The rank of a signed graph in terms of girth

TL;DR

This paper establishes a lower bound on the rank of a signed graph's adjacency matrix based on its girth and characterizes extremal cases where the bound is tight.

Contribution

It proves that the rank of a signed graph is at least its girth minus two and characterizes all graphs where equality holds.

Findings

Proved that r(G,σ) ≥ gr(G) - 2 for signed graphs.

Characterized extremal graphs satisfying r(G,σ) = gr(G) - 2 and r(G,σ) = gr(G).

Provided insights into the relationship between girth and adjacency matrix rank.

Abstract

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