Signed graphs with cut points whose positive inertia indexes are two

Abstract

A signed graph Gσ consists of an underlying graph G and a sign function σ, which assigns each edge uv of G a sign σ(uv), either positive or negative. The adjacency matrix of Gσ is defined as A(Gσ)=(au,vσ) with au,vσ=σ(uv)au,v, where au,v=1 if u,v∈V(G) are adjacent, and au,v=0 otherwise. The positive inertia index of Gσ, written as p(Gσ), is defined to be the number of positive eigenvalues of A(Gσ). Recently, Yu et al. (2016) [12] characterized the signed graphs Gσ with pendant vertices such that p(Gσ)=2. In this paper, we extend the above work to a more general case, characterizing the signed graphs Gσ with cut points whose positive inertia index is 2.