The non-commuting, non-generating graph of a non-simple group
Saul D. Freedman

TL;DR
This paper studies the structure of a graph associated with a group, revealing how the group's properties influence the graph's connectivity and diameter, with detailed classifications for certain finite groups.
Contribution
It characterizes the connectedness and diameter of the non-commuting, non-generating graph for groups where $G/Z(G)$ is not simple, including detailed classifications for finite groups with specific graph structures.
Findings
Graph is either connected with diameter ≤ 4, or has isolated vertices and a component with diameter ≤ 4, or two components each of diameter 2.
Finite groups with the graph consisting of two components of diameter 2 are fully described.
The study extends understanding of the relationship between group structure and associated graphs.
Abstract
Let be a (finite or infinite) group such that is not simple. The non-commuting, non-generating graph of has vertex set , with vertices and adjacent whenever and . We investigate the relationship between the structure of and the connectedness and diameter of . In particular, we prove that the graph either: (i) is connected with diameter at most ; (ii) consists of isolated vertices and a connected component of diameter at most ; or (iii) is the union of two connected components of diameter . We also describe in detail the finite groups with graphs of type (iii). In the companion paper arXiv:2212.01616, we consider the case where is finite and simple.
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Taxonomy
TopicsFinite Group Theory Research · Algebraic structures and combinatorial models · Advanced Graph Theory Research
