Author(s)

Jianming Zhu

Department of Applied Mathematics, Shanghai University of International Business and Economics, Shanghai, China.

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 2n with the first to the ninth smallest matching energies for all n≥211.

Matching Energy, Unicyclic Graph, Perfect Matching

Zhu, J. (2019) Ordering of Unicyclic Graphs with Perfect Matchings by Minimal Matching Energies. Open Journal of Discrete Mathematics, 9, 17-32. doi: 10.4236/ojdm.2019.91004.