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- W792346822 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 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 ${K_{n_v}}_{{vin V(G)| d_v 0}}$ 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." @default.
- W792346822 created "2016-06-24" @default.
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- W792346822 date "2015-04-23" @default.
- W792346822 modified "2023-09-27" @default.
- W792346822 title "Critical ideals of signed graphs with twin vertices" @default.
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