Minimality considerations for graph energy over a class of graphs

AbstractLet 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 λi(i=1,2,…n), are called to be its eigenvalues. The energy E(G) of a graph G, is the sum of absolute values of all eigenvalues, namely, E(G)=∑i=1n|λi|. Let Un 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 Un 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