TY - UNPB

T1 - Solvable conjugacy class graph of groups

AU - Bhowal, Parthajit

AU - Cameron, Peter J.

AU - Nath, Rajat Kanti

AU - Sambale, Benjamin

PY - 2021/12/5

Y1 - 2021/12/5

N2 - In this paper we introduce the graph \(\Gamma_{sc}(G)\) associated with a group \(G\), called the solvable conjugacy class graph (abbreviated as SCC-graph), whose vertices are the nontrivial conjugacy classes of \(G\) and two distinct conjugacy classes \(C, D\) are adjacent if there exist \(x \in C\) and \(y \in D\) such that \(\langle x, y\rangle\) is solvable. We discuss the connectivity, girth, clique number, and several other properties of the SCC-graph. One of our results asserts that there are only finitely many finite groups whose SCC-graph has given clique number~\(d\), and we find explicitly the list of such groups with \(d=2\).

AB - In this paper we introduce the graph \(\Gamma_{sc}(G)\) associated with a group \(G\), called the solvable conjugacy class graph (abbreviated as SCC-graph), whose vertices are the nontrivial conjugacy classes of \(G\) and two distinct conjugacy classes \(C, D\) are adjacent if there exist \(x \in C\) and \(y \in D\) such that \(\langle x, y\rangle\) is solvable. We discuss the connectivity, girth, clique number, and several other properties of the SCC-graph. One of our results asserts that there are only finitely many finite groups whose SCC-graph has given clique number~\(d\), and we find explicitly the list of such groups with \(d=2\).

KW - math.CO

KW - math.GR

KW - 05C25

M3 - Preprint

BT - Solvable conjugacy class graph of groups

ER -