Edge-transitivity of Cayley graphs generated by transpositions

TL;DR

This paper establishes a precise correspondence between the isomorphism and edge-transitivity properties of transposition graphs and their associated Cayley graphs of the symmetric group, deepening understanding of their structural symmetries.

Contribution

It proves that transposition graphs are isomorphic if and only if their Cayley graphs are, and that edge-transitivity of the transposition graph is equivalent to that of the Cayley graph.

Findings

Transposition graphs are isomorphic iff their Cayley graphs are isomorphic.

Edge-transitivity of transposition graphs is equivalent to edge-transitivity of Cayley graphs.

Provides a characterization linking graph symmetries to algebraic properties of the generating set.

Abstract

Let $S$ be a set of transpositions generating the symmetric group $S_n$. The transposition graph of $S$ is defined to be the graph with vertex set $\{1,\ldots,n\}$, and with vertices $i$ and $j$ being adjacent in $T(S)$ whenever $(i,j) \in S$. In the present note, it is proved that two transposition graphs are isomorphic if and only if the corresponding two Cayley graphs are isomorphic. It is also proved that the transposition graph $T(S)$ is edge-transitive if and only if the Cayley graph $Cay(S_n,S)$ is edge-transitive.