Extremal graphs for the sum of the first two largest signless Laplacian eigenvalues

For a graph $G$, let $S_2(G)$ be the sum of the first two largest signless Laplacian eigenvalues of $G$, and $f(G)=e(G)+3-S_2(G)$. Very recently, Zhou, He and Shan proved that $K_{1,n-1}^{+}$ (the star graph with an additional edge) is the unique graph with minimum value of $f(G)$ among graphs on $n$ vertices. In this paper, we prove that $K_{1,e(G)-1}^{+}$ is the unique graph with minimum value of $f(G)$ among graphs with $e(G)$ edges.