On the Sum of K Largest Laplacian Eigenvalues of Graphs

Consider a simple graph g(v; e) of order n, size m and having the vertex set v (g) = fv1; v2; : : : ; vngand edge set e(g) = fe1; e2; : : : ; emg. the adjacency matrix a = (aij) of g is a (0; 1)-square matrixof order n whose (i; j)-entry is equal to 1 if vi is adjacent to vj and equal to 0, otherwise. let d(g) =diag(d1; d2; : : : ; dn) be the diagonal matrix associated to g, where di = deg(vi); for all i = 1; 2; : : : ; n.the matrix l(g) = d(g) − a(g) is called the laplacian matrix and its eigenvalues are called the laplacian eigenvalues of the graph g. let 0 = µn ≤ µn−1 ≤ · · · ≤ µ1 be the laplacian eigenvalues of gand sk(g) =kpi=1µi, k = 1; 2; : : : ; n be the sum of k largest laplacian eigenvalues of g. for any k,k = 1; 2; : : : ; n, a. brouwer conjectured that sk(g) =kpi=1µi ≤ m + k+1 2 . we discuss the bounds forsk(g) and the recent developments of the brouwer’s conjecture. further, we investigate analogous conjectures (of the brouwer’s type) in other types of graphs.