Discrete Mathematics Letters www.dmlett.com Discrete Math. Lett. 2 (2019) 32–37 A note on the Laplacian resolvent energy, Kirchhoff index and their relations Emir Zogi´ c * , Edin Glogi´ c Department of Mathematical Sciences, State University of Novi Pazar, Novi Pazar, Serbia (Received: 27 July 2019. Received in revised form: 15 August 2019. Accepted: 3 September 2019. Published online: 9 September 2019.) c  2019 the authors. This is an open access article under the CC BY (International 4.0) license (https://creativecommons.org/licenses/by/4.0/). Abstract Let G be a simple graph of order n and let L be its Laplacian matrix. Eigenvalues of the matrix L are denoted by μ1,μ2, ··· ,μn and it is assumed that μ1  μ2  ···  μn. The Laplacian resolvent energy and Kirchhoff index of the graph G are defined as RL(G)= ∑ n i=1 1 n+1-μ i and Kf (G)= n ∑ n-1 i=1 1 μ i , respectively. In this paper, we derive some bounds on the invariant RL(G) and establish a relation between RL(G) and Kf (G). Keywords: Graph energy; Laplacian resolvent energy; Kirchhoff index. 2010 Mathematics Subject Classification: 05C50, 15A18. 1. Introduction Let G =(V (G),E(G)) be a simple graph with V (G)= {v 1 ,v 2 ,...,v n } and |E(G)| = m. Denote by A(G) the adjacency matrix of G, and by λ 1 ,λ 2 , ··· ,λ n its eigenvalues satisfying λ 1  λ 2  ···  λ n . The (ordinary) energy of the graph G is defined [7] as E (G)= n  i=1 |λ i |. Concept of graph energy has many applications, especially in chemistry. The most important properties of graph energy can be found in the monographs [11, 13] and in the references cited therein. There are graph energies which are based on different matrices associated with graphs. One of them is the Laplacian resolvent energy. Before defining this energy, we need some basic definitions. Let L be the Laplacian matrix of the graph G and μ 1 ,μ 2 , ··· ,μ n be its Laplacian eigenvalues satisfying μ 1  μ 2  ···  μ n . The Laplacian resolvent matrix R L (z) of the matrix L is defined as R L (z)=(zI n − L) -1 . Since all the Laplacain eigenvalues satisfy the condition μ i ≤ n, i =1, 2, ..., n, in the paper [3] it was proposed to choose z = n +1. The Laplacian resolvent energy is defined [3] as RL(G)= n  i=1 1 n +1 − μ i . Some of the basic properties and various bounds of the Laplacian resolvent energy can be found in the papers [3, 14, 20, 21]. In the paper [17], Klein and Randi´ c introduced the notion of resistance distance r ij which is defined as the resistance between the nodes i and j in an electrical network corresponding to the graph G in which all edges are replaced by unit resistors. The Kirchhoff index is defined as Kf (G)=  i<j r ij . Gutman and Mohar in the paper [8] and Zhy et al. in the paper [25] independently proved that Kirchhoff index can be represented as Kf (G)= n n-1  i=1 1 μ i . For some basic properties and various bounds of Kf (G), see the monograph [9]. Now, we recall some analytic inequalities for real number sequences that are of interest for the subsequent considera- tions. * Corresponding author (ezogic@np.ac.rs)