On the sum of the first two largest signless Laplacian eigenvalues of a graph

For a graph $G$, let $S_2(G)$ be the sum of the first two largest signless Laplacian eigenvalues of $G$, and $f(G)=e(G)+3-S_2(G)$. Oliveira, Lima, Rama and Carvalho conjectured that $K^+_{1,n-1}$ (the star graph with an additional edge) is the unique graph with minimum value of $f(G)$ on $n$ vertices. In this paper, we prove this conjecture, which also confirm a conjecture for the upper bound of $S_2(G)$ proposed by Ashraf et al.
Comment: 15 pages, 5 figures