Counterexamples to two conjectures about distance sequences

Abstract

It is shown that, contrary to a pair of well-known conjectures, there exist finite and infinite examples of: (1) vertex-transitive graphs whose distance sequences are not unimodal, and (2) graphs with primitive automorphism group whose distance sequences are not logarithmically convex. In particular, a family of finite graphs is presented whose automorphism groups are primitive and whose distance sequences are not unimodal. © 1987.