The rank of a signed graph in terms of the rank of its underlying graph

Let Gamma = (G, sigma) be a signed graph and A(Gamma) be its adjacency matrix, where G is the underlying graph of Gamma. The rank r(Gamma) of Gamma is the rank of A(Gamma). We know that for a signed graph Gamma = (G, sigma), Gamma is balanced if and only if Gamma = (G, sigma) similar to (G, +). That is, when Gamma is balanced, then r(Gamma) = r(G), where r(G) is the rank of the underlying graph G of Gamma. A natural problem is that: how about the relations between the rank of an unbalanced signed graph and the rank of its underlying graph? In this paper, we first prove that r(G)-2d(G) <= r(Gamma) <= r(G) + 2d(G) for an unbalanced signed graph with d(G) >= 1, where d(G) = vertical bar E(G)vertical bar-vertical bar V (G)vertical bar+ w(G) is the dimension of cycle spaces of G, w(G) is the number of connected components of G. As an application, we also prove that 1 - d(G) < r(Gamma)/r(G) <= 1+ d(G) for an unbalanced signed graph with d(G) >= 1. All corresponding extremal graphs are characterized. (C) 2017 Elsevier Inc. All rights reserved.