Solvable conjugacy class graph of groups
Parthajit Bhowal, Peter J. Cameron, Rajat Kanti Nath, Benjamin Sambale

TL;DR
This paper introduces the solvable conjugacy class graph (SCC-graph) for groups, analyzing its properties and classifying finite groups with specific clique numbers, notably providing a complete list for clique number 2.
Contribution
It defines the SCC-graph for groups, studies its properties, and classifies finite groups with a given clique number, especially for clique number 2.
Findings
Finite groups with a given SCC-graph clique number are finite in number.
Explicit classification of groups with SCC-graph clique number 2.
Properties such as connectivity, girth, and clique number of SCC-graphs are analyzed.
Abstract
In this paper we introduce the graph associated with a group , called the solvable conjugacy class graph (abbreviated as SCC-graph), whose vertices are the nontrivial conjugacy classes of and two distinct conjugacy classes are adjacent if there exist and such that 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~, and we find explicitly the list of such groups with .
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