On the sum of signless Laplacian spectra of graphs

Abstract

For a simple graph $G(V,E)$ with $n$ vertices, $m$ edges, vertex set $V(G)=\{v_1, v_2, \dots, v_n\}$ and edge set $E(G)=\{e_1, e_2,\dots, e_m\}$, the adjacency matrix $A=(a_{ij})$ of $G$ is a $(0, 1)$-square matrix of order $n$ whose $(i,j)$-entry is equal to 1 if $v_i$ is adjacent to $v_j$ and equal to 0, otherwise. Let $D(G)={diag}(d_1, d_2, \dots, d_n)$ be the diagonal matrix associated to $G$, where $d_i=\deg(v_i),$ for all $i\in \{1,2,\dots,n\}$. The matrices $L(G)=D(G)-A(G)$ and $Q(G)=D(G)+A(G)$ are respectively called the Laplacian and the signless Laplacian matrices and their spectra (eigenvalues) are respectively called the Laplacian spectrum ($L$-spectrum) and the signless Laplacian spectrum ($Q$-spectrum) of the graph $G$. If $0=\mu_n\leq\mu_{n-1}\leq\cdots\leq\mu_1$ are the Laplacian eigenvalues of $G$, Brouwer conjectured that the sum of $k$ largest Laplacian eigenvalues $S_{k}(G)$ satisfies $S_{k}(G)=\sum\limits_{i=1}^{k}\mu_i\leq m+{k+1 \choose 2}$ and this conjecture is still open. If $q_1,q_2, \dots, q_n$ are the signless Laplacian eigenvalues of $G$, for $1\leq k\leq n$, let $S^{+}_{k}(G)=\sum_{i=1}^{k}q_i$ be the sum of $k$ largest signless Laplacian eigenvalues of $G$. Analogous to Brouwer's conjecture, Ashraf et al. conjectured that $S^{+}_{k}(G)\leq m+{k+1 \choose 2}$, for all $1\leq k\leq n$. This conjecture has been verified in affirmative for some classes of graphs. We obtain the upper bounds for $S^{+}_{k}(G)$ in terms of the clique number $\omega$, the vertex covering number $\tau$ and the diameter of the graph $G$. Finally, we show that the conjecture holds for large families of graphs.