On the spectral radius of unicyclic graphs with perfect matchings

Let be the set of all unicyclic graphs on 2k (k 2) vertices with perfect matchings. Let U12k be the graph on 2k vertices obtained from C3 by attaching a pendant edge and k−2 paths of length 2 at one vertex of C3; Let U22k be the graph on 2k vertices obtained from C3 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 U12k and U22k have the largest and the second largest spectral radius among the graphs in when k≠3.