## Abstract

In this paper we introduce the graph Γ

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

_{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*∈*C*and y ∈*D*such that*x*and*y*generate a solvable group.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. We pose some problems on the relation of the SCC-graph to the solvable graph and to the NCC-graph, which we cannot solve.Original language | English |
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Article number | 113467 |

Number of pages | 8 |

Journal | Discrete Mathematics |

Volume | 346 |

Issue number | 8 |

Early online date | 20 Apr 2023 |

DOIs | |

Publication status | Published - 1 Aug 2023 |

## Keywords

- Clique number
- Non-solvable group
- Graph
- Girth
- Conjugacy class