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, . . . ,d_n) be its diagonal matrix, where d_i = deg(v_i), for all i = 1, 2, . . . ,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, . . . , 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. 438 (2013) 4539-4546.] conjectured that$s^{+}_{k} (g)leq m+{k+1 choose 2}$, for all k = 1, 2, . . . ,n. we present a survey about the developments of this conjecture.