On the Sum of k Largest Laplacian Eigenvalues of a Graph and Clique Number

For a simple graph G with order n and size m having Laplacian eigenvalues $$\mu _1, \mu _2, \dots , \mu _{n-1},\mu _n=0$$ , let $$S_k(G)=\sum _{i=1}^{k}\mu _i$$ , be the sum of k largest Laplacian eigenvalues of G. We obtain upper bounds for the sum of k largest Laplacian eigenvalues of two large families of graphs. As a consequence, we prove Brouwer’s Conjecture for large number of graphs which belong to these families of graphs.