On Harary energy and Reciprocal distance Laplacian energies1

Let G be an graph simple, undirected, connected and unweighted graphs. The Reciprocal distance energy of a graph G is equal to the sum of the absolute values of the reciprocal distance eigenvalues. In this work, we find a lower bound for the Harary energy, reciprocal distance Laplacian energy and reciprocal distance signless Laplacian energy of a graph. Moreover, we find relationship between the Harary energy and Reciprocal distance Laplacian energies.


Introduction and preliminaries
Let G = (V, E) be a connected simple undirected graph with vertex set V and edge set E. The distance d (v i , v j ) = d i,j between the vertices v i and v j of G is equal to the length of (number of edges in) the shortest path that connects v i and v j . The Harary matrix of graph G, which is also called as the Reciprocal Distance matrix, is an n × n matrix defined as [18] Henceforth, we consider i = j for d (v i , v j ). The transmission of a vertex v, denoted by T r G (v) and defined by T r G (v) = u∈V (G) d (u, v).
Definition 1 Let G be a simple connected graph with V (G) = {v 1 , v 2 , . . . , v n }. The reciprocal distance degree of a vertex v, denoted by RT r G (v), is given by .
Let RT (G) be the n × n diagonal matrix defined by RT i,i = RT r G (v i ).
Sometimes we use the notation RT i instead of RT r G (v i ) for i = 1, . . . , n.
Definition 2 A connected graph G is called a k-reciprocal distance degree regular graph if RT i = k for all i ∈ {1, 2, . . . , n}. 1 This work is partially supported by MINEDUC-UA project ANT20992 The Harary index of a graph G, denoted by H (G), is defined in [18] as Clearly, We recall that the spectral radius of a matrix A is ρ(A) = max In 2018, Bapat and Panda [3], defined the Reciprocal distance Laplacian matrix as RL (G) = RT (G) − RD (G) and in 2019, Alhevaz et al. [1], defined the Reciprocal distance signless Laplacian matrix as RQ(G) = RT (G) + RD(G).
We observe that RD (G), RL (G) and RQ (G) are real symmetric matrices, then we can write the eigenvalues in decreasing order, this is and Moreover, RD(G) and RQ(G) are irreducible nonnegative matrices, ρ(RD(G)) and ρ(RQ(G)) are a simple eigenvalues of RD(G) and RQ(G), respectively. In [14] the authors obtained upper bounds and lower bounds for the spectral radius of Reciprocal distance, Reciprocal distance Laplacian and Reciprocal distance signless Laplacian matrices of a graph, and they characterized the graphs that attained some of the bounds mentioned. The energy of a graph is a concept originating from theoretical chemistry and in 1978 Ivan Gutman defined the energy of a graph through the eigenvalues of the adjacency matrix of graph [8]. In particular: let A(G) be the adjacency matrix of a graph G of order n, then the energy of About the energy of graph, we highlight two classic bounds to a graph on n vertices and m edges: given by McClelland in [13] and given by Koolen and Moulton in [12], respectively. The energy of a graph has been extensively studied over the years. Although, in some cases it has been possible to determine the energy for certain graphs, but in general it is not possible to determine it exactly. Examples of some works about energy on special graphs, such as bipartite graphs, cyclic and acyclic graphs, regular graph, line graphs, trees with a given diameter [2,9,10,11,19,20,21,22,23].
The concept energy of a graph has been extended to different matrices associated with a graph: let M be a matrix associated with a graph G, then the energy of matrix M is defined in [4] by where λ(M (G)) is the average of eigenvalues of M . Several authors have defined the energy of different matrices coinciding or using the definition given above.
Definition 3 [7] The Harary energy of a graph G, denoted by E H (G), is defined as The Harary energy is also called Reciprocal distance energy.

Definition 4 [1]
Let G be a connected graph of order n. Then the Reciprocal distance signless Laplacian energy of G, denoted by E RQ (G) is defined as Definition 5 [15] Let G be a connected graph of order n. Then the Reciprocal distance Laplacian In [15], we found bounds on the Reciprocal Distance Energy, Reciprocal Distance Laplacian Energy and Reciprocal Distance signless Laplacian Energy, and we characterized the graphs that attained some of those bounds. Now, in this work we find a new bounds for the Harary energy and reciprocal distance signless Laplacian energy of a graph, and we obtain relationship between the Harary energy and Reciprocal distance Laplacian energies.
On the other hand, in [  To finish this section, we recall that the Frobenius norm of an n × n matrix M = (m i,j ) is the eigenvalues of M . In particular, this property is satisfied to RD(G), RL(G) and RQ(G) matrices.

Lower bounds for the Harary energy and Reciprocal distance Laplacian energies
In this section we obtain lower bounds for Harary energy and Reciprocal distance Laplacian energies.
We recall that RD(G) is a normal matrix. Therefore Theorem 2 Let G be a connected graph with n ≥ 2 vertices. Let λ p , λ r be RQ-eigenvalues

From binomial expansion, we obtain
Similarly, for the matrix RL(G)) we obtain the following result. Then Example 1 We consider the graphs G 1 , G 2 , G 3 given by Figure 1, G 4 is the star on 7 vertices, G 5 is the path on 7 vertices and G 6 is the cycle on 7 vertices, denoted by S 7 , P 7 and C 7 respectively. Figure 1.
The following tables show the bounds obtained for the above graphs.  Table 3. Lower bounds for the Reciprocal distance Laplacian energy.

Relationship between the Harary energy and Reciprocal distance Laplacian energies
In this section we find relationship between the Harary energy and Reciprocal distance Laplacian energies: first we study two particular cases, when G is a regular graph of diameter 2 and when the graph G is reciprocal distance regular; and finally we give general relations between the Harary energy and Reciprocal distance Laplacian energies.
Theorem 4 [5] Let G be an r-regular graph of diameter 2 on n vertices, its adjacency spectrum be spec(A(G)) = {r, λ 2 , . . . , λ n }. Then the RD-spectrum of G is Theorem 5 Let G be an r-regular graph on n vertices such that diam(G) = 2. Let r, λ 1 , . . . , λ n be the adjacency eigenvalues of G. Then eigenvalues of the reciprocal distance signless Laplacian matrix of G are n + r − 1 and where J n denote the all-1 matrix of order n and I n denote the identity matrix of order n.
Proof. If G is k-reciprocal distance regular graph then RQ(G) = kI n + RD(G) and RL(G) = kI n − RD(G).

Thus
λ i (RQ(G)) = k + λ i (RD(G)) and λ i (RQ(G)) = k − λ i (RD(G)). Then Therefore, the result is obtained. Let X, Y and Z be matrices, such that X + Y = Z. The Ky Fan theorem [6] establishes an inequality between the sum of the singular values of Z and the sum of the sum of the singular values of X and Y .
Lemma 2 [6] Let X, Y and Z be square matrices of order n, such that Z = X + Y . Then Equality holds if and only if there exists an orthogonal matriz P , such that P X and P Y are both positive semidefinite matrix.
Theorem 9 Let G be a graph of order n. Then Proof. We observe that RQ(G) − RL(G) = 2RD(G), then By the Lemma 2 we get the left inequality. Now, apply the same Lemma 2 on we get the right inequality.
Theorem 10 If G is a connected graph of order n. Then Corollary 1 If G is a connected graph, then Proof. By Cauchy-Schwarz inequality, we get Proof. The prove is similar to Corollary 1.