Twists and Twistability

TL;DR

This paper introduces twisted automorphisms in metrically homogeneous graphs, characterizes their associated permutations, and links these findings to known results in association schemes, expanding understanding of graph symmetries.

Contribution

It defines twisted automorphisms for metrically homogeneous graphs and characterizes the permutations involved, connecting graph automorphisms to association scheme permutations.

Findings

Identified all permutations associated with twisted automorphisms.

Characterized classes of graphs allowing twisted isomorphisms for each permutation.

Linked permutations to those found in finite association schemes.

Abstract

Metrically homogeneous graphs are connected graphs which, when endowed with the path metric, are homogeneous as metric spaces. In this paper we introduce the concept of twisted automorphisms, a notion of isomorphism up to a permutation of the language. We find all permutations of the language which are associated with twisted automorphisms of metrically homogeneous graphs. For each non-trivial permutation of this type we also characterize the class of metrically homogeneous graphs which allow a twisted isomorphism associated with that permutation. The permutations we find are, remarkably, precisely those found by Bannai and Bannai in an analogous result in the context of finite association schemes.