On Word and Gómez Graphs and Their Automorphism Groups in the Degree Diameter Problem

Abstract

One of the prominent areas of research in graph theory is the degree-diameter problem, in which we seek to determine how many vertices a graph may have when constrained to a given degree and diameter. Different variants of this problem are obtained by considering further restrictions on the graph, such as whether it is directed, undirected, mixed, vertex-transitive etc. The currently best known extremal constructions are the Gómez graphs, both when considered as directed and undirected. As the Gómez graphs are similar to the Faber-Moore-Chen graphs, we give a natural generalisation of their definition which we call word graphs. We then prove that the Gómez graphs are as large as possible word graphs for given degree and diameter. Further, we provide a test to determine the full automorphism group of a word graph, and apply it to the extremal directed Gómez graphs to settle the previously open question of when the Gómez graphs are also Cayley graphs. Finally, we conclude with a brief list of interesting related open problems.