Conjugate Graphs of Finite Groups

Abstract

In this paper we introduce the conjugate graph [Formula: see text] associated to a nonabelian group G with vertex set G\Z(G) such that two distinct vertices join by an edge if they are conjugate. We show if [Formula: see text], where S is a finite nonabelian simple group which satisfy Thompson's conjecture, then G ≅ S. Further, if central factors of two nonabelian groups H and G are isomorphic and |Z(G)| = |Z(H)|, then H and G have isomorphic conjugate graphs.

Spektrum Signless-Laplace dan Spektrum Detour Graf Konjugasi dari Grup Dihedral

“…Unsur g dan h di G dikatakan saling konjugasi jika ada x di G sehingga g = xhx -1 . Graf konjugasi dari grup G adalah graf yang memuat semua anggota grup G sebagai titik dan dua titik berbeda akan terhubung langsung jika keduanya saling konjugasi [8] [9].…”

“…Beberapa penelitian mengenai spektrum signless-Laplace dan spektrum detour sudah dilakukan, misalnya [5,7,[10][11][12]. Di lain pihak penelitian terkait graf konjugasi juga telah dilakukan, misalnya [8,9]. Meskipun demikian, belum ada yang mengkaji spektrum signless-Laplace dan spektrum detour graf konjugasi dari grup dihedral.…”

The probability that an element of a metacyclic 3-Group of negative type fixes a set and its orbit graph

“…Moreover, Erfanian and Tolue [20] introduced the conjugate graph in which two vertices of this graph are connected if they are conjugate. Later on, Omer et al [21] .…”

Generalized conjugate graph

“…Erfanian and Tolue [5] introduced a conjugate graph Γ associated to a non-abelian group with vertex set \ ( ) such that two distinct vertices are adjacent if they are conjugate, and the graph theoretical properties such as planarity, regularity and completeness of the conjugate graph were veri ed. The idea of de ning such a graph was obtained from the work of Blackburn, Britnell and Wildon [2] who presented ( ) as a probability that two elements of the group are conjugate.…”

“…So we may assume that is non-abelian. If = 1, then Γ ( ,1) coincides with the known conjugate graph denoted by Γ in [5]. Now assume that is the set of all subsets of with elements.…”