WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 23, 2024
Conjugate Graph and Conjugacy Class Graph related to Direct Product of Dihedral Groups
Authors: ,
Abstract: Let $$G$$ be a non-abelian group. The conjugate graph of $$G$$ is a graph whose vertices are the non-central elements of $$G$$ and two vertices are adjacent if they belong to the same conjugacy class. The conjugacy class graph of $$G$$ is a graph whose vertices are the non-central conjugacy classes of $$G$$ and two vertices a,b are adjacent if $$gcd(|a|, |b|)$$ is greater than one. In this paper, we explore the structures of these graphs for the groups $$D_n×D_m$$ for odd and even values of $$n$$ and $$m$$. The chromatic number and independence number of the conjugate graphs, their line graphs and complement graphs are found. We discuss various graph parameters like the existence of Eulerian and Hamiltonian cycles, planarity, connectedness, chromatic number, clique number, independence number, and diameter of the conjugacy class graphs of these groups.
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Keywords: Conjugate graph, conjugacy class graph, line graph, complement graph, direct product, non-abelian group, dihedral group
Pages: 458-466
DOI: 10.37394/23206.2024.23.48