| Abstract: | Let G be a graph on n vertices, and let CHP(G;λ) be the characteristic polynomial of its adjacency matrix A(G). All n roots of CHP(G;λ), denoted by  , are called to be its eigenvalues. The energy E(G) of a graph G, is the sum of absolute values of all eigenvalues, namely,  . Let  be the set of n-vertex unicyclic graphs, the graphs with n vertices and n edges. A fully loaded unicyclic graph is a unicyclic graph taken from  with the property that there exists no vertex with degree less than 3 in its unique cycle. Let  be the set of fully loaded unicyclic graphs. In this article, the graphs in  with minimal and second-minimal energies are uniquely determined, respectively. |
