On a relationship between Cayley graphs and G-graphs with some applications

Abstract

Like Cayley graphs, G-graphs are graphs that are constructed from groups but they correspond to alternative constructions. The purpose of this article is to study the connection between these two types of graphs. Such a connection opens up a possible pathway between these two theories and thus investigating certain problems from one of these areas might be easier to tackle when dealt with them as problems in the other. First, we show the existence of a link that connects classes of these two types of graphs, and then we investigate the implications of this result on certain open problems in the theory of Cayley graphs. In particular, we show that computing the spectra of a certain infinite family of Cayley graphs can be easily realized via the use of G-graphs. In the process, general results concerning G-graphs and the spectra of a hypergraph are presented. Finally, we use a certain graph operation to present a new alternative tool for constructing integral graphs.