Ordering the complements of trees by the number of maximum matchings

Abstract

ABSTRACT:A "perfect matching" of a graph G with n vertices is a set of n/2 independent edges of G. In the present study, we succeeded in determining the trees whose complements have the extremal number of "perfect matchings" for two different group of trees. Some further problems are also posed.

Extremal matching energy of complements of trees

“…3 Ordering complements of trees with respect to their matchings For convenience, we use the same definitions of trees which are defined in [16].…”

“…M E(P n ) = M E(T (l) ) > M E(T (l−1) ) > ... > M E(T (2) ) > M E(T (1) ) > M E(T ). In order to prove Theorem 1.2, we need the following two lemmas by Yan et al in [16]. Proof of Theorem 1.2.…”

“…Lemma 2.4. ( [16]) If T is a tree with n vertices and edge-independence number ν(T ) = p, then T has at most n − p vertices of degree one. In particular, if T has exactly n − p vertices of degree one, then every vertex of degree at least two in T is adjacent to at least one vertex of degree one.…”

“…In order to prove Theorem 1.2, we need the following two lemmas by Yan et al in [16]. n ) < M E(T ) < M E(P n ) and M E(T 1 n,2 ) < M E(P n ).…”

“…In this paper, inspired by the idea [16], we investigate the problem of the matching energy of the complement of trees and obtain the following main theorems.…”