The sum of the first two largest signless laplacian eigenvalues of trees and unicyclic graphs

Let G be a graph on n vertices with e(G) edges. The sum of eigenvalues of graphs has been receiving a lot of attention these years. Let S-2(G) be the sum of the first two largest signless Laplacian eigenvalues of G, and define f(G) = e(G) + 3 - S-2(G). Oliveira et al. (2015) conjectured that f(G) >= f(U-n) with equality if and only if G similar or equal to U-n, where U-n is the n-vertex unicyclic graph obtained by attaching n pendent vertices to a vertex of a triangle. In this paper, it is proved that S-2(G) < e(G) + 3 - 2/n when G is a tree, or a unicyclic graph whose unique cycle is not a triangle. As a consequence, it is deduced that the conjecture proposed by Oliveira et al. is true for trees and unicyclic graphs whose unique cycle is not a triangle.