Non-commuting graphs of nilpotent groups
Alireza Abdollahi, Hamid Shahverdi

TL;DR
This paper investigates the structure of non-commuting graphs of non-abelian nilpotent groups, proving that isomorphic irregular graphs imply equal group orders, revealing a link between graph properties and group order.
Contribution
It establishes that non-abelian nilpotent groups with irregular isomorphic non-commuting graphs must have the same order, a novel connection between graph isomorphism and group size.
Findings
Irregular isomorphic non-commuting graphs imply equal group orders.
Non-abelian nilpotent groups' structure is reflected in their non-commuting graphs.
Graph properties can determine algebraic group characteristics.
Abstract
Let be a non-abelian group and be the center of . The non-commuting graph associated to is the graph whose vertex set is and two distinct elements are adjacent if and only if . We prove that if and are non-abelian nilpotent groups with irregular isomorphic non-commuting graphs, then .
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