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

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.

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

signless Laplacian matrix , signless Laplacian spectrum , clique number , forest