Cyclic Permutability Graph of a Group.

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Bibliographic Details
Title:	Cyclic Permutability Graph of a Group.
Authors:	Sathiya, C.¹ csathiyamaths@gmail.com, Devi, P.² pdevigri@gmail.com
Source:	IAENG International Journal of Applied Mathematics. Feb2026, Vol. 56 Issue 2, p796-803. 8p.
Subjects:	Finite groups, Graph theory, Graph connectivity, Nonabelian groups, Cyclic groups, Representations of graphs
Abstract:	Let G be a finite group. The cyclic permutability graph of G, denoted by G(G), is defined as the graph whose vertex set consists of all non-trivial elements of G, where two distinct vertices a and b are adjacent if and only if the product ab forms a subgroup of G. This graph-theoretic framework provides valuable insights into the structural behaviour of groups through their cyclic subgroups. In this paper, we present a detailed classification of all finite groups for which the cyclic permutability graph G(G) assumes specific well-known graph structures such as a complete graph, a totally disconnected graph, a cycle, a path, a bipartite graph, a complete bipartite graph, and a star graph. We also determine the girth of G(G)--the length of its shortest cycle--for certain classes of finite non-abelian groups. Furthermore, we compute the adjacency spectrum and the Wiener index of G(G) for selected finite non-abelian groups, providing quantitative measures of their connectivity and structural complexity. In addition, we investigate the planarity and toroidality of G(G), identifying the conditions under which this graph can be embedded in the plane or on the torus without edge crossings. These results contribute to a deeper understanding of the interplay between group theory and graph theory, particularly in the context of cyclic subgroup structures. [ABSTRACT FROM AUTHOR]
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Database:	Engineering Source

Description
Abstract:	Let G be a finite group. The cyclic permutability graph of G, denoted by G(G), is defined as the graph whose vertex set consists of all non-trivial elements of G, where two distinct vertices a and b are adjacent if and only if the product ab forms a subgroup of G. This graph-theoretic framework provides valuable insights into the structural behaviour of groups through their cyclic subgroups. In this paper, we present a detailed classification of all finite groups for which the cyclic permutability graph G(G) assumes specific well-known graph structures such as a complete graph, a totally disconnected graph, a cycle, a path, a bipartite graph, a complete bipartite graph, and a star graph. We also determine the girth of G(G)--the length of its shortest cycle--for certain classes of finite non-abelian groups. Furthermore, we compute the adjacency spectrum and the Wiener index of G(G) for selected finite non-abelian groups, providing quantitative measures of their connectivity and structural complexity. In addition, we investigate the planarity and toroidality of G(G), identifying the conditions under which this graph can be embedded in the plane or on the torus without edge crossings. These results contribute to a deeper understanding of the interplay between group theory and graph theory, particularly in the context of cyclic subgroup structures. [ABSTRACT FROM AUTHOR]
ISSN:	19929978