A Note on Constructing Large Cayley Graphs of Given Degree and Diameter by Voltage Assignments

Abstract

Voltage graphs are a powerful tool for constructing large graphs (called lifts) with prescribed properties as covering spaces of small base graphs. This makes them suitable for application to the degree/diameter problem, which is to determine the largest order of a graph with given degree and diameter.

Many currently known largest graphs of degree $\le 15$ and diameter $\le 10$ have been found by computer search among Cayley graphs of semidirect products of cyclic groups. We show that all of them can in fact be described as lifts of smaller Cayley graphs of cyclic groups, with voltages in (other) cyclic groups. This opens up a new possible direction in the search for large vertex-transitive graphs of given degree and diameter.