy $L{E}^{+}\left(G\right)$ of G is given. In the following, we generalize this result by using the notion of clique partition of a graph and we provide a comparison between the energy of the clique partition graph $\mathcal{E}\left(P_G\right)$ of $P_G$ and $\mathcal{E}\left({\mathcal{Q}}_F\right)$. For this, we need the following lemma, which is obtained from Theorem 3 and is a generalization of Lemma 6.

Lemma 7. 

Let G be an s clique-uniform graph of order n associated with a clique partition F where $\left|F\right|=k$. Then, ${\lambda }_i\left({\mathcal{Q}}_F\right)={\lambda }_i\left(P_G\right)+s,fori\in \{1,\dots ,min\{n,k\}\}.$

Theorem 15. 

Let G be an s clique-uniform graph of order n associated with a clique partition F where $\left|F\right|=k$.

(i) 

If $k<n$, then $\mathcal{E}\left(P_G\right)\le \mathcal{E}\left({\mathcal{Q}}_F\right)+\frac{2ks}{n}-2s$.

(ii) 

If $k>n$, then $\mathcal{E}\left(P_G\right)\ge \mathcal{E}\left({\mathcal{Q}}_F\right)+\frac{2ks}{n}-2s$.

(iii) 

If $k=n$, then $\mathcal{E}\left(P_G\right)=\mathcal{E}\left({\mathcal{Q}}_F\right)$.

Proof. 

$\left(i\right)$ Let ${\nu }^{+}={\nu }^{+}\left(P_G\right)\le k<n$. By Lemma 7 we have

$$\sum _{i=1}^{{\nu }^{+}}{\lambda }_i\left(P_G\right)=\sum _{i=1}^{{\nu }^{+}}({\lambda }_i\left({\mathcal{Q}}_F\right)-s)=\sum _{i=1}^{{\nu }^{+}}{\lambda }_i\left({\mathcal{Q}}_F\right)-s{\nu }^{+}.$$

On the other hand, from (15) we have

$$\begin{array}{ccc}\hfill \mathcal{E}\left(P_G\right)=2\sum _{i=1}^{\nu +}{\lambda }_i\left(P_G\right)& =& 2\sum _{i=1}^{\nu +}{\lambda }_i\left({\mathcal{Q}}_F\right)-2s{\nu }^{+}-2{\nu }^{+}\frac{{\sum }_{i=1}^n{t}_i^F}{n}+2{\nu }^{+}\frac{{\sum }_{i=1}^n{t}_i^F}{n}\hfill \\ & \le & \mathcal{E}\left({\mathcal{Q}}_F\right)-2s{\nu }^{+}+2{\nu }^{+}\frac{ks}{n}as\left(19\right),\sum _{i=1}^n{t}_i^F=ks\hfill \\ & =& \mathcal{E}\left({\mathcal{Q}}_F\right)+2{\nu }^{+}\left(\frac{ks}{n}-s\right)as{\nu }^{+}\ge 1,k<n\hfill \\ & \le & \mathcal{E}\left({\mathcal{Q}}_F\right)+\frac{2ks}{n}-2s.\hfill \end{array}$$

$\left(ii\right)$ Recall that $\tau$ is the largest positive integer such that ${\lambda }_{\tau }\left({\mathcal{Q}}_F\right)\ge \overline{t}=\frac{ks}{n}$ and let $\tau <n<k$. Again by Lemma 7 we have ${\sum }_{i=1}^{\tau }{\lambda }_i\left({\mathcal{Q}}_F\right)={\sum }_{i=1}^{\tau }\left({\lambda }_i\left(P_G\right)\right)+s\tau .$ On the other hand, by (19) and Lemma 7 we have

$$\mathcal{E}\left({\mathcal{Q}}_F\right)=2\sum _{i=1}^{\tau }{\lambda }_i\left({\mathcal{Q}}_F\right)-\frac{2ks\tau }{n}=2\sum _{i=1}^{\tau }{\lambda }_i\left(P_G\right)+2s\tau -\frac{2ks\tau }{n}.$$

From (15) with the above equation we have

$$\mathcal{E}\left(P_G\right)\ge 2\sum _{i=1}^{\tau }{\lambda }_i\left(P_G\right)=\mathcal{E}\left({\mathcal{Q}}_F\right)+2\tau \left(\frac{ks}{n}-s\right)\ge \mathcal{E}\left({\mathcal{Q}}_F\right)+\frac{2ks}{n}-2s.$$

$\left(iii\right)$ If $k\ne n$, then $\mathcal{E}\left(P_G\right)\ne \mathcal{E}\left({\mathcal{Q}}_F\right)$ by $\left(i\right)$ and $\left(ii\right)$, i.e., if $\mathcal{E}\left(P_G\right)=\mathcal{E}\left({\mathcal{Q}}_F\right)$, then $k=n$.

It suffices to show that if $k=n$, then $\mathcal{E}\left(P_G\right)=\mathcal{E}\left({\mathcal{Q}}_F\right)$. Indeed, if $k=n$, then

$$\mathcal{E}\left({\mathcal{Q}}_F\right)=\sum _{i=1}^n|{\lambda }_i\left({\mathcal{Q}}_F\right)-\frac{{\sum }_{i=1}^n{t}_i^F}{n}|=\sum _{i=1}^n|{\lambda }_i\left({\mathcal{Q}}_F\right)-\frac{ks}{n}|.$$

Since $k=n$ with Lemma 7 we have $\mathcal{E}\left({\mathcal{Q}}_F\right)={\sum }_{i=1}^n\left|{\lambda }_i\left(P_G\right)\right|=\mathcal{E}\left(P_G\right).$ □

In the following, we present an upper bound for the energy of a graph G.

Theorem 16. 

Let G be a graph of order n and the negative inertia ${\nu }^{-}={\nu }^{-}\left(G\right)$ and let $t_i^F$ be the $i^{th}$ largest clique degree of G associated with the clique partition F, for $1\le i\le n$. Then,

$$\mathcal{E}\left(G\right)\le 2\underset{F}{min}\sum _{i=1}^{{\nu }^{-}}{t}_i^F,$$

(20)

where the minimum is given over all clique partitions F of G. Equality holds if G is a clique-regular graph associated with a minimum clique partition of size $cp\left(G\right)=n-{\nu }^{-}$.

Proof. 

From (15) and (10) we have $\mathcal{E}\left(G\right)=2{\sum }_{i=1}^{{\nu }^{-}}-{\lambda }_{n-i+1}\le 2{\sum }_{i=1}^{{\nu }^{-}}{t}_i^F,$ where $t_i^F$ is $i^{th}$ largest clique-degree of G associated with a clique partition F. Since this upper bound is valid for any clique partition of G, we select the optimal value, namely, $\underset{F}{min}2{\sum }_{i=1}^{{\nu }^{-}}{t}_i^F$. The second part of the proof follows directly from Theorem 8. □

The following result provides an upper bound on the energy of G in terms of the vertex degrees.

Theorem 17. 

Let G be a graph of order n with the vertex degrees $d_1\ge d_2\ge \cdots \ge d_n$. Then

$$\mathcal{E}\left(G\right)\le 2\sum _{i=1}^h{d}_i,$$

where $h=min\{{\nu }^{+},{\nu }^{-}\}$.

Proof. 

Since $t_i^F\le d_i$ for $i\in \left[n\right]$ along with Theorem 16 gives

$$\mathcal{E}\left(G\right)\le 2\sum _{i=1}^{{\nu }^{-}}{d}_i.$$

(21)

On the other hand, the Laplacian matrix $L=D-\mathcal{A}$ of G is a positive semi-definite matrix, so $\mathcal{A}\le D$. From this with Lemma 1 we obtain ${\lambda }_i\le d_i$ for $1\le i\le n$. Then, $\mathcal{E}\left(G\right)=2{\sum }_{i=1}^{{\nu }^{+}}{\lambda }_i\le 2{\sum }_{i=1}^{{\nu }^{+}}{d}_i$. Using the previous inequality with (21) completes the proof. □

From Theorem 16 with (7) we obtain the following upper bound for the energy of G:

$$\mathcal{E}\left(G\right)\le 2\sum _{i=1}^{n-\alpha }{t}_i^F\le 2\sum _{i=1}^{n-\alpha }{d}_i,$$

where $\alpha$ is the independence number of the graph G. By (15) and (14) applying a similar method carried out for the proof of Theorem 16, we obtain the next result.

Theorem 18. 

Let G be a graph of order n with a clique partition $F=\{C_1,\dots ,C_k\}$ and let $|C_i|=s_i^F$ for $1\le i\le k$ such that $s_1^F\ge s_2^F\ge \cdots \ge s_k^F$. For the clique partition graph $P_G$ of G, we have

$$\mathcal{E}\left(P_G\right)\le 2\underset{F}{min}\sum _{i=1}^{{\nu }^{-}\left(P_G\right)}{s}_i^F.$$

Equality holds if G is a clique-uniform graph associated with a minimum clique partition of size $cp\left(G\right)=n+{\nu }^{-}\left(P_G\right)$.

Next, we present an upper bound on the energy $\mathcal{E}\left(L_G\right)$ of the line graph $L_G$ with a full characterization of the corresponding extremal graphs.

Theorem 19. 

Let G be a graph with the line graph $L_G$. Then,

$$\mathcal{E}\left(L_G\right)\le 4{\nu }^{-}\left(L_G\right).$$

(22)

Equality holds if and only if G is a graph with connected components $G_i=(V_i,E_i)$ for $i\ge 1$ with $n_i=\left|V_i\right|$ and $|E_i|\ge 2$, and possibly some isolated vertices or single edges. Further, each non-bipartite connected component $G_i$ satisfies $|E_i|>|V_i|$ and $q_{n_i}\ge 2$, and each bipartite connected component $G_i$ is either a 4-cycle or satisfies $|E_i|>|V_i|$ and $q_{n_i-1}\ge 2$.

Proof. 

As previously noted, if the clique partition F of G is as same as the edge set E of G, then $s_i^F=2$ for $i\in \left[n\right]$ and $P_G\cong L_G$. Using Theorem 18, we have

$$\mathcal{E}\left(L_G\right)=2\sum _{i=1}^{{\nu }^{-}\left(L_G\right)}-{\lambda }_{m-i+1}\left(P_G\right)\le 2\sum _{i=1}^{{\nu }^{-}\left(L_G\right)}2=4{\nu }^{-}\left(L_G\right),$$

(23)

which gives the required result in (22).

To characterize these extreme graphs in (22), we assume equality holds in (23). Then, all negative eigenvalues of $P_G$ must be equal to $-2$ by (23). We then consider the following two cases:

Case (1) G is connected. First, assume that $m>n$. If G is non-bipartite, then by Lemma 6, ${\lambda }_i\left(L_G\right)=-2$ for $n+1\le i\le m$ and ${\lambda }_n\left(L_G\right)=q_n-2\ne -2$ as $q_n\ne 0$. Since ${\lambda }_n\left(L_G\right)$ must be nonnegative, we have $q_n\ge 2$. Otherwise G is bipartite and by Lemma 6 along with $q_n\left(G\right)=0$, ${\lambda }_i\left(L_G\right)=-2$ for $n\le i\le m$ and ${\lambda }_{n-1}\left(L_G\right)=q_{n-1}-2\ne -2$ as $q_{n-1}\ne 0$. Since ${\lambda }_{n-1}\left(L_G\right)$ must be nonnegative, it follows that $q_{n-1}\ge 2$. Next, assume that $m=n$. Since all negative eigenvalues of $L_G$ are equal to $-2$, we have ${\lambda }_m\left(L_G\right)={\lambda }_n\left(L_G\right)=-2$. If ${\nu }^{-}=1$, then $\mathrm{Spec}\left(L_G\right)=\{2,0,0,-2\}$ and $L_G$ is the cycle $C_4$ of order 4. Otherwise ${\nu }^{-}\ge 2$, and ${\lambda }_{n-1}=-2$, that is, $q_{n-1}=0$, which is a contradiction as G is connected. Finally, assume that $m<n$. Since G is connected it must be a tree and hence $m=n-1$. In this case we have ${\lambda }_m\left(L_G\right)={\lambda }_{n-1}\left(L_G\right)=-2$, that is, $q_{n-1}=0$, which again leads to a contradiction.

Case (2) Assume G is disconnected. Since isolated vertices and single edges do not affect the negative inertia of $L_G$, we may assume that G has connected components along with the possibility of some isolated vertices and single edges. Now each connected component of G can be characterized by the first case, and the proof is complete. □

4. Vertex-Clique Incidence Matrix of a Graph Associated with an Edge Clique Cover

In this section, we consider the vertex-clique incidence matrix, denoted by $M_F$, associated with an edge clique cover F of a graph G. Recall that the $(i,j)$-entry of $M_F$ is real and nonzero if and only if the vertex i belongs to the clique $C_j\in F$. A strategy for minimizing the number of distinct eigenvalues of $M_F{M}_F^T\in S\left(G\right)$, is to minimize the number of distinct eigenvalues of the related matrix $M_F^T{M}_F$. Consequently, we obtain an upper bound on the parameter $q\left(G\right)$. A key technique used here is to consider an extended version of $M_F$, by considering arbitrary real entries in the matrix $M_F$, but simultaneously paying careful attention to preserving the condition that $M_F{M}_F^T\in S\left(G\right)$.

4.1. Applications to the Minimum Number of Distinct Eigenvalues of a Graph

In this section, applying the tool of the vertex-clique incidence matrix of a graph associated with its edge clique cover, we characterize a few new classes of graphs with $q\left(G\right)=2$.

If G and H are graphs then the Cartesian product of G and H denoted by $G\square H$, is the graph on the vertex set $V\left(G\right)\times V\left(H\right)$ with $\{g_1,h_1\}$ and $\{g_2,h_2\}$ adjacent if and only if either $g_1=g_2$ and $h_1$ and $h_2$ are adjacent in H or $g_1$ and $g_2$ are adjacent in G and $h_1=h_2$. The first statement in the next theorem can also be found in [7], however, we include a proof here to aid in establishing the second claim.

Theorem 20. 

Let $G\cong K_s\square K_2$ with $s\ge 3$. Then, $q\left(G\right)=2$ and G has an SSP matrix realization with two distinct eigenvalues.

Proof. 

Let $M=\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right),$ where $M_1=J_s-(s-1)I_s$ and $M_2=J_s-I_s$. Then, we have

$$A=M{M}^T=\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)\left(M_1^T{M}_2^T\right)=\left(\begin{array}{cc}{M}_1{M}_1^T& M_1{M}_2^T\\ M_2{M}_1^T& M_2{M}_2^T\end{array}\right)=\left(\begin{array}{cc}{A}_1& (s-1)I_s\\ (s-1)I_s& A_2\end{array}\right),$$

(24)

where

$$A_1=M_1{M}_1^T=M_1^2={(s-1)}^2{I}_s+(2-s)J_s,A_2=M_2{M}_2^T=M_2^2=I_s+(s-2)J_s,$$

$$M_1{M}_2^T=M_1{M}_2=(s-1)I_s.$$

From the structure of A, we have $A\in S\left(G\right)$. On the other hand,

$$M^{T}M=\left(M_1^T{M}_2^T\right)\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right)=M_1^T{M}_1+M_2^T{M}_2=(2-s)J_s+{(s-1)}^2{I}_s+I_s+(s-2)J_s=c{I}_s,$$

where $c=s^2-2s+2$. Hence $\mathrm{Spec}\left(M{M}^T\right)=\{c^{\left[s\right]},0^{\left[s\right]}\}$ and $q\left(G\right)=2$.

Now, we show that the matrix A has SSP. We need to prove that the only symmetric matrix satisfying $A\circ X=O$, $I\circ X=O$, and $[A,X]=AX-XA=O$ is $X=O$.

From the two equations $A\circ X=O$, $I\circ X=O$, X must have the following form: $X=\left(\begin{array}{cc}O& X_1\\ X_1^T& O\end{array}\right)$, where $X_1=\left(\begin{array}{cccc}0& x_{12}& \dots & x_{1s}\\ x_{21}& 0& & x_{2s}\\ ⋮& \ddots & \ddots & ⋮\\ x_{s1}& x_{s2}& \dots & 0\end{array}\right)$. The equality $AX=XA$ gives $X_1=X_1^T$. Also, we have $A_1{X}_1=X_1{A}_2$, i.e., $[{(s-1)}^2{I}_s+(2-s)J_s]X_1=X_1[I_s+(s-2)J_s]$.

Hence $s{X}_1=X_1{J}_s+J_s{X}_1$. Then ${\left(s{X}_1\right)}_{ij}={(X_1{J}_s+J_s{X}_1)}_{ij}$ for $i,j\in \left[s\right]$. Considering $i=j=1$, we have ${\left(s{X}_1\right)}_{11}=0$ and ${(X_1{J}_s+J_s{X}_1)}_{ii}=2{\sum }_{j=1}^s{x}_{1j}$, and then ${\sum }_{j=1}^s{x}_{1j}=0$. Considering $(i,j)=(k,k)$ for $2\le k\le s$ we arrive at ${\sum }_{j=1}^s{x}_{kj}=0$ for $2\le k\le s$. This means that the row and column sums in $X_1$ are equal to zero. Now, consider $i,j\in \left[s\right]$ where $i\ne j$.

We have

$$s{x}_{ij}={\left(s{X}_1\right)}_{ij}={(X_1{J}_s+J_s{X}_1)}_{ij}={\left(X_1{J}_s\right)}_{ij}+{\left(J_s{X}_1\right)}_{ij}=\sum _{k=1}^s{x}_{ik}+\sum _{k=1}^s{x}_{jk}=0.$$

Thus, $X_1=O_s$ and, consequently, $X=O$. Hence, the proof is complete. □

Corollary 6. 

For even n, we have $q\left(\overline{C_n}\right)=2$.

Proof. 

Let $G\cong K_n\backslash H$ and let H be the graph obtained from the complete bipartite graph $K_{n/2,n/2}$ by removing a perfect matching. Then, by Theorem 20 and Lemma 3, for H or any subgraph of H, $q\left(G\right)=2$. Considering this and that $C_n$ is a subgraph of H, the result is obtained. □

Theorem 21. 

Let G be a graph obtained from $(K_s\square K_2)\vee s{K}_1$ by removing a perfect matching between $s{K}_1$ and a copy of $K_s$. Then $q\left(G\right)=2$ and G has an SSP matrix realization with two distinct eigenvalues.

Proof. 

Let $M=\left(\begin{array}{c}{M}_1\\ M_2\\ I_s\end{array}\right),$ where $M_1=J_s-(s-1)I_s$ and $M_2=J_s-I_s$. Considering the fact that $M_1$ and $M_2$ are symmetric, we have

$$A=M{M}^T=\left(\begin{array}{c}{M}_1\\ M_2\\ I_s\end{array}\right)\left(M_1^T{M}_2^T{I}_s\right)=\left(\begin{array}{ccc}{M}_1{M}_1^T& M_1{M}_2^T& M_1{I}_s\\ M_2{M}_1^T& M_2{M}_2^T& M_2{I}_s\\ M_1& M_2& I_s\end{array}\right)=\left(\begin{array}{ccc}{A}_1& (s-1)I_s& M_1\\ (s-1)I_s& A_2& M_2\\ M_1& M_2& I_s\end{array}\right),$$

where

$$A_1=M_1{M}_1^T=M_1^2={(s-1)}^2{I}_s+(2-s)J_s,A_2=M_2{M}_2^T=M_2^2=I_s+(s-2)J_s,$$

$$M_1{M}_2^T=M_1{M}_2=(s-1)I_s.$$

From the structure of A, we have $A\in S\left(G\right)$. On the other hand,

$$M^{T}M=\left(M_1^T{M}_2^T{I}_s\right)\left(\begin{array}{c}{M}_1\\ M_2\\ I_s\end{array}\right)=M_1^T{M}_1+M_2^T{M}_2+I_s^2=(2-s)J_s+{(s-1)}^2{I}_s+I_s+(s-2)J_s+I_s=c{I}_s,$$

where $c=s^2-2s+3$. This gives $\mathrm{Spec}\left(M{M}^T\right)=\{c^{\left[s\right]},0^{\left[2s\right]}\}$, which proves $q\left(G\right)=2$.

Now, we show that the matrix A has SSP. We need to prove that the only symmetric matrix satisfying $A\circ X=O$, $I\circ X=O$, and $[A,X]=AX-XA=O$ is $X=O$.

From the two equations $A\circ X=O$, $I\circ X=O$, X must have the following form: $X=\left(\begin{array}{ccc}O& X_1& O\\ X_1^T& O& X_2\\ O& X_2& X_3\end{array}\right)$, where $X_1=\left(\begin{array}{cccc}0& x_{12}& \dots & x_{1s}\\ x_{21}& 0& & x_{2s}\\ ⋮& \ddots & \ddots & ⋮\\ x_{s1}& x_{s2}& \dots & 0\end{array}\right)$, $X_2=diag(y_1,\dots ,y_s)$ and $X_3=\left(\begin{array}{cccc}0& z_{12}& \dots & z_{1s}\\ z_{12}& 0& & z_{2s}\\ ⋮& \ddots & \ddots & ⋮\\ z_{1s}& z_{2s}& \dots & 0\end{array}\right)$. The matrix equation

$$AX=XA$$

(25)

gives $X_1=X_1^T$. From (25) we also have $M_2{X}_2+X_3=X_2{M}_2+X_3$, i.e., $(J_s-I_s)X_2=X_2(J_s-I_s)$, i.e., $J_s{X}_2=X_2{J}_s$. This gives $y_1=y_2=\cdots =y_s$, i.e., $X_2=y_1{I}_s$.

Again from (25), we have $A_1{X}_1+M_1{X}_2=X_1{A}_2$, that is, $M_1{X}_2=X_1{A}_2-A_1{X}_1$, that is, $(J_s-(s-1)I_s)\left(y_1{I}_s\right)=X_1(I_s+(s-2)J_s)-({(s-1)}^2{I}_s+(2-s)J_s)X_1$, i.e.,

$$y_1(2-s)I_s+y_1{J}_s=(2s-s^2)X_1+(s-2)X_1{J}_s+(s-2)J_s{X}_1.$$

Considering a main diagonal entry, say $(i,i)$, in the above matrix equation, we obtain

$$\sum _{j=1}^s{x}_{ij}=-\frac{y_1}{2}.$$

(26)

Considering the $(i,j)$-entry in the above matrix equation, we obtain $x_{ij}=-y_1\frac{s-1}{s-2}$. From the above and (26), $y_1=0$, that is, $X_2=O$. Using the equation $A_1{X}_1+M_1{X}_2=X_1{A}_2$, we arrive at the matrix equation $A_1{X}_1=X_1{A}_2$. Following a similar argument as in the proof of Theorem 20 we obtain $X_1=O$.

Again from (25), we have $M_1{X}_1+X_2=X_2{A}_2+X_3{M}_2$. Since $X_1=X_2=O$, we get $X_3{M}_2=O$, i.e., $X_3=X_3{J}_s$. Considering both the $(i,i)$ and $(i,j)$ entries from the matrix equation, we arrive at ${\sum }_{k=1}^s{z}_{ik}=0$ and $z_{ij}={\sum }_{k=1}^s{z}_{ik}=0$, that is, $X_3=O$, which gives $X=O$. □

Corollary 7. 

Consider the complete bipartite graph $K_{s,s}$ by removing a perfect matching. Define a new graph H by adding a copy of $K_s$ to this graph such that each vertex in $K_s$ is adjacent to the corresponding vertex in a copy of $s{K}_1$. Then, $q\left(\overline{H}\right)=2$. Moreover, the result holds for any subgraph of H on the same vertex set.

In [36], the authors studied the problem of graphs requiring property $p(r,s)$. A graph G has $p(r,s)$ if it contains a path of length r and every path of length r is contained in a cycle of length s. They prove that the smallest integer m so that every graph on n vertices with m edges has $p(2,4)$ (or each path of length 2 is contained either in a 3-cycle, or a 4-cycle) is $\left(\genfrac{}{}{0pt}{}{n}{2}\right)-(n-4)$ for all $n\ge 5$. Using this, it was noted in [37] that the above equation from [36] implies that the smallest number of edges required to guarantee that all graphs G on n vertices satisfy $q\left(G\right)=2$ is at least $\left(\genfrac{}{}{0pt}{}{n}{2}\right)-(n-3)$. For small values of n, it is known that in fact, equality holds in the previous claim. Namely, if at most $n-3$ edges are removed from the complete graph $K_n$ with $n\le 7$, then the resulting graph has a matrix realization with two distinct eigenvalues. Along these lines and based on [37] the following is a natural conjecture:

Conjecture 1. 

Removing up to $n-3$ edges from $K_n$ does not change the number of distinct eigenvalues of $K_n$. That is, for any subgraph H of $K_n$ with $\left|E\right(H\left)\right|\le n-3$, $q(K_n\backslash H)=2.$

We confirm Conjecture 1 for $n=7,8$ and note that our analysis of the case $n=7$ differs slightly from [37]. For this, we need the next few lemmas.

Lemma 8. 

Let $T_1$ be the tree given in Figure 3. We have $q\left(\overline{T_1}\right)=2$ and $\overline{T_1}$ has an SSP matrix realization with two distinct eigenvalues.

Figure 3. Tree $T_1$.

Proof. 

Consider the $7\times 4$ matrix $M_1$ as follows:

$$M_1=\left(\begin{array}{cccc}1& -2& 2& 1\\ 2& -1& -2& 2\\ 2& 2& 1& 2\\ 1& 2& 2& 0\\ -2& -1& 2& 0\\ 2& -2& 1& 0\\ 1& 0& 0& 0\end{array}\right).$$

Using the Gram–Schmidt method we arrive at a column orthonormal matrix $M_2$. In this case, we have $A=M_2{M}_2^T\in S\left(\overline{T_1}\right)$. In addition, $M_2^T{M}_2=I_4$ and $\mathrm{Spec}\left(A\right)=\{1^{\left[4\right]},0^{\left[3\right]}\}$. This proves that $q\left(\overline{T_1}\right)=2$. Furthermore, A has SSP (this can be confirmed using SageMath [38]), and by Lemma 3, the complement of any subgraph of $T_1$ on the same vertex set also has a matrix realization with two distinct eigenvalues. □

Lemma 9. 

Let $G\cong K_{1,3}\cup K_3$. Then, $q\left(\overline{G}\right)=2$ and $\overline{G}$ has an SSP matrix realization with two distinct eigenvalues.

Proof. 

Consider the $7\times 3$ matrix $M_1$ corresponding to the labeled graph G given in Figure 4 as follows:

$$M_1=\left(\begin{array}{ccc}1& 2& 2\\ 2& 1& -2\\ 2& -2& 1\\ 1& 1& 1\\ 1& -1& 1\\ -\sqrt{2}& 0& \sqrt{2}\\ 0& \sqrt{2}& 0\end{array}\right).$$

Figure 4. The graph G.

$A=M_1{M}_1^T\in S\left(\overline{G}\right)$. Also $M_1^T{M}_1=13I_3$ and $\mathrm{Spec}\left(A\right)=\{{13}^{\left[3\right]},0^{\left[4\right]}\}$. This proves that $q\left(\overline{G}\right)=2$. Furthermore, A has SSP (a computation that can be verified by SageMath [38]), and by Lemma 3, the complement of any subgraph of G on the same vertex set also has a matrix realization with two distinct eigenvalues. □

We now verify that Conjecture 1 holds for $n=7$.

Theorem 22. 

Removing up to four edges from $K_7$ does not change the number of distinct eigenvalues of $K_7$, i.e., for any subgraph H of $K_7$ on seven vertices, with $\left|E\right(H\left)\right|\le 4$ we have $q(K_7\backslash H)=2.$

Proof. 

To establish this result, it is sufficient to prove the complement of any graph H in Figure 5 has a matrix realization with two distinct eigenvalues. Suppose that the graphs in Figure 5 are denoted by $H_i$ for $i\in \left[10\right]$ from left to right in each row. Then, the graphs $H_i$ for $i=1,3,7,8,10$ are the union of complete bipartite graphs with some isolated vertices. By Lemma 4 (2), the complements of these graphs and any subgraph of these graphs have a matrix realization with two distinct eigenvalues. Additionally, $q\left(\overline{H_i}\right)=2$ for $i=4,5,9$ and for any subgraph $H_i^{\prime }$ of $H_i$, $q\left(\overline{H_i^{\prime }}\right)=2$ by Lemma 8. Moreover, $q\left(\overline{H_6}\right)=2$ and for any subgraph $H_6^{\prime }$ of $H_6$, $q\left(\overline{H_6^{\prime }}\right)=2$ by Lemma 9. Additionally, from Lemmas 8 and 9 such realizations exist with the SSP. Hence any subgraph of these graphs has a matrix realization with two distinct eigenvalues. To complete the proof, we need to show the complement graph of $H_2$ has a matrix realization with two distinct eigenvalues with the SSP. To this end, consider the $7\times 3$ matrix $M_1$ as follows:

$$M_1=\left(\begin{array}{ccc}1& -2& 1\\ 2& -1& 2\\ 2& 2& 2\\ 1& 2& 0\\ -2& -1& 0\\ 2& -2& 0\\ 1& 0& 0\end{array}\right).$$

Figure 5. All graphs with 7 vertices and 4 edges.

Using the Gram–Schmidt method we arrive at a column orthonormal matrix $M_2$. We have $A=M_2{M}_2^T\in S\left(\overline{H_2}\right)$. In addition, $M_2^T{M}_2=I_3$ and $\mathrm{Spec}\left(A\right)=\{1^{\left[3\right]},0^{\left[4\right]}\}$. Hence, $q\left(\overline{H_2}\right)=2$. Furthermore, A has SSP (a computation that can be verified by SageMath [38]), and by Lemma 3, the complement of any subgraph of $H_2$ on the same vertex set also has a matrix realization with two distinct eigenvalues. □

We require the following results to confirm Conjecture 1 for $n=8$.

Lemma 10. 

Let $G\cong H_1\cup 2K_1$, where $H_1$ is the graph on the left given in Figure 6. Then $q\left(\overline{G}\right)=2$ and $\overline{G}$ has an SSP matrix realization with two distinct eigenvalues.

Figure 6. The graphs $H_1$ (left) and $H_2$ (right).

Proof. 

Given G as assumed it can be shown without too much difficulty that $\overline{G}\cong (H_2\vee K_3)-e$, where $H_2$ is the graph on the right given in Figure 6 minus an edge e with one endpoint in $K_3$ and the other endpoint in $H_2$ with degree three.

Suppose $M=\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right),$ is a vertex-clique incidence matrix of $\overline{G}$, where the blocks $M_1$ and $M_2$ are vertex-clique incidence matrices corresponding to graphs $H_2$ and $K_3$, that is, $M{M}^T\in S\left(\overline{G}\right).$ From (24) we have $M_1{M}_1^T\in S\left(H_2\right)$ and $M_2{M}_2^T\in S\left(K_3\right)$. On the other hand, we have

$$M^{T}M=M_1^T{M}_1+M_2^T{M}_2.$$

(27)

Consider a vertex-clique incidence matrix $M_1$ as follows:

$$M_1=\left(\begin{array}{ccc}1& 0& 0\\ 1& 0& 1\\ 1& 1& 0\\ 0& \sqrt{2}& 0\\ 0& 0& \sqrt{2}\end{array}\right).$$

Then we have $M_1{M}_1^T\in S\left(H_2\right)$ and $M_1^T{M}_1=\left(\begin{array}{ccc}3& 1& 1\\ 1& 3& 0\\ 1& 0& 3\end{array}\right)$. Given $M_1$ above, the remainder of the proof is devoted to constructing a matrix $M_2$ so that following (27) we have $M^{T}M=c{I}_3$, for some scalar c. Consider a matrix $M_2$ so that

$$M_2^T{M}_2=\left(\begin{array}{ccc}a& -1& -1\\ -1& a& 0\\ -1& 0& a\end{array}\right),$$

(28)

where a is a constant. Suppose the matrix $M_2=\left(\begin{array}{ccc}{x}_1& y_1& z_1\\ x_2& y_2& z_2\\ x_3& y_3& z_3\end{array}\right).$ This with (28) leads to the following equations:

$$x_1^2+x_2^2+x_3^2=y_1^2+y_2^2+y_3^2=z_1^2+z_2^2+z_3^2=a,$$

$$x_1{y}_1+x_2{y}_2+x_3{y}_3=-1,x_1{z}_1+x_2{z}_2+x_3{z}_3=-1,y_1{z}_1+y_2{z}_2+y_3{z}_3=0.$$

Solving this system of non-linear equations, we have a candidate matrix $M_2$: $M_2=\left(\begin{array}{ccc}1& -1& z_1\\ -1& 2& z_2\\ 2& 1& z_3\end{array}\right)$, where $z_1=\frac{1}{7}(2\sqrt{51}-1)$, $z_2=\frac{1}{35}(6\sqrt{51}+4)$, and $z_3=\frac{-1}{35}(2\sqrt{51}+13)$.

Thus

$$M=\left(\begin{array}{ccc}1& 0& 0\\ 1& 0& 1\\ 1& 1& 0\\ 0& \sqrt{2}& 0\\ 0& 0& \sqrt{2}\\ 1& -1& z_1\\ -1& 2& z_2\\ 2& 1& z_3\end{array}\right).$$

It is obvious that $M{M}^T\in S\left(\overline{G}\right)$ and $M^{T}M=9I_3$. Since the matrices $AB$ and $BA$ have same nonzero eigenvalues, we have $\mathrm{Spec}\left(M{M}^T\right)=\{9^{\left[3\right]},0^{\left[5\right]}\}$, and then $q\left(\overline{G}\right)=2$. Moreover, applying a basic computation from SageMath [38], we can confirm that $M{M}^T$ has SSP and this completes the proof. □

By Lemma 10, $\overline{G}$ has an SSP realization $A=M{M}^T$ with two distinct eigenvalues. Then by Lemma 3, any supergraph on the same vertex set as G has a realization with the same spectrum as A. In particular, $q(H_2\vee K_3)=2$. This is stated in the following corollary.

Corollary 8. 

Let $G\cong H_2\cup 3K_1$, where $H_2$ is the right graph given in Figure 6. Then, $q\left(\overline{G}\right)=2$ and $\overline{G}$ has an SSP matrix realization with two distinct eigenvalues.

Lemma 11. 

Let $G\cong H_3\cup 3K_1$, where $H_3$ is obtained from $C_5$ by joining a vertex to any vertex in $C_5$. Then, $q\left(\overline{G}\right)=2$ and $\overline{G}$ has an SSP matrix realization with two distinct eigenvalues.

Proof. 

We know that $\overline{G}\cong (C_5\vee K_3)-e$, where e is an edge with one endpoint in $K_3$ and the other in $C_5$. Suppose $M=\left(\begin{array}{c}{M}_1\\ M_2\end{array}\right),$ is a vertex-clique incidence matrix of $\overline{G}$, where blocks $M_1$ and $M_2$ are vertex-clique incidence matrices corresponding to graphs $C_5$ and $K_3$, that is, $M{M}^T\in S\left(\overline{G}\right).$ From (24) we have $M_1{M}_1^T\in S\left(C_5\right)$ and $M_2{M}_2^T\in S\left(K_3\right)$. On the other hand, we also have the equations in (27). Now, we consider a vertex-clique incidence matrix $M_1$ as follows:

$$M_1=\left(\begin{array}{ccc}1& 0& 0\\ 1& 1& 0\\ -1& 1& 1\\ 0& -1& 1\\ 0& 0& 1\end{array}\right).$$

Then, $M_1{M}_1^T\in S\left(C_5\right)$ and $M_1^T{M}_1=\left(\begin{array}{ccc}3& 0& -1\\ 0& 3& 0\\ -1& 0& 3\end{array}\right)$. Given $M_1$ above, the remainder of the proof is devoted to constructing a matrix $M_2$ so that following (27) we have $M^{T}M=c{I}_3$, for some scalar c. We need to create a matrix $M_2$ so that

$$M_2^T{M}_2=\left(\begin{array}{ccc}a& 0& 1\\ 0& a& 0\\ 1& 0& a\end{array}\right),$$

(29)

where a is a constant. Suppose $M_2=\left(\begin{array}{ccc}{x}_1& y_1& z_1\\ x_2& y_2& z_2\\ x_3& y_3& z_3\end{array}\right).$ This with (29) leads to the following equations:

$$x_1^2+x_2^2+x_3^2=y_1^2+y_2^2+y_3^2=z_1^2+z_2^2+z_3^2=a,$$

$$x_1{y}_1+x_2{y}_2+x_3{y}_3=0,x_1{z}_1+x_2{z}_2+x_3{z}_3=1,y_1{z}_1+y_2{z}_2+y_3{z}_3=0.$$

Solving these non-linear equations we have $M_2=\left(\begin{array}{ccc}\frac{1}{\sqrt{3}}& 0& \frac{1}{\sqrt{3}}\\ \frac{1}{\sqrt{3}}& \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{3}}\\ \frac{1}{\sqrt{3}}& \frac{-1}{\sqrt{2}}& \frac{1}{\sqrt{3}}\end{array}\right)$. Thus, we have

$$M=\left(\begin{array}{ccc}1& 0& 0\\ 1& 1& 0\\ -1& 1& 1\\ 0& -1& 1\\ 0& 0& 1\\ \frac{1}{\sqrt{3}}& 0& \frac{1}{\sqrt{3}}\\ \frac{1}{\sqrt{3}}& \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{3}}\\ \frac{1}{\sqrt{3}}& \frac{-1}{\sqrt{2}}& \frac{1}{\sqrt{3}}\end{array}\right).$$

It is clear that $M{M}^T\in S\left(\overline{G}\right)$ and $M^{T}M=4I_3$. Since the matrices $AB$ and $BA$ have same nonzero eigenvalues, we have $\mathrm{Spec}\left(M{M}^T\right)=\{4^{\left[3\right]},0^{\left[5\right]}\}$, and $q\left(\overline{G}\right)=2$. Moreover, applying a basic computation from SageMath [38], it follows that $M{M}^T$ has SSP and this completes the proof. □

By Lemma 11, $\overline{G}$ has an SSP realization $A=M{M}^T$ with two distinct eigenvalues. By Lemma 3, any supergraph on the same set of vertices as G has a matrix realization with the same spectrum as A. Thus, $q(C_5\vee K_3)=2$. This is stated in the following corollary.

Corollary 9. 

Let $G\cong C_5\cup 3K_1$. Then, $q\left(\overline{G}\right)=2$ and $\overline{G}$ has an SSP matrix realization with two distinct eigenvalues.

Proposition 1. 

Let $G\cong K_3\cup K_{1,n-4}$, where $n\ge 7$. Then $q\left(\overline{G}\right)=2$ and $\overline{G}$ has an SSP matrix realization with two distinct eigenvalues.

Proof. 

We show that the complement of G has a matrix realization with two distinct eigenvalues with the SSP. Consider $n\times 3$ matrix $M_1$ with rows labeled as given in Figure 7 for $n=8$:

$$M_1=\left(\begin{array}{ccc}1& 2& 2\\ 2& 1& -2\\ 2& -2& 1\\ -\sqrt{2}& 0& \sqrt{2}\\ 0& \sqrt{\frac{2}{n-4}}& 0\\ ⋮& ⋮& ⋮\\ 0& \sqrt{\frac{2}{n-4}}& 0\end{array}\right).$$

Figure 7. The graph G.

We have $A=M_1{M}_1^T\in S\left(\overline{G}\right)$. Also $M_1^T{M}_1=11I_3$ and $\mathrm{Spec}\left(A\right)=\{{11}^{\left[3\right]},0^{[n-3]}\}$. This proves that $q\left(\overline{G}\right)=2$. To verify that A has SSP, suppose X is a symmetric matrix such that $A\circ X=O$, $I\circ X=O$, and $[A,X]=AX-XA=O$. Note to verify $[A,X]=AX-XA=O$ it is equivalent to prove that $AX$ is symmetric. Now assume that X has the form:

$$X=\left(\begin{array}{ccc}0& O& x^T\\ O& X_1& O\\ x& O& O\end{array}\right),\mathrm{where}{X}_1=\left(\begin{array}{ccc}0& a& b\\ a& 0& c\\ b& c& 0\end{array}\right),$$

and x is a (possibly) nonzero vector of size $n-4$. Since $AX$ is symmetric, comparing the (1,3) and (3,1) blocks of $AX$ we note that $\alpha Jx=4x$. So if we set $\beta ={𝟙}^{T}x$, then $x=\frac{\alpha }{4}\beta 𝟙$. Comparing the (1,2) and (2,1) blocks of $AX$ gives

$$2\sqrt{\alpha }\beta =-4\sqrt{2}a-\sqrt{2}b=-\sqrt{2}b+4\sqrt{2}c,\mathrm{and}\sqrt{\alpha }\beta =\sqrt{2}a-\sqrt{2}c.$$

Hence, it follows that $a=-c$ and $\beta =\frac{2\sqrt{2}a}{\sqrt{\alpha }}$. Finally, comparing the (2,3) and (3,2) blocks of $AX$, we have

$$a\sqrt{\alpha }-2b\sqrt{\alpha }=2a\sqrt{\alpha }-2c\sqrt{\alpha }=2b\sqrt{\alpha }+c\sqrt{\alpha }={\left(\frac{\alpha }{4}\beta \right)}^2=\frac{a^2}{2\alpha }.$$

From the above equations we deduce that $b=-\frac{3}{2}a$. Substituting the equations $a=-c$, $\beta =\frac{2\sqrt{2}a}{\sqrt{\alpha }}$, and $b=-\frac{3}{2}a$ into the equation $2\sqrt{\alpha }\beta =-\sqrt{2}b+4\sqrt{2}c$, yields $4\sqrt{2}a=\frac{3}{\sqrt{2}}a-4\sqrt{2}a$. Assuming $a\ne 0$, implies an immediate contradiction. Thus $a=0$, and it follows, based on the analysis above that $X=0$. Hence A has the SSP. Using the fact that this matrix realization has the SSP together with Lemma 3, it follows that the complement of any subgraph of G on the same vertex set also realizes distinct eigenvalues. □

Lemma 12. 

Let G be the graph given in Figure 8. Then, $q\left(\overline{G}\right)=2$ and $\overline{G}$ has an SSP matrix realization with two distinct eigenvalues.

Figure 8. The graph G.

Proof. 

We show that the complement graph of G has a matrix realization with two distinct eigenvalues with the SSP. To do this, first we consider $8\times 3$ matrix M as follows:

$$M=\left(\begin{array}{ccc}\sqrt{\frac{15}{2}}& 0& 0\\ 0& 1& 1\\ 0& 1& 1\\ 0& 1& 2\\ 0& -2& 1\\ 1& -1& 0\\ 1& 0& 1\\ \sqrt{\frac{2}{2}}& \sqrt{2}& -\sqrt{2}\end{array}\right).$$

We have $A=M{M}^T\in S\left(\overline{G}\right)$. Also $M^{T}M=10I_3$ so $\mathrm{Spec}\left(A\right)=\{{10}^{\left[3\right]},0^{\left[5\right]}\}$. This proves that $q\left(\overline{G}\right)=2$. Furthermore, A has SSP (observed using SageMath [38]) and by Lemma 3, the complement of any subgraph of G on the same vertex has a matrix realization having two distinct eigenvalues. □

Now we are in a position to establish that Conjecture 1 holds for $n=8$.

Theorem 23. 

Removing up to five edges from $K_8$ does not change the number of distinct eigenvalues of $K_8$, i.e., for any subgraph H on eight vertices of $K_8$ with $\left|E\right(H\left)\right|\le 5$, $q(K_8\backslash H)=2.$

Proof. 

To establish this result, it is sufficient to prove the complement of any graph H in Figure 9 has a matrix realization with two distinct eigenvalues. Suppose that the graphs in Figure 9 are denoted by $H_i$ for $i\in \left[24\right]$ from left to right in each row. The graphs $H_i$ for $i=1,2,9,10,15,22,23$ are the union of complete bipartite graphs with some isolated vertices. By Lemma 4 (2), the complements of these graphs and any subgraph of these graphs have a matrix realization with two distinct eigenvalues. Additionally, $q\left(\overline{H_i}\right)=2$ for $i=5,11,12,16,17,18,19,20,24$ and for any subgraph $H_i^{\prime }$ of $H_i$, $q\left(\overline{H_i^{\prime }}\right)=2$ by Theorem 20. For $i=3,7,8,13,14$, we have $q\left(\overline{H_i}\right)=2$ and for any subgraph $H_i^{\prime }$ of $H_i$, $q\left(\overline{H_i^{\prime }}\right)=2$ by Lemma 12. Additionally, from Theorem 20 and Lemma 12 such realizations exist with the SSP. Hence, any subgraph of these graphs has a matrix realization with two distinct eigenvalues.

Figure 9. All graphs with 8 vertices and 5 edges.

Further, $q\left(\overline{H_{21}}\right)=q\left(\overline{(2K_2\cup K_1)\cup K_3}\right)=q(G\vee 3K_1)=2$ by Lemma 5, where the graph $G=\overline{2K_2\cup K_1}=K_{2,2}\vee K_1$ is connected. If we remove any edges in $H_{21}$ from the triangle, then the complement of the result graph has at least two distinct eigenvalues by Lemma 4 (2), and if we remove any edges in $H_{21}$ from out of the triangle, again by Lemma 5, we have that the complement of the result graph has a matrix realization with at least two distinct eigenvalues. We have $q\left(\overline{H_4}\right)=2$ and the complement of any subgraph of this graph has a matrix realization with two distinct eigenvalues, by Corollary 8. Moreover, $q\left(\overline{H_6}\right)=2$, and the complement of any subgraph of this graph also has a matrix realization with two distinct eigenvalues, by Corollary 9. This completes the proof of the theorem. □

5. Concluding Remarks and Open Problems

In this work, we utilized the notions of a clique partition and an edge clique cover of a graph to introduce and explore the various properties of a vertex-clique incidence matrix of the graph, which can be viewed as a generalization of the vertex-edge incidence matrix. Using these incidence matrices, we obtained sharp interesting lower bounds concerning the negative eigenvalues and thus the negative inertia of a graph, and we generalized the notion of the line graph of a graph by introducing the clique partition graph of the given graph. Additionally, we determined the relations between the spectrum of a graph and its clique partition graph. Further, we generalized the notion of incidence energy and signless Laplacian energy of a graph and provided some novel upper bounds for the energies of a graph, its clique partition graph, and the line graph. Finally, applying a general version of a vertex-clique incidence matrix of a graph associated with its edge clique cover, we were able to characterize a few classes of graphs with $q\left(G\right)=2$. To close, we list two important and unresolved issues related to some of the content of the current work.

Problem 1: 

Characterize the corresponding extremal graphs for which the inequalities given in (4), (6), (10), (13), and (14) hold with equality.

Problem 2: 

Prove that Conjecture 1 is valid for any graph G of an order of at least nine.