On Zagreb index, signless Laplacian eigenvalues and signless Laplacian energy of a graph

Abstract

Let G be a simple graph with order n and size m. The quantity M 1 (G) = n i=1 d 2 v i is called the first Zagreb index of G, where d v i is the degree of vertex v i , for all i = 1, 2, . . . , n. The signless Laplacian matrix of a graph G is Q(G) = D(G) + A(G), where A(G) and D(G)denote, respectively, the adjacency and the diagonal matrix of the vertex degrees of G. Let0 be the signless Laplacian eigenvalues of G. The largest signless Laplacian eigenvalue q 1 is called the signless Laplacian spectral radius or Q-index of G and is denoted bywhere 1 ≤ k ≤ n, respectively denote the sum of k largest and smallest signless Laplacian eigenvalues of G. The signless Laplacian energy of G is defined as QE(G) = n i=1 |q i − d|, where d = 2m n is the average vertex degree of G. In this article, we obtain upper bounds for the first Zagreb index M 1 (G) and show that each bound is best possible. Using these bounds, we obtain several upper bounds for the graph invariant S + k (G) and characterize the extremal cases. As a consequence, we find upper bounds for the Q-index and lower bounds for the graph invariant L k (G) in terms of various graph parameters and determine the extremal cases. As an application, we obtain upper bounds for the signless Laplacian energy of a graph and characterize the extremal cases.