The Extremal Graphs of Some Topological Indices with Given Vertex k-Partiteness

Abstract

The vertex k-partiteness of graph G is defined as the fewest number of vertices whose deletion from G yields a k-partite graph. In this paper, we characterize the extremal value of the reformulated first Zagreb index, the multiplicative-sum Zagreb index, the general Laplacian-energy-like invariant, the general zeroth-order Randić index, and the modified-Wiener index among graphs of order n with vertex k-partiteness not more than $m$.

1. Introduction

All graphs considered in this paper are simple, undirected, and connected. Let G be a graph with vertex set $V(G)=\{v_1,\cdots ,v_n\}$ and edge set $E(G)=\{e_1,\cdots ,e_m\}.$ The degree of a vertex $u\in V(G)$ is the number of edges incident to $u,$ denoted by $d_G(u).$ The distance between two vertices u and v is the length of the shortest path connecting u and $v,$ denoted by $d_G(u,v).$ The complement of $G,$ denoted by $\overline{G},$ is the graph with vertex set $V(\overline{G})=V(G)$ and edge set $E(\overline{G})=\{uv:uv\notin E(G)\}.$ A subgraph of G induced by $H,$ denoted by $\langle H\rangle ,$ is the subgraph of G that has the vertex set $H,$ and for any two vertices $u,v\in V(H),$ they are adjacent in H iff they are adjacent in $G.$ The adjacency matrix of G is a square $n\times n$ matrix such that its element $a_{ij}$ is one when there is an edge from vertex $u_i$ to vertex $u_j,$ and zero when there is no edge, denoted by $A(G).$ Let $D(G)=diag(d_1,d_2,\cdots ,d_n)$ be the diagonal matrix of vertex degrees of $G.$ The Laplacian matrix of G is defined as $L(G)=D(G)-A(G),$ and the eigenvalues of $L(G)$ are called Laplacian eigenvalues of $G,$ denoted by ${\mu }_1,\cdots ,{\mu }_n$ with ${\mu }_1\ge \cdots \ge {\mu }_n.$ It is well known that ${\mu }_n=0,$ and the multiplicity of zero corresponds to the number of connected components of $G$.

A bipartite graph is a graph whose vertex set can be partitioned into two disjoint sets $U_1$ and $U_2,$ such that each edge has an end vertex in $U_1$ and the other one in $U_2.$ A complete bipartite graph, denoted by $K_{s,t},$ is a bipartite graph with $|U_1|=s$ and $|U_2|=t,$ where any two vertices $u\in U_1$ and $v\in U_2$ are adjacent. If every pair of distinct vertices in G is connected by a unique edge, we call G a complete graph. The complete graph with n vertices is denoted by $K_n.$ An independent set of G is a set of vertices, no two of which are adjacent. A graph G is called k-partite if its vertex-set can be partitioned into k different independent sets $U_1,\cdots ,U_k.$ When $k=2,$ they are the bipartite graphs, and $k=3$ the tripartite graphs. The vertex k-partiteness of graph $G,$ denoted by $v_k(G),$ is the fewest number of vertices whose deletion from G yields a k-partite graph. A complete k-partite graph, denoted by $K_{s_1,\cdots ,s_k},$ is a k-partite graph with k different independent sets $|U_1|=s_1,\cdots ,\left|U_k\right|=s_k,$ where there is an edge between every pair of vertices from different independent sets.

A topological index is a numerical value that can be used to characterize some properties of molecule graphs in chemical graph theory. Recently, many researchers have paid much attention to studying different topological indices. Dimitrov [1] studied the structural properties of trees with minimal atom-bond connectivity index. Li and Fan [2] obtained the extremal graphs of the Harary index. Xu et al. [3] determined the eccentricity-based topological indices of graphs. Hayat et al. [4] studied the valency-based topological descriptors of chemical networks and their applications. Let $G+uv$ be the graph obtained from G by adding an edge $uv\in E(\overline{G}).$ Let $I(G)$ be a graph invariant, if $I(G+uv)>I(G)$ (or $I(G+uv)<I(G),$ respectively) for any edge $uv\in E(\overline{G}),$ then we call $I(G)$ a monotonic increasing (or decreasing, respectively) graph invariant with the addition of edges [5]. Let ${\mathcal{G}}_{n,m,k}$ be the set of graphs with order n and vertex k-partiteness $v_k(G)\le m,$ where $1\le m\le n-k.$ In [5,6,7], the authors have researched several monotonic topological indices in ${\mathcal{G}}_{n,m,2},$ such as the Kirchhoff index, the spectral radius, the signless Laplacian spectral radius, the modified-Wiener index, the connective eccentricity index, and so on. Inspired by these results, we extend the parameter of graph partition from two-partiteness to arbitrary k-partiteness. Moreover, we study some monotonic topological indices and characterize the graphs with extremal monotonic topological indices in ${\mathcal{G}}_{n,m,k}$.

2. Preliminaries

The join of two-vertex-disjoint graphs $G_1,G_2,$ denoted by $G=G_1\vee G_2,$ is the graph obtained from the disjoint union $G_1\cup G_2$ by adding edges between each vertex of $G_1$ and each of $G_2.$ It is to say that $V(G)=V(G_1)\cup V(G_2)$ and $E(G)=E(G_1)\cup E(G_2)\cup \{uv:u\in V(G_1),v\in V(G_2)\}.$

The join operation can be generalized as follows. Let $\mathit{F}=\{G_1,\cdots ,G_k\}$ be a set of vertex-disjoint graphs and H be an arbitrary graph with vertex set $V(H)=\{1,\cdots ,k\}.$ Each vertex $i\in V(H)$ is assigned to the graph $G_i\in \mathit{F}.$

The H-join of the graphs $G_1,\cdots ,G_k$ is the graph $G=H[G_1,\cdots ,G_k],$ such that $V(G)=\bigcup _{j=1}^{k}V(G_j)$ and:

$$E(G)=\bigcup _{j=1}^{k}E(G_j)\bigcup (\bigcup _{ij\in E(H)}\{uv:u\in V(G_i),v\in V(G_j)\}).$$

If $H=K_2,$ the H-join is the usual join operation of graphs, and the complete k-partite graph $K_{s_1,\cdots ,s_k}$ can be seen as the $K_k$-join graph $K_k[O_{s_1},\cdots ,O_{s_k}]$, where $O_{s_i}$ is an empty graph of order $s_i,$ $1\le i\le k.$

For $U\subseteq V(G),$ let $G-U$ be the graph obtained from G by deleting the vertices in U and the edges incident with them.

Lemma 1.

Let G be an arbitrary graph in ${\mathcal{G}}_{n,m,k}$ and $I(G)$ be a monotonic increasing graph invariant. Then, there exists k positive integers $s_1,\cdots ,s_k$ satisfying ${\displaystyle \sum _{i=1}^k}{s}_i=n-m,$ such that $I(G)\le I(\widehat{G})$ holds for all graphs $G\in {\mathcal{G}}_{n,m,k},$ where $\widehat{G}=K_m\vee (K_k[O_{s_1},\cdots ,O_{s_k}])\in {\mathcal{G}}_{n,m,k},$ with equality holds if and only if $G\cong \widehat{G}.$

Proof. 

Choose $\widehat{G}\in {\mathcal{G}}_{n,m,k}$ with the maximum value of a monotonic increasing graph invariant such that $I(G)\le I(\widehat{G})$ for all $G\in {\mathcal{G}}_{n,m,k}.$ Assume that the k-partiteness of graph $\widehat{G}$ is $m^{\prime },$ then there exists a vertex set U of graph $\widehat{G}$ with order $m^{\prime }$ such that $\widehat{G}-U$ is a k-partite graph with k-partition $\{U_1,\cdots ,U_k\}.$ For $1\le i\le k,$ let $s_i$ be the order of $U_i$; hence, $n={\displaystyle \sum _{i=1}^k}{s}_i+m^{\prime }.$

Firstly, we claim that $\widehat{G}-U=K_k[O_{s_1},\cdots ,O_{s_k}].$ Otherwise, there exists at least two vertices $u\in U_{s_i}$ and $v\in U_{s_j}$ for some $i\ne j,$ which are not adjacent in $\widehat{G}.$ By joining the vertices u and $v,$ we get a new graph $\widehat{G}+uv,$ obviously, $\widehat{G}+uv\in {\mathcal{G}}_{n,m,k}.$ Then, $I(\widehat{G})<I(\widehat{G}+uv),$ which is a contradiction.

Secondly, we claim that U is the complete graph $K_{m^{\prime }}.$ Otherwise, there exists at least two vertices $u,v\in U$, which are not adjacent. By connecting the vertices u and $v,$ we arrive at a new graph $\widehat{G}+uv,$ obviously, $\widehat{G}+uv\in {\mathcal{G}}_{n,m,k}.$ Then, we have $I(\widehat{G})<I(\widehat{G}+uv),$ a contradiction again.

Using a similar method, we can get $\widehat{G}=K_{m^{\prime }}\vee (K_k[O_{s_1},\cdots ,O_{s_k}]).$

Finally, we prove that $m^{\prime }=m.$ If $m^{\prime }\le m-1,$ then ${\displaystyle \sum _{i=1}^k}{s}_i=n-m^{\prime }\ge n-m+1>n-m\ge k;$ thus, ${\displaystyle \sum _{i=1}^k}{s}_i>k.$ Without loss of generality, we assume that $s_1\ge 2.$ By moving a vertex $u\in O_{s_1}$ to the set of U and adding edges between u and all the other vertices in $O_{s_1},$ we get a new graph $\tilde{G}=K_{m^{\prime }+1}\vee (K_k[O_{s_1-1},O_{s_2},\cdots ,O_{s_k}]).$ It is easy to check that $\tilde{G}\in {\mathcal{G}}_{n,m,k}$ has $s_1-1$ edges more than the graph $\widehat{G}.$ By the definition of the monotonic increasing graph invariant, we get $I(\widehat{G})<I(\tilde{G}),$ which is obviously another contradiction.

Therefore, $\widehat{G}$ is the join of a complete graph with order m and a complete k-partite graph with order $n-m.$ That is to say $\widehat{G}=K_m\vee (K_k[O_{s_1},\cdots ,O_{s_k}]).$

The proof of the lemma is completed.  □

Lemma 2.

Let G be an arbitrary graph in ${\mathcal{G}}_{n,m,k}$ and $I(G)$ be a monotonic decreasing graph invariant. Then, there exists k positive integers $s_1,\cdots ,s_k$ satisfying ${\displaystyle \sum _{i=1}^k}{s}_i=n-m,$ such that $I(G)\ge I(\widehat{G})$ holds for all graphs $G\in {\mathcal{G}}_{n,m,k},$ where $\widehat{G}=K_m\vee (K_k[O_{s_1},\cdots ,O_{s_k}])\in {\mathcal{G}}_{n,m,k},$ with equality holds if and only if $G\cong \widehat{G}.$

3. Main Results

In this section, we will characterize the graphs with an extremal monotonic increasing (or decreasing, respectively) graph invariant in ${\mathcal{G}}_{n,m,k}.$ Assume that $n-m=sk+t,$ where s is a positive integer and t is a non-negative integer with $0\le t<k.$

3.1. The Reformulated First Zagreb Index, Multiplicative-Sum Zagreb Index, and k-Partiteness

The first Zagreb index is used to analyze the structure-dependency of total $\pi$-electron energy on the molecular orbitals, introduced by Gutman and Trinajstć [8]. It is denoted by:

$$M_1(G)={\displaystyle \sum _{uv\in E(G)}}(d_G(u)+d_G(v)),$$

which can be also calculated as:

$$M_1(G)={\displaystyle \sum _{u\in V(G)}}{d}_G{(u)}^2.$$

Todeschini and Consonni [9] considered the multiplicative version of the first Zagreb index in 2010, defined as:

$${\Pi }_1(G)=\prod _{u\in V(G)}{d}_G{(u)}^2.$$

For an edge $e=uv\in E(G),$ we define the degree of e as $d_G(e)=d_G(u)+d_G(v)-2.$ Millic̆ević et al. [10] introduced the reformulated first Zagreb index, defined as:

$${\tilde{M}}_1(G)=\sum _{e\in E(G)}{d}_G{(e)}^2={\displaystyle \sum _{uv\in E(G)}}{(d_G(u)+d_G(v)-2)}^2.$$

Eliasi et al. [11] introduced another multiplicative version of the first Zagreb index, which is called the multiplicative-sum Zagreb index and defined as:

$${\Pi }_1^{*}(G)=\prod _{uv\in E(G)}(d_G(u)+d_G(v)).$$

They are widely used in chemistry to study the heat information of heptanes and octanes. For some recent results on the fourth Zagreb indices, one can see [12,13,14,15,16,17].

Lemma 3.

Let G be a graph with $u,v\in V(G).$ If $uv\in E(\overline{G}),$ then ${\tilde{M}}_1(G)<{\tilde{M}}_1(G+uv).$

Lemma 4.

Let G be a graph with $u,v\in V(G).$ If $uv\in E(\overline{G}),$ then ${\Pi }_1^{*}(G)<{\Pi }_1^{*}(G+uv).$

Note that $s_1,\cdots ,s_k$ are k positive integers with ${\displaystyle \sum _{i=1}^k}{s}_i=n-m.$

Theorem 1.

Let $\widehat{G}$ be a graph of order $n>2,$ and the join of a complete graph with order m and a complete k-partite graph with order $n-m$ in ${\mathcal{G}}_{n,m,k},$ i.e., $\widehat{G}=K_m\vee (K_k[O_{s_1},\cdots ,O_{s_k}]).$ By moving one vertex from the part of $O_{s_1}$ to the part of $O_{s_2},$ we get a new graph $\tilde{G}=K_m\vee (K_k[O_{s_1-1},O_{s_2+1},\cdots ,O_{s_k}]).$ If $s_1-1\ge s_2+1,$ then ${\tilde{M}}_1(\tilde{G})>{\tilde{M}}_1(\widehat{G}).$

Proof. 

By the definition of the reformulated first Zagreb index ${\tilde{M}}_1(G)$, we can calculate as follows:

$${\tilde{M}}_1(\widehat{G})=\frac{m(m-1)}{2}{(2n-4)}^2+\sum _{i=1}^{k}m{s}_i{(2n-s_i-3)}^2+\sum _{1\le i<j\le k}{s}_i{s}_j{(2n-s_i-s_j-2)}^2.$$

Therefore,

$$\begin{array}{cc}\hfill {\tilde{M}}_1(\tilde{G})-{\tilde{M}}_1(\widehat{G})& =m(s_1-1){(2n-s_1-2)}^2+m(s_2+1){(2n-s_2-4)}^2\hfill \\ & +(s_1-1)(s_2+1){(2n-s_1-s_2-2)}^2-m{s}_1{(2n-s_1-3)}^2\hfill \\ & -m{s}_2{(2n-s_2-3)}^2-s_1{s}_2{(2n-s_1-s_2-2)}^2\hfill \\ & +\sum _{i=3}^k(s_1-1)s_i{(2n-s_1-s_i-1)}^2+\sum _{i=3}^k(s_2+1)s_i{(2n-s_2-s_i-3)}^2\hfill \\ & -\sum _{i=3}^k{s}_1{s}_i{(2n-s_1-s_i-2)}^2-\sum _{i=3}^k{s}_2{s}_i{(2n-s_2-s_i-2)}^2\hfill \\ & =(s_1-s_2-1)[(5n+3p-12)p+{(n+p-2)}^2\hfill \\ & +(7n+8m-12)\sum _{i=3}^k{s}_i+{(\sum _{i=3}^k{s}_i)}^2+{\displaystyle \sum _{i=3}^k}{s}_i(3\sum _{i=3}^k{s}_i-4s_i)\hfill \\ & =(s_1-s_2-1)[{(n-2)}^2+(7n-16)m+4m^2\hfill \\ & +(7n+8m-12)\sum _{i=3}^k{s}_i+4{({\displaystyle \sum _{i=3}^k}{s}_i)}^2-4\sum _{i=3}^k{s}_i^2]\hfill \\ & >(s_1-s_2-1)[{(n-2)}^2+(4n-8)m+4m^2]\hfill \\ & =(s_1-s_2-1){(n-2+2m)}^2>0.\square \hfill \end{array}$$

Note that we have $n-m=sk+t=(k-t)s+t(s+1),$ where s is a positive integer and t is a non-negative integer with $0\le t<k.$ For simplicity, we write $K_m\vee (K_k[\{k-t\}{O}_s,\left\{s\right\}{O}_{s+1}])=K_m\vee (K_k[{\underbrace{O_s,\cdots ,O_s}}_{k-t},{\underbrace{O_{s+1},\cdots ,O_{s+1}}}_t]).$ Then, the extremal value and the corresponding graph of the reformulated first Zagreb index ${\tilde{M}}_1(G)$ can be shown as follows.

Theorem 2.

Let G be an arbitrary graph in ${\mathcal{G}}_{n,m,k}.$ Then:

$$\begin{array}{cc}\hfill {\tilde{M}}_1(G)& \le \frac{m(m-1)}{2}{(2n-4)}^2+m(n-m)(6n-3s-11)\hfill \\ & +2(n-m)(n-m-s){(n-s-1)}^2\hfill \\ & +t(s+1)[-6{(n-s-1)}^2+n+2m(5-2n+s)+(t-2)(s+1)],\hfill \end{array}$$

with the equality holding if and only if $G\cong K_m\vee (K_k[\{k-t\}{O}_s,\left\{s\right\}{O}_{s+1}]).$

Proof. 

By Lemmas 1, 3, and Theorem 1, the extremal graph having the maximum reformulated first Zagreb index in ${\mathcal{G}}_{n,m,k}$ is the graph $K_m\vee (K_k[\{k-t\}{O}_s,\left\{s\right\}{O}_{s+1}]).$

Let $\widehat{G}=K_m\vee (K_k[\{k-t\}{O}_s,\left\{s\right\}{O}_{s+1}]).$

Then, we obtain that:

$$\begin{array}{cc}\hfill {\tilde{M}}_1(\widehat{G})& =\frac{m(m-1)}{2}{(2n-4)}^2+(k-t)ms{(2n-s-3)}^2\hfill \\ & +tm(s+1){(2n-s-4)}^2+\frac{t(t-1)}{2}{(s+1)}^2{(2n-2s-4)}^2\hfill \\ & +\frac{(k-t)(k-t-1)}{2}{s}^2{(2n-2s-2)}^2+t(k-t)s(s+1){(2n-2s-3)}^2\hfill \\ & =\frac{m(m-1)}{2}{(2n-4)}^2+m(n-m)(6n-3s-11)\hfill \\ & +2(n-m)(n-m-s){(n-s-1)}^2\hfill \\ & +t(s+1)[-6{(n-s-1)}^2+n+2m(5-2n+s)+(t-2)(s+1)].\square \hfill \end{array}$$

Theorem 3.

Let $\widehat{G}$ be a graph of order $n>2,$ and the join of a complete graph with order m and a complete k-partite graph with order $n-m$ in ${\mathcal{G}}_{n,m,k},$ i.e., $\widehat{G}=K_m\vee (K_k[O_{s_1},\cdots ,O_{s_k}]).$ If $s_1-1\ge s_2+1,$ by moving one vertex from the part of $O_{s_1}$ to the part of $O_{s_2},$ we get a new graph $\tilde{G}=K_m\vee (K_k[O_{s_1-1},O_{s_2+1},\cdots ,O_{s_k}]).$ Then, ${\Pi }_1^{*}(\tilde{G})>{\Pi }_1^{*}(\widehat{G}).$

Proof. 

By the definition of the multiplicative-sum Zagreb index ${\Pi }_1^{*}(G)$, it is easy to see that:

$${\Pi }_1^{*}(\widehat{G})={(2n-2)}^{\frac{m(m-1)}{2}}{\Pi }_{i=1}^k{(2n-s_i-1)}^{m{s}_i}{\Pi }_{1\le i<j\le k}{(2n-s_i-s_j)}^{s_i{s}_j}.$$

Hence,

$$\begin{array}{cc}\hfill \frac{{\Pi }_1^{*}(\tilde{G})}{{\Pi }_1^{*}(\widehat{G})}& ={(2n-s_1-s_2)}^{(s_1-s_2-1)}\frac{2n-s_2-2}{2n-s_1-1}{a}^{m(s_1-1)}{b}^{m{s}_2}\hfill \\ & {\Pi }_{i=3}^k{c}^{(s_1-1)s_i}{\Pi }_{i=3}^k{d}^{s_2{s}_i}{\Pi }_{i=3}^k{(\frac{2n-s_2-s_i-1}{2n-s_1-s_i})}^{s_i}\hfill \\ & >{(ab)}^{m{s}_2}{\Pi }_{i=3}^k{(cd)}^{s_2{s}_i},\hfill \end{array}$$

where $a=\frac{2n-s_1}{2n-s_1-1},b=\frac{2n-s_2-2}{2n-s_2-1},c=\frac{2n-s_1-s_i+1}{2n-s_1-s_i},d=\frac{2n-s_2-s_i-1}{2n-s_2-s_i}.$

By a simple calculation, we have:

$$(2n-s_1)(2n-s_2-2)-(2n-s_1-1)(2n-s_2-1)=s_1-s_2-1>0,$$

$$(2n-s_1-s_i+1)(2n-s_2-s_i-1)-(2n-s_1-s_i)(2n-s_2-s_i)=s_1-s_2-1>0.$$

Therefore, $\frac{{\Pi }_1^{*}(\tilde{G})}{{\Pi }_1^{*}(\widehat{G})}>1.$ □

Theorem 4.

Let G be an arbitrary graph in ${\mathcal{G}}_{n,m,k}.$ Then:

$$\begin{array}{cc}\hfill {\Pi }_1^{*}(G)& \le {(2n-2)}^{\frac{m(m-1)}{2}}{(2n-s-1)}^{ms(k-t)}{(2n-s-2)}^{m(s+1)t}\hfill \\ & {(2n-2s)}^{\frac{s^2(k-t)(k-t-1)}{2}}{(2n-2s-2)}^{\frac{{(s+1)}^{2}t(t-1)}{2}}{(2n-2s-1)}^{s(s+1)t(k-t)},\hfill \end{array}$$

with the equality holding if and only if $G\cong K_m\vee (K_k[\{k-t\}{O}_s,\left\{s\right\}{O}_{s+1}]).$

Proof. 

By Lemmas 1, 4, and Theorem 3, the extremal graph having the maximum multiplicative-sum Zagreb index in ${\mathcal{G}}_{n,m,k}$ should be the graph $K_m\vee (K_k[\{k-t\}{O}_s,\left\{s\right\}{O}_{s+1}]).$

Let $\widehat{G}=K_m\vee (K_k[\{k-t\}{O}_s,\left\{s\right\}{O}_{s+1}]).$ We get that,

$$\begin{array}{cc}\hfill {\Pi }_1^{*}(\widehat{G})& ={(2n-2)}^{\frac{m(m-1)}{2}}{(2n-s-1)}^{ms(k-t)}{(2n-s-2)}^{m(s+1)t}\hfill \\ & {(2n-2s)}^{\frac{s^2(k-t)(k-t-1)}{2}}{(2n-2s-2)}^{\frac{{(s+1)}^{2}t(t-1)}{2}}{(2n-2s-1)}^{s(s+1)t(k-t)}.\square \hfill \end{array}$$

3.2. The General Laplacian-Energy-Like Invariant and k-Partiteness

The general Laplacian-energy-like invariant (also called the sum of powers of the Laplacian eigenvalues) of a graph G is defined by Zhou [18] as:

$$S_{\alpha }(G)=\sum _{i=1}^{n-1}{\mu }_i^{\alpha },$$

where $\alpha$ is an arbitrary real number.

$S_{\alpha }(G)$ is the Laplacian-energy-like invariant [19], and the Laplacian energy [20] when $\alpha =\frac{1}{2}$ and $\alpha =2,$ respectively. For $\alpha =-1,$ $n{S}_{-1}(G)$ is equal to the Kirchhoff index [21], and $\alpha =1,$ $\frac{1}{2}{S}_1(G)$ is equal to the number of edges in $G.$ For some recent results on the general Laplacian-energy-like invariant, one can see [22,23,24,25].

Lemma 5.

[18] Let G be a graph with $u,v\in V(G).$ If $uv\in E(\overline{G}),$ then $S_{\alpha }(G)>S_{\alpha }(G+uv)$ for $\alpha <0,$ and $S_{\alpha }(G)<S_{\alpha }(G+uv)$ for $\alpha >0.$

Lemma 6.

[26] If ${\mu }_1\ge \cdots \ge {\mu }_{i-1}\ge {\mu }_i=0$ are the Laplacian eigenvalues of graph G and ${\mu }_1^{\prime }\ge \cdots \ge {\mu }_{j-1}^{\prime }\ge {\mu }_j^{\prime }=0$ are the Laplacian eigenvalues of graph $G^{\prime },$ then the Laplacian eigenvalues of $G\vee G^{\prime }$ are:

$$i+j,{\mu }_1+j,{\mu }_2+j,\cdots ,{\mu }_{i-1}+j,{\mu }_1^{\prime }+i,{\mu }_2^{\prime }+i,\cdots ,{\mu }_{j-1}^{\prime }+i,0.$$

It is well known that Laplacian eigenvalues of the complete graph $K_p$ are $0,p,\cdots ,p,$ and Laplacian eigenvalues of $O_p$ are $0,0,\cdots ,0.$ Then, the Laplacian eigenvalues of $K_{s_1,s_2}=O_{s_1}\vee O_{s_2}$ are $s_1+s_2,s_2,\cdots ,s_2,s_1,\cdots ,s_1,0,$ where the multiplicity of $s_2$ is $s_1-1$ and the multiplicity of $s_1$ is $s_2-1.$ The Laplacian eigenvalues of $K_{s_1,s_2,s_3}=K_{s_1,s_2}\vee O_{s_3}$ are $s_1+s_2+s_3,s_1+s_2+s_3,s_2+s_3,\cdots ,s_2+s_3,s_1+s_3,\cdots ,s_1+s_3,0,$ where the multiplicity of $s_2+s_3$ is $s_1-1$ and the multiplicity of $s_1+s_3$ is $s_2-1.$

By induction, we have that the Laplacian eigenvalues of $K_{s_1,\cdots ,s_k}$ are ${\displaystyle \sum _{i=1}^k}{s}_i,\cdots ,{\displaystyle \sum _{i=1}^k}{s}_i,{\displaystyle \sum _{i=1}^k}{s}_i-s_1,\cdots ,{\displaystyle \sum _{i=1}^k}{s}_i-s_1,\cdots ,{\displaystyle \sum _{i=1}^k}{s}_i-s_k,\cdots ,{\displaystyle \sum _{i=1}^k}{s}_i-s_k,0,$ where the multiplicity of ${\displaystyle \sum _{i=1}^k}{s}_i$ is $k-1$ and the multiplicity of ${\displaystyle \sum _{i=1}^k}{s}_i-s_j$ is $s_j-1,$ for $1\le j\le k.$

From Lemma 6 and the above analysis, we obtain the following lemma.

Lemma 7.

Let $\widehat{G}$ be a graph of order $n,$ and the join of a complete graph with order m and a complete k-partite graph with order $n-m$ i.e., $\widehat{G}=K_m\vee (K_k[O_{s_1},\cdots ,O_{s_k}]).$ Then, the Laplacian eigenvalues of $\widehat{G}$ are $n,\cdots ,n,n-s_1,\cdots ,n-s_1,\cdots ,n-s_k,\cdots ,n-s_k,0,$ where the multiplicity of n is $m+k-1$ and the multiplicity of $n-s_j$ is $s_j-1,$ for $1\le j\le k.$

Theorem 5.

Let $\widehat{G}$ be a graph of order $n>2,$ and the join of a complete graph with order m and a complete k-partite graph with order $n-m$ in ${\mathcal{G}}_{n,m,k},$ i.e., $\widehat{G}=K_m\vee (K_k[O_{s_1},\cdots ,O_{s_k}]).$ If $s_1-1\ge s_2+1,$ by moving one vertex from the part of $O_{s_1}$ to the part of $O_{s_2},$ we get a new graph $\tilde{G}=K_m\vee (K_k[O_{s_1-1},O_{s_2+1},\cdots ,O_{s_k}]).$ Then, $S_{\alpha }(\tilde{G})<S_{\alpha }(\widehat{G})$ for $\alpha <0,$ and $S_{\alpha }(\tilde{G})>S_{\alpha }(\widehat{G})$ for $0<\alpha <1.$

Proof. 

By the definition of the general Laplacian-energy-like invariant $S_{\alpha }(G)$ and Lemma 7, we conclude that:

$$S_{\alpha }(\widehat{G})=(m+k-1)n^{\alpha }+{\displaystyle \sum _{i=1}^k}(s_i-1){(n-s_i)}^{\alpha }.$$

Therefore:

$$\begin{array}{cc}\hfill S_{\alpha }(\tilde{G})-S_{\alpha }(\widehat{G})& =(s_1-2){(n-s_1+1)}^{\alpha }+s_2{(n-s_2-1)}^{\alpha }\hfill \\ & -(s_1-1){(n-s_1)}^{\alpha }-(s_2-1){(n-s_2)}^{\alpha }\hfill \\ & =(s_1-2)[{(n-s_1+1)}^{\alpha }-{(n-s_1)}^{\alpha }]\hfill \\ & +(s_2-1)[{(n-s_2-1)}^{\alpha }-{(n-s_2)}^{\alpha }]+{(n-s_2-1)}^{\alpha }-{(n-s_1)}^{\alpha }.\hfill \end{array}$$

For $\alpha <0,$ we have:

$$\begin{array}{cc}\hfill S_{\alpha }(\tilde{G})-S_{\alpha }(\widehat{G})& <(s_1-2)[{(n-s_1+1)}^{\alpha }-{(n-s_1)}^{\alpha }]+(s_2-1)[{(n-s_2-1)}^{\alpha }-{(n-s_2)}^{\alpha }]\hfill \\ & <(s_1-2)[{(n-s_1+1)}^{\alpha }-{(n-s_1)}^{\alpha }+{(n-s_2-1)}^{\alpha }-{(n-s_2)}^{\alpha }]\hfill \\ & =(s_1-2)[f(n-s_1)-f(n-s_2-1)],\hfill \end{array}$$

where $f(x)={(x+1)}^{\alpha }-x^{\alpha },$ $f^{\prime }(x)=\alpha {(x+1)}^{\alpha -1}-\alpha x^{\alpha -1}>0.$

Then, $f(n-s_1)<f(n-s_2-1),$ and $S_{\alpha }(\tilde{G})<S_{\alpha }(\widehat{G}).$

For $0<\alpha <1,$ we have:

$$\begin{array}{cc}\hfill S_{\alpha }(\tilde{G})-S_{\alpha }(\widehat{G})& >(s_1-2)[{(n-s_1+1)}^{\alpha }-{(n-s_1)}^{\alpha }]+(s_2-1)[{(n-s_2-1)}^{\alpha }-{(n-s_2)}^{\alpha }]\hfill \\ & >(s_2-1)[{(n-s_1+1)}^{\alpha }-{(n-s_1)}^{\alpha }+{(n-s_2-1)}^{\alpha }-{(n-s_2)}^{\alpha }]\hfill \\ & =(s_2-1)[f(n-s_1)-f(n-s_2-1)],\hfill \end{array}$$

where $f(x)={(x+1)}^{\alpha }-x^{\alpha },$ $f^{\prime }(x)=\alpha {(x+1)}^{\alpha -1}-\alpha x^{\alpha -1}<0.$

Then, $f(n-s_1)>f(n-s_2-1),$ and $S_{\alpha }(\tilde{G})>S_{\alpha }(\widehat{G}).$ □

Theorem 6.

Let G be an arbitrary graph in ${\mathcal{G}}_{n,m,k}.$ Then, for $\alpha <0,S_{\alpha }(G)\ge (m+k-1)n^{\alpha }+(k-t)(s-1){(n-s)}^{\alpha }+ts{(n-s-1)}^{\alpha },$ for $0<\alpha <1,S_{\alpha }(G)\le (m+k-1)n^{\alpha }+(k-t)(s-1){(n-s)}^{\alpha }+ts{(n-s-1)}^{\alpha },$ with the equality holding if and only if $G\cong K_m\vee (K_k[\{k-t\}{O}_s,\left\{s\right\}{O}_{s+1}]).$

Proof. 

By Lemmas 1, 2, and Theorem 5, the extremal graph having the extremal value of the general Laplacian-energy-like invariant in ${\mathcal{G}}_{n,m,k}$ should be the graph $K_m\vee (K_k[\{k-t\}{O}_s,\left\{s\right\}{O}_{s+1}]).$

Let $\widehat{G}=K_m\vee (K_k[\{k-t\}{O}_s,\left\{s\right\}{O}_{s+1}]),$ then we can verify that $S_{\alpha }(\widehat{G})=(m+k-1)n^{\alpha }+(k-t)(s-1){(n-s)}^{\alpha }+ts{(n-s-1)}^{\alpha }.$  □

3.3. The General Zeroth-Order Randić Index and k-Partiteness

The general zeroth-order Randić index is introduced by Li [27] as:

$${}^0{R}_{\alpha }(G)=\sum _{u\in V(G)}{(d_G(u))}^{\alpha },$$

where $\alpha$ is a non-zero real number.

${}^0{R}_{\alpha }(G)$ is the inverse degree [28], the zeroth-Randić index [29], the first Zagreb index [30], and the forgotten index [31] when $\alpha =-1,\alpha =-\frac{1}{2},\alpha =2$, and $\alpha =3,$ respectively. For some recent results on the general zeroth-order Randić index, one can see [32,33,34].

Lemma 8.

Let G be a graph with $u,v\in V(G).$ If $uv\in E(\overline{G}),$ then ${}^0{R}_{\alpha }(G)>$${}^0{R}_{\alpha }(G+uv)$ for $\alpha <0,$ and ${}^0{R}_{\alpha }(G)<$${}^0{R}_{\alpha }(G+uv)$ for $\alpha >0.$

Theorem 7.

Let $\widehat{G}$ be a graph of order $n>2,$ and the join of a complete graph with order m and a complete k-partite graph with order $n-m$ in ${\mathcal{G}}_{n,m,k},$ i.e., $\widehat{G}=K_m\vee (K_k[O_{s_1},\cdots ,O_{s_k}]).$ If $s_1-1\ge s_2+1,$ by moving one vertex from the part of $O_{s_1}$ to the part of $O_{s_2},$ we get a new graph $\tilde{G}=K_m\vee (K_k[O_{s_1-1},O_{s_2+1},\cdots ,O_{s_k}]).$ Then, ${}^0{R}_{\alpha }(\tilde{G})<$${}^0{R}_{\alpha }(\widehat{G})$ for $\alpha <0,$ and ${}^0{R}_{\alpha }(\tilde{G})>$${}^0{R}_{\alpha }(\widehat{G})$ for $0<\alpha <1.$

Proof. 

By the definition of the general zeroth-order Randić index ${}^0{R}_{\alpha }(G),$ we have:

$${}^0{R}_{\alpha }(\widehat{G})=m{(n-1)}^{\alpha }+\sum _{i=1}^k{s}_i{(n-s_i)}^{\alpha }$$

Then,

$$\begin{array}{cc}\hfill {}^0{R}_{\alpha }(\tilde{G}){-}^0{R}_{\alpha }(\widehat{G})& =(s_1-1){(n-s_1+1)}^{\alpha }-s_1{(n-s_1)}^{\alpha }\hfill \\ & +(s_2+1){(n-s_2-1)}^{\alpha }-s_2{(n-s_2)}^{\alpha }\hfill \\ & ={(n-s_2-1)}^{\alpha }-{(n-s_1)}^{\alpha }\hfill \\ & +(s_1-1)[{(n-s_1+1)}^{\alpha }-{(n-s_1)}^{\alpha }]+s_2[{(n-s_2-1)}^{\alpha }-{(n-s_2)}^{\alpha }].\hfill \end{array}$$

For $\alpha <0$, we have:

$$\begin{array}{cc}\hfill {}^0{R}_{\alpha }(\tilde{G}){-}^0{R}_{\alpha }(\widehat{G})& <(s_1-1)[{(n-s_1+1)}^{\alpha }-{(n-s_1)}^{\alpha }+{(n-s_2-1)}^{\alpha }-{(n-s_2)}^{\alpha }]\hfill \\ & =(s_1-1)[f(n-s_1)-f(n-s_2-1)],\hfill \end{array}$$

where $f(x)={(x+1)}^{\alpha }-x^{\alpha },$ $f^{\prime }(x)=\alpha {(x+1)}^{\alpha -1}-\alpha x^{\alpha -1}>0.$ Then, $f(n-s_1)<f(n-s_2-1)$, ${}^0{R}_{\alpha }(\tilde{G}){<}^0{R}_{\alpha }(\widehat{G}).$

For $0<\alpha <1$, we have:

$$\begin{array}{cc}\hfill {}^0{R}_{\alpha }(\tilde{G}){-}^0{R}_{\alpha }(\widehat{G})& >s_2[{(n-s_1+1)}^{\alpha }-{(n-s_1)}^{\alpha }+{(n-s_2-1)}^{\alpha }-{(n-s_2)}^{\alpha }]\hfill \\ & =s_2[f(n-s_1)-f(n-s_2-1)],\hfill \end{array}$$

where $f(x)={(x+1)}^{\alpha }-x^{\alpha },$ $f^{\prime }(x)=\alpha {(x+1)}^{\alpha -1}-\alpha x^{\alpha -1}<0.$

Then, $f(n-s_1)>f(n-s_2-1),$ $R_{\alpha }(\tilde{G})>R_{\alpha }(\widehat{G}).$  □

Theorem 8.

Let G be an arbitrary graph in ${\mathcal{G}}_{n,m,k}.$ Then, for $\alpha <0,$ ${}^0{R}_{\alpha }(G)\ge m{(n-1)}^{\alpha }+(k-t)s{(n-s)}^{\alpha }+t(s+1){(n-s-1)}^{\alpha },$ for $0<\alpha <1,$ ${}^0{R}_{\alpha }(G)\le m{(n-1)}^{\alpha }+(k-t)s{(n-s)}^{\alpha }+t(s+1){(n-s-1)}^{\alpha },$ with the equality holding if and only if $G\cong K_m\vee (K_k[\{k-t\}{O}_s,\left\{s\right\}{O}_{s+1}]).$

Proof. 

By Lemma 8 and Theorem 7, in view of Lemmas 1 and 2, the extremal graph having the extremal value of the general zeroth-order Randić index in ${\mathcal{G}}_{n,m,k}$ should be the graph $K_m\vee (K_k[\{k-t\}{O}_s,\left\{s\right\}{O}_{s+1}]).$

Let $\widehat{G}=K_m\vee (K_k[\{k-t\}{O}_s,\left\{s\right\}{O}_{s+1}]).$ By a simple calculation, we have ${}^0{R}_{\alpha }(\widehat{G})=m{(n-1)}^{\alpha }+(k-t)s{(n-s)}^{\alpha }+t(s+1){(n-s-1)}^{\alpha }.$  □

3.4. The Modified-Wiener Index and k-Partiteness

The modified-Wiener index is defined by Gutman [35] as:

$$W_{\lambda }(G)=\sum _{u,v\in V(G)}{d}_G^{\lambda }(u,v),$$

where $\lambda$ is a non-zero real number.

Lemma 9.

Let G be a graph with $u,v\in V(G).$ If $uv\in E(\overline{G}),$ then $W_{\lambda }(G)<W_{\lambda }(G+uv)$ for $\lambda <0,$ and $W_{\lambda }(G)>W_{\lambda }(G+uv)$ for $\lambda >0.$

Theorem 9.

Let $\widehat{G}$ be a graph of order $n>2,$ and the join of a complete graph with order m and a complete k-partite graph with order $n-m$ in ${\mathcal{G}}_{n,m,k},$ i.e., $\widehat{G}=K_m\vee (K_k[O_{s_1},\cdots ,O_{s_k}]).$ If $s_1-1\ge s_2+1,$ by moving one vertex from the part of $O_{s_1}$ to the part of $O_{s_2},$ we get a new graph $\tilde{G}=K_m\vee (K_k[O_{s_1-1},O_{s_2+1},\cdots ,O_{s_k}]).$ Then, $W_{\lambda }(\tilde{G})>W_{\lambda }(\widehat{G})$ for $\lambda <0,$ and $W_{\lambda }(\tilde{G})<W_{\lambda }(\widehat{G})$ for $\lambda >0.$

Proof. 

By the definition of the modified-Wiener index $W_{\lambda }(G),$ we have the following result.

$$W_{\lambda }(\widehat{G})=\frac{m(m-1)}{2}+{\displaystyle \sum _{i=1}^k}\frac{s_i(s_i-1)}{2}{2}^{\lambda }+{\displaystyle \sum _{i=1}^k}m{s}_i+{\displaystyle \sum _{1\le i<j\le k}}{s}_i{s}_j$$

Then,

$$\begin{array}{cc}\hfill W_{\lambda }(\tilde{G})-W_{\lambda }(\widehat{G})& =\frac{(s_1-1)(s_1-2)}{2}{2}^{\lambda }+\frac{(s_2+1)s_2}{2}{2}^{\lambda }+m(s_1-1)\hfill \\ & +m(s_2+1)+(s_1-1)(s_2+1)+{\displaystyle \sum _{i=3}^k}(s_1-1)s_i+{\displaystyle \sum _{i=3}^k}(s_2+1)s_i\hfill \\ & -\frac{s_1(s_1-1)}{2}{2}^{\lambda }-\frac{s_2(s_2-1)}{2}{2}^{\lambda }-m{s}_1-m{s}_2-s_1{s}_2-{\displaystyle \sum _{i=3}^k}{s}_1{s}_i-{\displaystyle \sum _{i=3}^k}{s}_2{s}_i\hfill \\ & =(s_1-s_2-1)(1-2^{\lambda }).\hfill \end{array}$$

For $\lambda >0,$ we have $W_{\lambda }(\tilde{G})<W_{\lambda }(\widehat{G}).$ For $\lambda <0,$ we have $W_{\lambda }(\tilde{G})>W_{\lambda }(\widehat{G}).$  □

Theorem 10.

Let G be an arbitrary graph in ${\mathcal{G}}_{n,m,k}.$ Then, for $\alpha <0,$ $W_{\lambda }(G)\le \frac{1}{2}[m(m-1)+(n-m)(n+m-s)-(s+1)t+s(n-m+t-k)2^{\lambda }],$ for $\alpha >0,$ $W_{\lambda }(G)\ge \frac{1}{2}[m(m-1)+(n-m)(n+m-s)-(s+1)t+s(n-m+t-k)2^{\lambda }],$ with the equality holding if and only if $G\cong K_m\vee (K_k[\{k-t\}{O}_s,\left\{s\right\}{O}_{s+1}]).$

Proof. 

By Lemma 9 and Theorem 9, in view of Lemmas 1 and 2, the extremal graph having the extremal value of the modified-Wiener index in ${\mathcal{G}}_{n,m,k}$ should be the graph $K_m\vee (K_k[\{k-t\}{O}_s,\left\{s\right\}{O}_{s+1}]).$

Let $\widehat{G}=K_m\vee (K_k[\{k-t\}{O}_s,\left\{s\right\}{O}_{s+1}]).$ Consequently, we have that:

$$\begin{array}{cc}\hfill W_{\lambda }(\widehat{G})& =\frac{m(m-1)}{2}+(k-t)\frac{s(s-1)}{2}{2}^{\lambda }+t\frac{s(s+1)}{2}{2}^{\lambda }+tm(s+1)+(k-t)ms\hfill \\ & =\frac{1}{2}[m(m-1)+(n-m)(n+m-s)-(s+1)t+s(n-m+t-k)2^{\lambda }].\square \hfill \end{array}$$

4. Conclusions

In this paper, we consider connected graphs of order n with vertex k-partiteness not more than m and characterize some extremal monotonic graph invariants such as the reformulated first Zagreb index, the multiplicative-sum Zagreb index, the general Laplacian-energy-like invariant, the general zeroth-order Randić index, and the modified-Wiener index among these graphs, and we investigate the corresponding extremal graphs of these invariants.