Two-connected signed graphs with maximum nullity at most two

Abstract

A signed graph is a pair (G,Σ), where G=(V,E) is a graph (in which parallel edges are permitted, but loops are not) with V={1,…,n} and Σ⊆E. The edges in Σ are called odd and the other edges of E even. By S(G,Σ) we denote the set of all symmetric n×n matrices A=[ai,j] with ai,j<0 if i and j are adjacent and connected by only even edges, ai,j>0 if i and j are adjacent and connected by only odd edges, ai,j∈R if i and j are connected by both even and odd edges, ai,j=0 if i≠j and i and j are non-adjacent, and ai,i∈R for all vertices i. The parameters M(G,Σ) and ξ(G,Σ) of a signed graph (G,Σ) are the largest nullity of any matrix A∈S(G,Σ) and the largest nullity of any matrix A∈S(G,Σ) that has the Strong Arnold Property, respectively. In a previous paper, we gave a characterization of signed graphs (G,Σ) with M(G,Σ)≤1 and of signed graphs with ξ(G,Σ)≤1. In this paper, we characterize the 2-connected signed graphs (G,Σ) with M(G,Σ)≤2 and the 2-connected signed graphs (G,Σ) with ξ(G,Σ)≤2.