Critical ideals of signed graphs with twin vertices

TL;DR

This paper investigates the properties of critical ideals in graphs with twin vertices, revealing patterns and relationships with algebraic co-rank, and introduces conjectures about the distribution of algebraic co-rank in such graphs.

Contribution

It introduces a framework for analyzing critical ideals in graphs with twin vertices, establishing their behavior under duplication and replication, and relates algebraic co-rank to twin structures.

Findings

More than half of the critical ideals in certain graph families are determined by the original graph.

The algebraic co-rank of graphs with twin vertices equals that of their associated replicated or duplicated graphs.

Conjectures suggest twin-free graphs tend to have larger algebraic co-rank, while graphs with small algebraic co-rank contain twin vertices.

Abstract

This paper studies critical ideals of graphs with twin vertices, which are vertices with the same neighbors. A pair of such vertices are called replicated if they are adjacent, and duplicated, otherwise. Critical ideals of graphs having twin vertices have good properties and show regular patterns. Given a graph $G=(V,E)$ and ${\bf d}\in \mathbb{Z}^{|V|}$, let $G^{\bf d}$ be the graph obtained from $G$ by duplicating ${\bf d}_v$ times or replicating $-{\bf d}_v$ times the vertex $v$ when ${\bf d}_v>0$ or ${\bf d}_v<0$, respectively. Moreover, given $\delta\in \{0,1,-1\}^{|V|}$, let \[ \mathcal{T}_{\delta}(G)=\{G^{\bf d}: {\bf d}\in \mathbb{Z}^{|V|} \text{ such that } {\bf d}_v=0 \text{ if and only if }\delta_v=0 \text{ and } {\bf d}_v\delta_v>0 \text{ otherwise}\} \] be the set of graphs sharing the same pattern of duplication or replication of vertices. More than one half of the critical ideals of a graph in $\mathcal{T}_{\delta}(G)$ can be determined by the critical ideals of $G$. The algebraic co-rank of a graph $G$ is the maximum integer $i$ such that the $i$-{\it th} critical ideal of $G$ is trivial. We show that the algebraic co-rank of any graph in $\mathcal{T}_{\delta}(G)$ is equal to the algebraic co-rank of $G^{\delta}$. For a large enough ${\bf d}\in \mathbb{Z}^{V(G)}$, we show that the critical ideals of $G^{\bf d}$ have similar behavior to the critical ideals of the disjoint union of $G$ and some set $\{K_{n_v}\}_{\{v\in V(G)| d_v<0\}}$ of complete graphs and some set $\{T_{n_v}\}_{\{v\in V(G) \, |\, {\bf d}_v>0\}}$ of trivial graphs. Additionally, we pose important conjectures on the distribution of the algebraic co-rank of the graphs with twins vertices. These conjectures imply that twin-free graphs have a large algebraic co-rank, meanwhile a graph having small algebraic co-rank has at least one pair of twin vertices.