Minimal Energies Of Bicyclic Graphs

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. A connected graph with n vertices and n + 1 edges is called a bicyclic graph. Let G(n) be the class of bicyclic graphs G on n vertices and containing no disjoint odd cycles of lengths k and I with k+ 1 = 2 (mod 4). Let G'(n) be the set of bicyclic bipartite graphs with n vertices, and exactly two cycles. In this paper, we determine the graphs with minimal, second-minimal and third-minimal energies in G(n), and the graphs with minimal energies in G'(n).