Minimality Considerations For Graph Energy Over A Class Of Graphs

摘要

Let C 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 λ_i (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) = Σ_(i=1)~n |λ_i|. Let U_n 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 U_n with the property that there exists no vertex with degree less than 3 in its unique cycle. Let U_n~1 be the set of fully loaded unicyclic graphs. In this article, the graphs in U_n~1 with minimal and second-minimal energies are uniquely determined, respectively.