On Laplacian resolvent energy of graphs

Abstract. 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 d1 ≥ d2 ≥ · · · dn be the vertex degree sequence and µ1 ≥ µ2 ≥ · · · ≥ µn−1 µn = 0 be the Laplacian eigenvalues. The Laplacian resolvent energy RL(G) of a graph G is defined as RL(G) ... . In this paper, we obtain an upper bound for i=1 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.