Circulants and the Characterization of Vertex-Transitive Graphs

Abstract

In this paper, we extend the notion of a circulant to a broader class of vertex.transitive graphs, which we call multidimensional circulants. This new class of graphs is shown to consist precisely of those vertextransitive graphs with an automorphism group containing a regular abelian subgroup. The result is proved using a theorem of Sabidussi which shows how to recover any vertex-transitive graph from any transitive subgroup of its automorphism group. The approach also allows a short proof of Turner's theorem that every vertex-transitive graph on a prime number of nodes is a circulant.

On the Decomposition of Vertex-Transitive Graphs into Multicycles

“…PROOF: We know from [7] that every vertex-transitive graph with a prime number of nodes is a circulant and that the automorphism group of a multidimensional circulant contains a regular abelian subgroup. Thus, such graphs are Cayley graphs.C There are some vertex-transitive graphs, however, with automorphism groups which do not contain a regular subgroup.…”

“…In [7], we show that X is a multidimensional circulant if and only if G(X) contains a regular abelian subgroup. Since G(X )= G(X), we conclude that X is also a multidimensional circulant.…”

“…For such a graph, the pair (n,S) is called a symbol for X. In [7], we generalize this concept and define a multidimensional circulant to be a graph X whose nodes can be labeled so that there exist an integer k, a k-vector a, and a set SCZ& such that V(X)= In this paper, we investigate the decomposition of vertex-transitive graphs into edge-disjoint symmetric graphs. In particular, we prove in section 2 that every vertex-transitive graph can be expressed as the edge-disjoint union of symmetric graphs.…”

“…This paper and the paper preceding were written while he served as a summer intern at NBS and in part while he was an undergraduate at Princeton University working under the guidance of Professor Stephen B. Of particular interest is the class of circulants [1,2,7,8,10,11] and a generalization thereof, the class of multidimensional circulants [7,8]. A circulant or, equivalently, a starred polygon is a graph whose nodes can be labeled so that there exists a set SCZ, where Z is the set of integers, such that V(X)={vO,,..., v-} and E(X)={ (v,vi)IO<ij<n and mod(i-jn)ES}.…”

“…Following the notation of [7,9],' we denote the set of nodes of a finite, simple graph X by V(X), the set of edges by E(X), and the automorphism group of X by G(X). Throughout, we regard G(X) as a permutation group on the nodes, and sometimes the edges, of X.…”