Vertex-transitive Haar graphs that are not Cayley graphs

TL;DR

This paper constructs an infinite family of vertex-transitive Haar graphs that are not Cayley graphs, providing explicit examples and addressing a question in algebraic graph theory.

Contribution

It introduces the first known infinite family of such graphs, including the smallest example with 40 vertices, expanding understanding of graph symmetry classes.

Findings

Constructed an infinite family of non-Cayley vertex-transitive Haar graphs.

Identified the smallest example as a 40-vertex graph, the Kronecker cover of the dodecahedron.

Demonstrated existence of these graphs in the context of algebraic graph theory.

Abstract

In a recent paper (arXiv:1505.01475 ) Est\'elyi and Pisanski raised a question whether there exist vertex-transitive Haar graphs that are not Cayley graphs. In this note we construct an infinite family of trivalent Haar graphs that are vertex-transitive but non-Cayley. The smallest example has 40 vertices and is the well-known Kronecker cover over the dodecahedron graph $G(10,2)$, occurring as the graph $40$ in the Foster census of connected symmetric trivalent graphs.