On the spectral radius of unicyclic graphs with perfect matchings

AbstractLet U+(2k) 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 U+(2k) when k≠3