Cayley-type graphs for group-subgroup pairs

TL;DR

This paper introduces a new class of Cayley-type graphs for group-subgroup pairs, exploring their properties, eigenvalues, and conditions under which they form Ramanujan graphs, with examples illustrating the results.

Contribution

It defines Cayley-type graphs for group-subgroup pairs and analyzes their properties, including conditions for Ramanujan graphs, expanding understanding of algebraic graph structures.

Findings

Connectedness, degree, and vertex-transitivity are characterized.

Eigenvalues, including the largest, are determined from group properties.

Conditions for Ramanujan graphs are established, with counterexamples provided.

Abstract

In this paper we introduce a Cayley-type graph for group-subgroup pairs and present some elementary properties of such graphs, including connectedness, their degree and partition structure, and vertex-transitivity. We relate these properties to those of the underlying group-subgroup pair. From the properties of the group, subgroup and generating set some of the eigenvalues can be determined, including the largest eigenvalue of the graph. In particular, when this construction results in a bipartite regular graph we show a sufficient condition on the size of the generating sets that results on Ramanujan graphs for a fixed group-subgroup pair. Examples of Ramanujan pair-graphs that do not satisfy this condition are also provided, to show that the condition is not necessary.