Signed graphs with stable maximum nullity at most two

A signed graph is a pair (G, S), where G =(V, E) is a graph (in which parallel edges are permitted, but loops are not) with V={1,..., n} and S. E. The edges in Sare called odd and the other edges of Eeven. By S(G, S) we denote the set of all symmetric n xnmatrices A =[ ai,j] with ai,j< 0if iand jare adjacent and connected by only even edges, ai,j> 0if iand jare adjacent and connected by only odd edges, ai,j. Rif iand jare connected by both even and odd edges, ai,j= 0if i= jand iand jare non-adjacent, and ai,i. Rfor all vertices i. For a graph G =(V, E) with V={1,..., n}, we denote by N(G) the set of all symmetric matrices X=[ xi,j] with xi,i= 0for all i. V, and xi,j= 0if iand jare adjacent. A matrix A. S(G, S) has the Strong Arnold Property if for all X. N(G), AX= 0implies that X= 0. The parameters.of a signed graph (G, S) are thelargest nullity of any matrix A. S(G, S) that has the Strong Arnold Property. In a previous paper, we gave a characterization of 2-connected signed graphs (G, S) with.(G, S) = 2. In this paper, we give a complete characterization of the signed graphs (G, S) with.(G, S) = 2. (C) 2021 Elsevier Inc. All rights reserved.