The spread of unicyclic graphs with given size of maximum matchings

Abstract

The spread s(G) of a graph G is defined as s(G) = maxi,j|λi − λj|, where the maximum is taken over all pairs of eigenvalues of G. Let U(n,k) denote the set of all unicyclic graphs on n vertices with a maximum matching of cardinality k, and U*(n,k) the set of triangle-free graphs in U(n,k). In this paper, we determine the graphs with the largest and second largest spectral radius in U*(n,k), and the graph with the largest spread in U(n,k).