Isomorphic tetravalent cyclic Haar graphs

TL;DR

This paper establishes necessary and sufficient conditions for determining when two connected cyclic Haar graphs of valency 4 are isomorphic, advancing understanding of their structural classification.

Contribution

It provides a complete characterization of isomorphism conditions for connected cyclic Haar graphs with degree 4, filling a gap in graph theory.

Findings

Derived necessary and sufficient conditions for isomorphism.

Characterized structural properties of valency 4 Haar graphs.

Enhanced classification framework for cyclic Haar graphs.

Abstract

Let $S$ be a subset of the cyclic group $\Z_n$. The cyclic Haar graph $H(\Z_n,S)$ is the bipartite graph with color classes $\Z_n^+$ and $\Z_n^-,$ and edges $\{x^+,y^-\},$ where $x,y \in \Z_n$ and $y - x \in S$. In this paper we give sufficient and necessary conditions for the isomorphism of two connected cyclic Haar graphs of valency 4.