On the spectral radius of unicyclic graphs with perfect matchings

Abstract

Let U +(2k) be the set of all unicyclic graphs on 2k (k⩾2) vertices with perfect matchings. Let U 1 2 k be the graph on 2 k vertices obtained from C 3 by attaching a pendant edge and k−2 paths of length 2 at one vertex of C 3; Let U 2 2 k be the graph on 2 k vertices obtained from C 3 by adding a pendant edge at each vertex together with k−3 paths of length 2 at one of three vertices. In this paper, we prove that U 1 2 k and U 2 2 k have the largest and the second largest spectral radius among the graphs in U +(2k) when k≠3.