Relation between the Hermitian energy of a mixed graph and the matching number of its underlying graph

Abstract

ABSTRACT Given a graph G, the mixed graph is obtained from G by orienting some of its edges (G is also called the underlying graph of ). Let be the Hermitian energy of and let be the matching number of the underlying graph G. In this paper, we first establish the inequality and characterize all the mixed graphs which make hold. Furthermore, we obtain , where and are, respectively, the vertex cover number, the number of odd cycles and the maximum degree of graph G. The lower bound is attained if and only if is switching equivalent to its underlying graph G, where G is the disjoint union of some complete bipartite graphs each of which contains a perfect matching, together with some isolated vertices. The upper bound is best possible. By our results in this paper, some main results in Tian FL, Wong D. [Relation between the skew energy of an oriented graph and its matching number. Discrete Appl Math. 2017;222:179–184]; Wang L, Ma XB. [Bounds of graph energy in terms of vertex cover number. Linear Algebra Appl. 2017;517:207–216]; Wong D, Wang XL, Chu R. [Lower bounds of graph energy in terms of matching number. Linear Algebra Appl. 2018;549:276–286] can be deduced in a unified approach.