The connectivity of commuting graphs

Abstract

A necessary and sufficient condition is proven for the connectivity of commuting graphs C(G, X), where G is Sym(n), the symmetric group of degree n, and X is any G-conjugacy class.

A novel method to construct NSSD molecular graphs

“…Recall that in a commuting graph, two elements are joined by an edge if they commute with each other. For further study of commuting graphs see [17][18][19][20][21][22]. Let J be a non-empty set.…”

On resolvability of a graph associated to a finite vector space

“…If V be a n-dimensional vector space over a finite field F with q elements, then the order of Γ(V) is q n − 1 and the size of Γ(V) is q 2n − q n + 1 − (2q − 1) n 2 . The study of graphs associated to algebraic structures has, recently, got much attention from researchers to expose the relation between algebra and graph theory: zero divisor graph associated to a commutative ring was discussed in [1,3]; commuting graphs for groups were studied in [2,4,17]; power graphs for groups and semigroups were focused in [5,8,21]; the papers [13,25] are devoted to investigate intersection graphs assigned to a vector space; and so on.…”

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

“…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.…”

“…Theorem 1. [1] Let G = Sym(n), a ∈ G be of cycle type e f , and X = a G . Then, C(G, X) is connected if and only if b(1) = 1, or e ≤ 3 and f = 1.…”