GI-graphs and their groups

TL;DR

This paper extends the class of generalized Petersen graphs to GI-graphs, classifies their automorphism groups, and identifies which are vertex-transitive or Cayley graphs, including a notable unit-distance drawing of GI(7;1,2,3).

Contribution

It introduces GI-graphs as a broader class, characterizes their automorphism groups, and determines conditions for vertex-transitivity and Cayley graph properties.

Findings

Automorphism groups of GI-graphs are classified.

Criteria for GI-graphs being vertex-transitive or Cayley graphs are established.

A specific GI-graph (GI(7;1,2,3)) is depicted with a unit-distance drawing.

Abstract

The class of generalized Petersen graphs was introduced by Coxeter in the 1950s. Frucht, Graver and Watkins determined the automorphism groups of generalized Petersen graphs in 1971, and much later, Nedela and \v{S}koviera and (independently) Lovre\v{c}i\v{c}-Sara\v{z}in characterised those which are Cayley graphs. In this paper we extend the class of generalized Petersen graphs to a class of GI-graphs. For any positive integer n and any sequence j_0,j_1,....,j_{t-1} of integers mod n, the GI-graph GI(n;j_0,j_1,....,j_{t-1}) is a (t+1)-valent graph on the vertex set Z_t x Z_n, with edges of two kinds: - an edge from (s,v) to (s',v), for all distinct s,s' in Z_t and all v in Z_n, - edges from (s,v) to (s,v+j_s) and (s,v-j_s), for all s in Z_t and v in Z_n. By classifying different kinds of automorphisms, we describe the automorphism group of each GI-graph, and determine which GI-graphs are vertex-transitive and which are Cayley graphs. A GI-graph can be edge-transitive only when t < 4 or equivalently, for valence at most 4. We present a unit-distance drawing of a remarkable GI(7;1,2,3).