Minimality considerations for graph energy over a class of graphs

Let GG be a graph on nn vertices, and let CHP(G;λ)CHP(G;λ) be the characteristic polynomial of its adjacency matrix A(G)A(G). All nn roots of CHP(G;λ)CHP(G;λ), denoted by λi(i=1,2,…n), are called to be its eigenvalues. The energy E(G)E(G) of a graph GG, is the sum of absolute values of all eigenvalues, namely, E(G)=∑i=1n|λi|. Let UnUn be the set of nn-vertex unicyclic graphs, the graphs with nn vertices and nn edges. A fully loaded unicyclic graph is a unicyclic graph taken from UnUn with the property that there exists no vertex with degree less than 3 in its unique cycle. Let Un1 be the set of fully loaded unicyclic graphs. In this article, the graphs in Un1 with minimal and second-minimal energies are uniquely determined, respectively.