TITLE:
Ordering of Unicyclic Graphs with Perfect Matchings by Minimal Matching Energies
AUTHORS:
Jianming Zhu
KEYWORDS:
Matching Energy, Unicyclic Graph, Perfect Matching
JOURNAL NAME:
Open Journal of Discrete Mathematics,
Vol.9 No.1,
January
28,
2019
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 wibe a non-isolated
vertex of graph Giwhere 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,…, Gkby 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.