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 wi be a non-isolated vertex of graph Gi where i=1, 2, …, k. We use Gu(k) (respectively, Hv(k)) to denote the graph which is the coalescence of G (respectively, H) and G1, G2,…, Gk by identifying the vertices u (respectively, v) and w1, w2,…, wk. In this paper, we first present a new technique of directly comparing the matching energies of Gu(k) and Hv(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 2n with the first to the ninth smallest matching energies for all n≥211.