Sandeep Bhatnagar - Department of Applied Mathematics, Aligarh Muslim University, Aligarh, India
Siddiqui Merajuddin - Department of Applied Mathematics, Aligarh Muslim University, Aligarh, India
Shariefuddin Pirzada - Department of Mathematics, University of Kashmir, Srinagar, India

Let G be a simple connected graph of order n and size m. The matrix L(G)=D(G)-A(G) is the Laplacian matrix of G, where D(G) and A(G) are the degree diagonal matrix and the adjacency matrix, respectively. For the graph G, let d_{۱}\geq d_{۲}\geq \cdots d_{n} be the vertex degree sequence and \mu_{۱}\geq \mu_{۲}\geq \cdots \geq \mu_{n-۱}>\mu_{n}=۰ be the Laplacian eigenvalues. The Laplacian resolvent energy RL(G) of a graph G is defined as RL(G)=\sum\limits_{i=۱}^{n}\frac{۱}{n+۱-\mu_{i}}. In this paper, we obtain an upper bound for the Laplacian resolvent energy RL(G) in terms of the order, size and the algebraic connectivity of the graph. Further, we establish relations between the Laplacian resolvent energy RL(G) with each of the Laplacian-energy-Like invariant LEL, the Kirchhoff index Kf and the Laplacian energy LE of the graph.

Laplacian resolvent energy, Laplacian energy, Laplacian-energy-like invariant, Kirchhoff index