Generalized cayley graphs and group structure: Insights from the direct products of P₂ and C₃
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Keywords:
Cayley graph, Caym (Ψ, S).Abstract
This work introduces a generalization of Cayley graphs, denoted Caym(ψ, S), where ψ is a finite group and S is a non-empty subset of ψ. In this construction, vertices are represented by m-dimensional column vectors with entries in ψ, and adjacency is determined by a matrix-based condition involving the inverse elements and a matrix of elements from S. We focus on elucidating the structure and fundamental properties of Caym(ψ, S) when the classical Cayley graph Cay(ψ, S) corresponds to the direct products P2 × P2
and P2 × C2. Through rigorous analysis, we reveal distinct structural characteristics arising from these specific group structures and their associated generating sets. The generalized Cayley graph, denoted as Caym(ψ, S), is a graph where the vertex set consists of all column matrices Xm, with each matrix having elements from the set Ψ. Two vertices Xm and Ym are adjacent if and only if Ym
−1, the inverse of Ym, is a column matrix where each entry corresponds to the inverse of the associated element in Ψ. Our findings provide valuable insights into the interplay between algebraic properties of groups and the topological features of their generalized Cayley graph representations. This study contributes to a deeper understanding of generalized Cayley graphs and their potential
applications in diverse fields such as network theory, coding theory, and cryptography.
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