On the conjecture for the sum of the largest signless Laplacian eigenvalues of a graph- a survey

Abstract

Let $G$ be a simple graph with order $n$ and size $m$. Let $D(G)=$ diag$(d_1, d_2, \dots, d_n)$ be its diagonal matrix, where $d_i=\deg(v_i),$ for all $i=1,2,\dots,n$ and $A(G)$ be its adjacency matrix. The matrix $Q(G)=D(G)+A(G)$ is called the signless Laplacian matrix of $G$. Let $q_1,q_2,\dots,q_n$ be the signless Laplacian eigenvalues of $Q(G)$ and let $S^{+}_{k}(G)=\sum_{i=1}^{k}q_i$ be the sum of the $k$ largest signless Laplacian eigenvalues. Ashraf et al. [F. Ashraf, G. R. Omidi, B. Tayfeh-Rezaie, On the sum of signless Laplacian eigenvalues of a graph, Linear Algebra Appl. {\bf 438} (2013) 4539-4546.] conjectured that $S^{+}_{k}(G)\leq m+{k+1 \choose 2}$, for all $k=1,2,\dots,n$. We present a survey about the developments of this conjecture.