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.

Let G be a connected simple graph with n vertices. The distance Laplacian matrix D L (G) is defined as D L (G) = Diag(T r) − D(G), where Diag(T r) is the diagonal matrix of vertex transmissions and D(G) is the distance matrix of G. The eigenvalues of D L (G) are the distance Laplacian eigenvalues of G and are denoted byis called the distance Laplacian spectral radius. Lu et al. (2017), Fernandes et al. (2018) and Ma et al. (2018) completely characterized the graphs having some distance Laplacian eigenvalue of multiplicity n − 3. In this paper, we characterize the graphs having distance Laplacian spectral radius of multiplicity n − 4 together with one of the distance Laplacian eigenvalue as n of multiplicity either 3 or 2. Further, we completely determine the graphs for which the distance Laplacian eigenvalue n is of multiplicity n − 4.

For a connected graph G of order n, let Diag(T r) be the diagonal matrix of vertex transmissions and D(G) be the distance matrix of G. The distance Laplacian matrix of G is