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 -