On diameter two Cayley graphs

Let X be a Cayley graph whose diameter is 2. Set R := Aut(X) and w is an element of V(X). In this paper, it is shown that: for every positive integer m at least 6, there is a such Cayley graph X of m points such that R-w acts transitively in X-2(w ) but not in X(w); for every positive integer k at least 3, there is a such graph X of valency k such that Rw is transitive in X-2(w ) but not in X(w). (C) 2022 Elsevier Inc. All rights reserved.

Automated summary

This paper investigates the properties of Cayley graphs and proves that there exist Cayley graphs satisfying certain conditions for specific positive integers m and k, where the automorphism group exhibits different levels of transitivity on different subsets.