On the spectral radius of unicyclic graphs with perfect matchings

Let U+(2k) be the set of all unicyclic graphs on 2k (k greater than or equal to 2) vertices with perfect matchings. Let U-2k(1) be the graph on 2k 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-2k(2) be the graph on 2k vertices obtained from 2k 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-2k(1) and U-2k(2) have the largest and the second largest spectral radius among the graphs in U+ (2k) when k not equal 3. (C) 2003 Elsevier Inc. All rights reserved.