On the maximum (signless) Laplacian spectral radius of the cacti

摘要

Suppose that the vertex set of a graph G is V(G) {v(1) v(2),..,v(n)} Then we denote by d(vi)(G) the degree of v(i) in G. Let A(G) he the adjacent matrix of G and D(G) be the n x n diagonal matrix with its (i, i)-entry equal to d(vi) (G). Then Q(A)(G) = D(G) + A(G) and L-A(G) = D(G) - A(G) are the signless Laplacian matrix and Laplacian matrix of G, respectively. The signless Laplacian and Laplacian spectral radius of G are respectively the largest eigenvalue of Q(A)(G) and L-A(G). In this paper we characterize the graphs with the maximum signless Laplacian spectral radius and the maximum Laplacian spectral radius respectively among all cacti of order n with given k cycles or r pendent vertices.