For a graph G, let S2(G) be the sum of the first two largest signless Laplacian eigenvalues of G, and f(G) =e(G) + 3-S2(G). Very recently, Zhou et al. (2024) proved that K+1,n-1 (the star graph with an additional edge) is the unique graph with minimum value of f(G) among the graphs on n vertices. In this paper, we prove that the vertex-disjoint union of K+1,e(G)-1 and possibly some isolated vertices is the unique graph with minimum value of f(G) among the graphs with e(G) edges. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.