Note on an upper bound for sum of the Laplacian eigenvalues of a graph

For a simple graph G with n vertices and m edges having Laplacian eigenvalues μ1(G)≥μ2(G)≥⋯≥μn(G), let Sk(G) be the sum of k largest Laplacian eigenvalues of G. In this note, we prove that if G is a connected graph of order n≥2 with m edges having clique number ω and vertex covering number τ, thenSk(G)≤k(τ+1)+m−ω(ω−1)2, with equality if k≤ω−1 and G is the graph obtained by joining n−ω pendant vertices with one of the vertices in Kω. Our work improves a recent work of Ganie et al.