Small vertex-transitive and Cayley graphs of girth six and given degree: an algebraic approach

Abstract

Abstract:We examine the existing constructions of the smallest known vertex-transitive graphs of a given degree and girth 6. It turns out that most of these graphs can be described in terms of regular lifts of suitable quotient graphs. A further outcome of our analysis is a precise identification of which of these graphs are Cayley. We also investigate higher level of transitivity of the smallest known vertex-transitive graphs of a given degree and girth 6 and relate their constructions to near-difference sets. ᭧

Distance-regular Cayley graphs with least eigenvalue $$-2$$ - 2

“…For q > 1, the classical construction of a projective plane of order q uses the finite field GF(q) and gives the so-called Desarguesian plane of order q. We note that Loz et al [21] showed that the (distance-regular) incidence graph of a Desarguesian plane is a Cayley graph. Here we will consider the line graph, however.…”

Distance-regular Cayley graphs with small valency

“…Because the Desarguesian projective plane (over GF(q)) is a symmetric 2-(q 2 + q + 1, q + 1, 1) design, and can be obtained from a (Singer) difference set in the cyclic group, it follows that the incidence graph of a Desarguesian projective plane is a Cayley graph. It was shown by Loz et al [18] that this Cayley graph is 4-arc-transitive. We note that all projective planes of order at most 8 are Desarguesian, and hence all incidence graphs of projective planes with valency at most 9 are Cayley graphs.…”

Power graphs: A survey

“…Cayley graphs of semigroups are closely related to automata theory, as explained in the monograph [37] and papers [35,39,45]. This is a very large topic, and so without trying to be complete we indicate only a few pertinent papers [20,34,38,58,59,61,62,63,64,65,66,67,68,71,72,80,81] just to illustrate.…”