Ordering of Unicyclic Graphs with Perfect Matchings by Minimal Matching Energies

Abstract

In 2012, Gutman and Wagner proposed the concept of the matching energy of a graph and pointed out that its chemical applications can go back to the 1970s. The matching energy of a graph is defined as the sum of the absolute values of the zeros of its matching polynomial. Let u and v be the non-isolated vertices of the graphs G and H with the same order, respectively. Let w_i be a non-isolated vertex of graph G_i where i=1, 2, …, k . We use G_u(k) (respectively, H_v(k) ) to denote the graph which is the coalescence of G (respectively, H ) and G₁, G₂,…, G_k by identifying the vertices u (respectively, v ) and w₁, w₂,…, w_k . In this paper, we first present a new technique of directly comparing the matching energies of G_u(k) and H_v(k) , which can tackle some quasi-order incomparable problems. As the applications of the technique, then we can determine the unicyclic graphs with perfect matchings of order 2 n with the first to the ninth smallest matching energies for all n≥211 .