More constructions of vertex-transitive non-Cayley graphs based on counting closed walks

IS rel)re:se!otlng vertex-transitive as coset to allow us to obtain new results eral older results concerning the numbers transitive that are not 1 studied for more than a now, vertex-transitive graphs have quite great deal of activity. Their rich groups of automorphisms make from both the point of view of groups as well as of combinatorics. Among the most recent achievements, we can mention the search for vertex-transitive graphs that are edge-but not arc-transitive in [1], or the discovery of vertex-, but not arc-transitive with primitive automorphism groups in [14]. Our paper focuses on a slightly different problem. Aside from the notoriously known Cayley graphs, not many families of vertex-transitive graphs were known and studied before 1990 (as pointed out, for instance, by Watkins in [15], who also