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 $M_1(G)=\displaystyle\sum_{i=1}^{n}d^2_{v_i}$ is called the first Zagreb index of $G$, where $d_{v_i}$ is the degree of vertex $v_i$, for all $i=1,2,\dots,n$. 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 $q_1\geq q_2\geq \dots\geq q_n\geq 0$ 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 $S^+_k(G)=\displaystyle\sum_{i=1}^{k}q_i$ and $L_k(G)=\displaystyle\sum_{i=0}^{k-1}q_{n-i}$, where $1\leq k\leq n$, respectively denote the sum of $k$ largest and smallest signless Laplacian eigenvalues of $G$. The signless Laplacian energy of $G$ is defined as $QE(G)=\displaystyle\sum_{i=1}^{n}|q_i-\overline{d}|$, where $\overline{d}=\frac{2m}{n}$ 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.