On the connectivity of the non-generating graph

2021/08/28 by Andrea Lucchini, Lucchini, Andrea, Daniele Nemmi +1 · 1 citation
Computer Science · Mathematics · #Advanced Graph Theory Research #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Interconnection Networks and Systems

paper · pdf · doi:10.48550/arxiv.2108.12569

openalex publication_date 2021/08/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a 2-generated finite group G, the non-generating graph of G has as vertices the elements of G and two vertices are adjacent if and only if they are distinct and do not generate G. We consider the graph Σ(G) obtained from the non-generating graph of G by deleting the universal vertices. We prove that if the derived subgroup of G is not nilpotent, then this graph is connected, with diameter at most 5. Moreover we give a complete classification of the finite groups G such that Σ(G) is disconnected.

Cited by

• Finite groups with the minimal generating set exchange property

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• Connectivity of generating graphs of nilpotent groups
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