Relations between the inertia indices of a mixed graph and those of its underlying graph

Abstract

Given a graph G, the mixed graph DG is obtained from G by orienting some of its edges, where G is called the underlying graph of DG. Let p(DG), n(DG) (resp. p(G), n(G)) be the positive inertia index and negative inertia index of DG (resp. G). In this paper, we first establish the inequalities −d(G)⩽p(DG)−p(G)⩽d(G) and −d(G)⩽n(DG)−n(G)⩽d(G), where d(G) is the dimension of cycle space of G. Furthermore, all the corresponding extremal graphs are characterized.