On the Sum and Spread of Reciprocal Distance Laplacian Eigenvalues of Graphs in Terms of Harary Index

: The reciprocal distance Laplacian matrix of a connected graph G is deﬁned as RD L ( G ) = RT ( G ) − RD ( G ) , where RT ( G ) is the diagonal matrix of reciprocal distance degrees and RD ( G ) is the Harary matrix. Clearly, RD L ( G ) is a real symmetric matrix, and we denote its eigenvalues as λ 1 ( RD L ( G )) ≥ λ 2 ( RD L ( G )) ≥ . . . ≥ λ n ( RD L ( G )) . The largest eigenvalue λ 1 ( RD L ( G )) of RD L ( G ) , denoted by λ ( G ) , is called the reciprocal distance Laplacian spectral radius. In this paper, we obtain several upper bounds for the sum of k largest reciprocal distance Laplacian eigenvalues of G in terms of various graph parameters, such as order n , maximum reciprocal distance degree RT max , minimum reciprocal distance degree RT min , and Harary index H ( G ) of G . We determine the extremal cases corresponding to these bounds. As a consequence, we obtain the upper bounds for reciprocal distance Laplacian spectral radius λ ( G ) in terms of the parameters as mentioned above and characterize the extremal cases. Moreover, we attain several upper and lower bounds for reciprocal distance Laplacian spread RDLS ( G ) = λ 1 ( RD L ( G )) − λ n − 1 ( RD L ( G )) in terms of various graph parameters. We determine the extremal graphs in many cases.


Introduction
Let G = (V(G), E(G)) be a connected simple graph with vertex set V(G) and edge set E(G). The order and size of G are |V(G)| = n and |E(G)| = m, respectively. The degree of a vertex v, denoted by d(v), is the number of edges incident on the vertex v. Other undefined notations and terminology can be seen in [1].
The adjacency matrix A(G) = (a ij ) of G is an n × n matrix in which (i, j)-entry is equal to 1 if there is an edge between vertex v i and vertex v j and equal to 0 otherwise. Let Deg(G) = diag(d(v 1 ), d(v 2 ), . . . , d(v n )) be the diagonal matrix of vertex degrees d G (v i ), i = 1, 2, . . . , n. The positive semi-definite matrix L(G) = Deg(G) − A(G) is the Laplacian matrix of G. The eigenvalues of L(G) are called the Laplacian eigenvalues of G, which are denoted by µ 1 (G), µ 2 (G), . . . , µ n (G) and are ordered as µ 1 (G) ≥ µ 2 (G) ≥ . . . ≥ µ n (G).
In G, the distance between two vertices v i , v j ∈ V(G), denoted by d(v i , v j ), is defined as the length of a shortest path between v i and v j . The diameter of G, denoted by d(G), is the length of a longest path among the distance between every two vertices of G. The distance matrix of G is denoted by D(G) and is defined as The transmission Tr G (v i ) (or briefly, Tr i if graph G is understood) of a vertex v i is defined as the sum of the distances from v i to all other vertices in G: Let Tr(G) = diag(Tr 1 , Tr 2 , . . . , Tr n ) be the diagonal matrix of vertex transmissions of G. In [2], Aouchiche and Hansen introduced the Laplacian for the distance matrix of a connected graph. The matrix D L (G) = Tr(G) − D(G) is called the distance Laplacian matrix of G.
The Harary matrix of graph G, which is also called as the reciprocal distance matrix, denoted by RD(G), is an n by n matrix defined as [3] The reciprocal distance degree of a vertex v i , denoted by RTr G (v i ) (or shortly RT i ), is given by The Harary index of a graph G, denoted by H(G), is defined in [3] as Clearly, To see more work performed on the Harary matrix, we refer the reader to [4][5][6] and the references therein.
In [7], the authors defined the reciprocal distance Laplacian matrix as RD L (G) = RT(G) − RD(G). Since RD L (G) is a real symmetric matrix, we can denote by the eigenvalues of RD L (G). Since RL(G) is a positive semidefinite matrix, we will denote the spectral radius of RD L (G) by λ(G) = λ 1 (RD L (G)), called the reciprocal distance Laplacian spectral radius. More work on the matrix RD L (G) can be seen in [8][9][10][11].
µ i (G) be the sum of the k largest Laplacian eigenvalues of G. Several researchers have been investigating the parameter S k (G) because of its importance in dealing with many problems in the theory, for instance, Brouwer's conjecture and Laplacian energy. We refer the reader to [12][13][14][15] for recent work conducted on the graph invariant S k (G). Motivated by the parameter S k (G) of the Laplacian matrix, we define the following. For 1 ≤ k ≤ n − 1, let RU k (G) denote the sum of the k largest reciprocal distance Laplacian eigenvalues: The Laplacian spread of a graph G is defined as LS(G) = µ 1 (G) − µ n−1 (G), where µ 1 (G) and µ n−1 (G) are, respectively, the largest and second smallest Laplacian eigenvalues of G. More on LS(G) can be found in [16][17][18].
Since 0 is always a simple eigenvalue of the the reciprocal distance Laplacian matrix, we define the reciprocal distance Laplacian spread of a connected graph G such as the Laplacian spread as where λ 1 (RD L (G) and λ n−1 (RD L (G) are, respectively, the largest and second smallest reciprocal distance Laplacian eigenvalues of G.
The rest of the paper is organized as follows. In Section 2, we obtain several upper bounds for the graph invariant RU k (G) in terms of various graph parameters, such as order n, maximum reciprocal distance degree RT max , minimum reciprocal distance degree RT min , and Harary index H(G) of G. We characterize the extremal cases corresponding to these bounds as well. As a consequence, we obtain the upper bounds for reciprocal distance Laplacian spectral radius λ(G) in terms of the same parameters as mentioned above and determine the extremal graphs. In Section 3, we find several upper and lower bounds for reciprocal distance Laplacian spread RDLS(G) in terms of various graph parameters. We characterize the extremal graphs in many cases.

Sum of the Reciprocal Distance Laplacian Eigenvalues
We begin with the following lemma.

Lemma 1. [7]
For any connected graph G, 0 is a simple eigenvalue of RD L (G). Proposition 1. Let G be a connected graph with n vertices. Then, Proof. (i) Using the fact that the sum of eigenvalues is equal to the trace of a matrix and using Lemma 1, we have The proof for (ii) follows arguments similar to those for (i).

Proposition 2.
Let G be a connected graph with n vertices. Then, with equality if and only if G ∼ = K n .
which proves the required inequality. Assume that the equality holds in the above inequality. Then, each d ij = 1, whenever For the converse, we observe that the equality holds for K n .

Lemma 2. [19]
Let x = (x 1 , x 2 . . . , x n ) and y = (y 1 , y 2 . . . , y n ) be n-tuples of real numbers Let RT max = max{RT i : i = 1, 2, . . . , n} and RT min = min{RT i : i = 1, 2, . . . , n} be the maximum reciprocal distance degree and the minimum reciprocal distance degree of the graph G, respectively. Using Lemma 2, we obtain an upper bound for the graph invariant i in terms of Harary index H(G) and order n of graph G.
Lemma 3. Let G be a connected graph with n vertices. Then, Moreover, inequality is sharp, as shown by all of the reciprocal distance degree regular graphs.
Proof. In Lemma 2, we take Assume that G is k-reciprocal distance degree regular. Then, the left hand side of Inequality 2 becomes nk 2 and the right hand side becomes 4H 2 (G) n = k 2 n 2 n = nk 2 , which shows that the equality holds for reciprocal distance degree regular graphs. Now, we obtain an upper bound for the graph invariant RU k (G) in terms of various graph parameters.

Theorem 1. Let G be a connected graph with n vertices and Harary index H(G). For
with equality if and only if G ∼ = K n . For k = n − 1, equality always holds.
Proof. Let RU k (G) = R k . For 1 ≤ k ≤ n − 2, using Proposition 1 and Cauchy-Schwarz inequality, we have Further simplification gives Therefore, Using Proposition 2, Lemma 3 in Inequality 3 and after simplifications, we have which proves the required inequality. Assume that equality holds in the above inequality. Then, equality must hold simultaneously in the Cauchy-Schwarz inequality, Proposition 2, and Lemma 3, which is only possible if G ∼ = K n .
Conversely, if G ∼ = K n , then the left hand side of the main equality is equal to kn. After performing the necessary calculations, the right-hand side reduces to 2H(K n )k n−1 + 0 = n(n−1)k n−1 = kn, which proves the converse part.
Using the fact that traces of a matrix are equal to the sum of its eigenvalues and noting that 2H(G) = R n−1 , we easily see that equality always holds when k = n − 1 in the main inequality.
Taking k = 1 in Theorem 1, we obtain an upper bound for the reciprocal distance Laplacian spectral radius λ(G) of a connected graph G in terms of the maximum reciprocal distance degree RT max , minimum reciprocal distance degree RT min , order n, and Harary index H(G).

Theorem 2. Let G be a connected graph with n vertices and Harary index H(G). Then,
with equality if and only if G ∼ = K n .

Lemma 4.
[20] Let [n] = {1, 2, . . . , n} be the canonical n-element set, and let [n] (2) denote the set of two-element subsets of [n], that is, the edge set of K n . To each entry {i, j} = ij in [n] (2) , associate a real variable z ij ; then, for n ≥ 2, and for all z ij s, we have Now, we obtain an upper bound for the sum of the squares of the reciprocal distance degrees in terms of the Harary index H(G) and the order n of the graph G.

Lemma 5.
Let G be a connected graph with order n and having diameter d. Then with equality if and only if G ∼ = K n .

Proof. Put 1 d ij
for z ij in Lemma 4 and observe that with each 1 Simplifying further, we have proving the required inequality. Assume that the equality holds in the above inequality. Then, each 1 Conversely, assume that G ∼ = K n . Then, we observe that H(G) = n(n−1) 2 and ∑ i RT 2 i = n(n − 1) 2 . Substituting these values in the main inequality, we see that the equality holds.
A similar argument has been adopted in studying Estrada index [21]. Using Lemma 5, we have the following upper bound for the graph invariant RU k (G) in terms of order n and Haray index H(G). This bound seems to be more elegant than the bound in Theorem 1 since it uses relatively less number of parameters.

Theorem 3. Let G be a connected graph with n vertices and Harary index H(G). For
with equality if and only if G ∼ = K n . For k = n − 1, equality always holds.
Proof. We proceed exactly as in Theorem 1 upto Inequality 3, then use Lemma 5 and Proposition 2, and obtain Simplifying further, we have which is the inequality in the statement of theorem. The remaining part of the proof follows by using similar arguments as in Theorem 1.
As a consequence of Theorem 3, we obtain the following upper bound for reciprocal distance Laplacian spectral radius λ(G) of a connected graph G in terms of the Harary index H(G) and order n of the graph G.

Theorem 4. Let G be a connected graph with n vertices and Harary index H(G). Then,
with equality if and only if G ∼ = K n .

Reciprocal Distance Laplacian Spread
We begin this section with the following observations. Lemma 6. [7] Let G be a connected graph on n vertices with diameter d = 2. Then, and µ i (G) both have the same multiplicity for i = 1, 2, . . . , n.
A special case of the well-known min-max theorem is the following result.

Lemma 7. [22]
If M is a symmetric n × n matrix with eigenvalues δ 1 ≥ δ 2 ≥ . . . ≥ δ n , then for any x ∈ R n (x = 0), Equality holds if and only if x is an eigenvector of M corresponding to the largest eigenvalue δ 1 .

Lemma 8. [7]
If G is a graph on n > 2 vertices, then the multiplicity of λ(G) is always less than or equal to n − 1 with equality if and only if G is the complete graph.
Theorem 5. Let G be a connected graph with n vertices having Harary index H(G). Then, Equality holds if and only if G ∼ = K n .
Since the reciprocal distance Laplacian spectrum of K n is {n (n−1) , 0}, therefore, equality holds in Inequality 4 if and only if G ∼ = K n .
If we use Theorem 2 instead of Theorem 4 in the above result, we have the following theorem: Theorem 6. Let G be a connected graph with n vertices having Wiener index W(G). Then, Equality holds if and only if G ∼ = K n .
The following lemma gives the lower bound for the reciprocal distance Laplacian spectral radius in terms of order n, diameter d, and S d .

Lemma 10.
Let G be a connected graph on n vertices having diameter d. Then, Consider the n-vector x = (x 1 , x 2 , . . . , x d−1 , x d , x d+1 , . . . , x n ) T defined by It can be easily seen that Similarly, On substituting inequalities 7, 8 in Inequality 6, we have Theorem 7. Let G be a connected graph with order n having diameter d. Then, Proof. Note that . From this equality, we see that λ n−1 (RD L (G)) ≤ 2H(G) n − 1 .
Author Contributions: Investigation, S.K., S.P. and Y.S.; writing-original draft preparation, S.K., S.P. and Y.S.; writing-review and editing, S.K., S.P. and Y.S. All authors have read and agreed to the published version of the manuscript.
Funding: The research of S. Pirzada is supported by the SERB-DST research project number CRG/2020 /000109.