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

Let G be a simple graph with order n and size m. The quantity M-1(G) = Sigma(n)(i=1) d(vi)(2) is called the first Zagreb index of G, where dvi 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. Let q(1) >= q(2) >= . . . >= q(n) >= 0 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 by q(G). Let Sigma(+)(k)(G) = Sigma(K)(i=1) q(i) and L-k(G) = Sigma(K-1)(i=0) q(n-i), where 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) = Sigma(n)(i=1) |q(i) - (d) over bar|, where (d) over bar = 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.