The independence graph of a finite group

Abstract

Given a finite group G, we denote by $$\Delta (G)$$ Δ ( G ) the graph whose vertices are the elements G and where two vertices x and y are adjacent if there exists a minimal generating set of G containing x and y. We prove that $$\Delta (G)$$ Δ ( G ) is connected and classify the groups G for which $$\Delta (G)$$ Δ ( G ) is a planar graph.

Subgroup Graphs of Finite Groups

“…Afterwards, in the last few decades, his view of diagraph has since been extended to different and modified graph of algebraic structures. Hence, more algebraic studies through the properties of these modified graphs have become topics of interest to many around the globe (See [8], [9], [10], [11], [12], [13], [14], [15]).…”

“…This study, the subgroup graph of finite groups like [8], [9], [10], [11], [12], [13], [14], [15], will focus on finite groups , however, the choice of it vertex set is the subgroups of . In the literature, vertex set of graphs of finite groups are always the elements n G, a deviation from this norm is the motivation for this study.…”

The generating graph of a profinite group

“…We denote by Γ(G) the graph whose vertices are the elements G and in which two vertices x and y are joined by an edge if and only if x = y and there exists a finite minimal generating set of G containing x and y. Roughly speaking, x and y are adjacent vertices of Γ(G) if they are 'independent', so we call Γ(G) the independence graph of G. In Section 5, extending results proved in [8] in the case of finite groups, we show that if G is a finitely generated profinite group and g is an isolated vertex of Γ(G), then either g belongs to the Frattini subgroup of G or it is a topological generator of G. Moreover the graph obtained from Γ(G) by removing the isolated vertices is always connect. Also in this case instead of generating subsets, we may consider virtually generating subsets, i.e.…”

“…In the second part of the paper we extend to the profinite groups the notion of independence graph, given in [8] for finite groups, and we introduce the new definition of virtually independence graph of a profinite group. Let G be a finitely generated profinite group.…”

The t-Graphs Over Finitely-Generated Groups and the Minkowski Metric

“…These differences lie in the adjacency criterion used to relate two group elements constituting the set of vertices of such a graph. Some essential authors in this context are A. Abdollahi [1], A. Lucchini [6,7], and D. Hai-Reuven [4], among others.…”