The rank of a signed graph in terms of girth

2021/09/07 by Lu, Yong, Wu, Qi
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2109.02830

Abstract

Let Γ=(G,σ) be a signed graph and A(G,σ) be its adjacency matrix. Denote by gr(G) the girth of G, which is the length of the shortest cycle in G. Let r(G,σ) be the rank of (G,σ). In this paper, we will prove that r(G,σ)≥ gr(G)-2 for a signed graph (G,σ). Moreover, we characterize all extremal graphs which satisfy the equalities r(G,σ)=gr(G)-2 and r(G,σ)=gr(G).

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