The non‐F graph of a finite group

Abstract

Given a formation 𝔉, we consider the graph whose vertices are the elements of 𝐺 and where two vertices 𝑔, ℎ ∈ 𝐺 are adjacent if and only if ⟨𝑔, ℎ⟩ ∉ 𝔉. We are interested in the two following questions. Is the set of the isolated vertices of this graph a subgroup of 𝐺? Is the subgraph obtained by deleting the isolated vertices a connected graph?

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

“…In the following, to study the t-graphs associated with a finite group G, we will consider only finite two-generator groups, which can be expressed in the form (10). These numbers m and n depend exclusively on the structure, namely on the group's presentation and the order of G.…”

“…Let G be a two-generator group in the form (10) with n, m ≥ 2, and G is the corresponding t-graph of G. Then, G has no isolated points if and only if t ≤ m+n−2…”

“…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 [3], A. Ballester-Bolinches et al [4][5][6][7][8], A. Lucchini [9,10], and D. Hai-Reuven [11], among others.…”

“…This fact leads us to state the first theorem in the next section, which allows us to characterize first the t-graphs associated with two-generator groups in the form (10), concerning the number of connected components.…”

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.…”