Finite 2‐Distance Transitive Graphs

Abstract

A noncomplete graph Γ is said to be (G, 2)‐distance transitive if G is a subgroup of the automorphism group of Γ that is transitive on the vertex set of Γ, and for any vertex u of Γ, the stabilizer Gu is transitive on the sets of vertices at distances 1 and 2 from u. This article investigates the family of (G, 2)‐distance transitive graphs that are not (G, 2)‐arc transitive. Our main result is the classification of such graphs of valency not greater than 5. We also prove several results about (G, 2)‐distance transitive, but not (G, 2)‐arc transitive graphs of girth 4.

Two-distance transitive normal Cayley graphs

“…These graphs are 2-distance transitive but are neither distance transitive nor 2-arc transitive. The extensive study of 2-distance transitive graphs has gained momentum in recent years, as evidenced by works such as those cited in [6,8,18,21,22]. This paper aims to contribute to this evolving topic.…”

“…By definition, a 2-arc-transitive graph is 2-distance-transitive, but a 2-distance-transitive graph may not be 2-arc-transitive; an example is the Kneser graph KG 6,2 , see [16]. Furthermore, Corr, Jin and Schneider [4] investigated properties of a connected (G, 2)-distance-transitive but not (G, 2)-arc-transitive graph of girth 4, and they applied the properties to classify such graphs with prime valency. For more information about 2-distance-transitive graphs, we refer to [6,7].…”

Finite Two-Distance-Transitive Dihedrants

“…Devillers et al [7] studied the class of locally s -distance-transitive graphs using the normal quotient strategy developed for s -arc-transitive graphs in [29]. Corr et al [6] investigated the family of -distance-transitive graphs, and they determined the family of -distance-transitive but not -arc-transitive graphs of valency at most five. Then the authors [21] gave a classification of the class of -distance-transitive but not -arc-transitive graphs of valency six.…”

Two-distance transitive normal Cayley graphs

“…By definition, a 2-arc-transitive graph is 2-distance-transitive, but a 2-distance-transitive graph may not be 2-arc-transitive; a simple example is the complete multipartite graph K 3,2 . Furthermore, Corr, Jin and Schneider [4] investigated properties of a connected (G, 2)-distance-transitive but not (G, 2)-arc-transitive graph of girth 4, and they applied the properties to classify such graphs with prime valency. For more information about 2-distance-transitive graphs, we refer to [6,7].…”