ON THE SIGNLESS LAPLACIAN SPECTRAL RADIUS OF UNICYCLIC GRAPHS WITH FIXED MATCHING NUMBER

Abstract

We determine the graph with the largest signless Laplacian spec- tral radius among all unicyclic graphs with fixed matching number. respectively. The largest eigenvalues of A(G) and Q(G) are called the spectral radius and the signless Laplacian spectral radius of G, denoted by �(G) and q(G), respectively. When G is connected, A(G) and Q(G) are nonegative irreducible matrix. By the Perron-Frobenius theory, �(G) is simple and has a unique positive unit eigenvector, so does q(G). We refer to such the eigenvector corresponding to q(G) as the Perron vector of G. Two distinct edges in a graph G are independent if they are not adjacent in G. A set of pairwise independent edges of G is called a matching in G. A matching of