A Note on Commuting Graphs for Symmetric Groups

Abstract

The commuting graph ${\cal C}(G,X)$, where $G$ is a group and $X$ a subset of $G$, has $X$ as its vertex set with two distinct elements of $X$ joined by an edge when they commute in $G$. Here the diameter and disc structure of ${\cal C}(G,X)$ is investigated when $G$ is the symmetric group and $X$ a conjugacy class of $G$.

Commuting Graphs, C(G, X) in Symmetric Groups Sym(n) and Its Connectivity

“…By using Theorems 1 and 2 where appropriate, we can now put these to good use to partially prove the following result (note that this observation is based on the results in Table 1 of [2] and Table 1 of [3]): Theorem 3. Let G = Sym(n) and t ∈ G be of cycle type 3 r with r ≥ 1.…”

“…Note again (1, 2) is the only edge of Γ. Now, by employing the definition given in the Introduction, (1,2) is an exact edge if h 1 = f 1 and h 2 = f 2 are the only possibility to get the following equality:…”

“…An edge (i, j) ∈ E(Γ) is called an exact edge if h i = f i and h j = f j . A special edge (i, j) with source i of Γ satisfies e j f j = e i and b(i) = e i where b(i) := e i lcm{d : d | e i , d ≤ f i } (2) for 1 ≤ i ≤ m. Given this notation, it is worth noting the definition of commuting graph and two related results in [1]. These two theorems are extremely useful to prove the connectivity of commuting graphs C(G, X) in symmetric groups for elements of order three described in Section 2.…”

“…It shows an absolute contrast to the case having G as the symmetric group of degree 6, Sym(6) and X = (1, 2, 3) G , where C(G, X) is disconnected even though both cases deal with the set of elements of cycle type 3 1 as the vertex set. More specific examples of connected and disconnected commuting graphs can be referred from the results in Table 1 of [2] and Table 1 of [3]. Evidently, there are certain cases where C(G, X) is not connected or disconnected, which further motivate us to categorize them.…”

“…Commuting graphs have been studied for a large varieties of groups and subset of the groups [2,4]. Woodcock [5] analyzed commuting graphs C(G, X) for symmetric group, Sym(n) and having the set of nontrivial elements of the group as the vertex set.…”