Two-distance transitive normal Cayley graphs

Abstract

In this paper, we construct an infinite family of normal Cayley graphs, which are 2distance-transitive but neither distance-transitive nor 2-arc-transitive. This answers a question proposed by Chen, Jin and Li in 2019.

Two-geodesic transitive graphs of order $p^n$ with $n\leq3$

“…There are two infinite families of 2-geodesic but not 2-arc transitive normal Cayley graphs on E(p 3 ), of which one family is given in the following example which was first constructed in [20]. Another family will be constructed and studied in the next section.…”

“…Another motivation for us to consider 2-geodesic transitive graphs of order p n with p a prime and n ≤ 3 comes from the work in [6,20]. Before proceeding, we introduce the following definitions.…”

“…In [20], we answered this question in positive by constructing an infinite family of 2-distance transitive Cayley graphs of order a prime cube which are neither distance transitive nor 2-arc transitive. This family of graphs are also 2-geodesic transitive.…”

“…p } were first given in [20] while the graphs Cay(G, S) with S = {a i , b i , (b j ab j ) i | i, j ∈ Z * p } are a new family of 2-geodesic transitive but not 2-arc transitive graphs. The layout of this paper is as follows.…”