2013/02/28 by Alison Thomson, Sanming Zhou · 4 citations
Computer Science · Engineering · Mathematics · #Automorphism #Bijection #Cayley graph #Circulant graph #Circulant matrix #Combinatorics #Discrete mathematics #Finite Group Theory Research #Frobenius group #Graph #Interconnection Networks and Systems #Line graph #Mathematics #Permutation (music) #Permutation group #Product (mathematics) #Vertex (graph theory) #Voltage graph #Wreath product #graph theory and CDMA systems #math.CO #msc:05C25 #msc:68M10 #msc:68R10
paper · pdf · doi:10.1016/j.dam.2013.08.044
published in Discrete Applied Mathematics 162, 296-305 (Elsevier BV) · Final version
openalex publication_date 2013/09/23 · arxiv created 2018/09/26 · arxiv updated 2018/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
A Frobenius group is a transitive permutation group which is not regular but only the identity element can fix two points. Such a group can be expressed as the semi-direct product G = K \rtimes H of a nilpotent normal subgroup K and another group H fixing a point. A first-kind G-Frobenius graph is a connected Cayley graph on K with connection set an H-orbit aH on K that generates K, where H has an even order or a is an involution. It is known that the first-kind Frobenius graphs admit attractive routing and gossiping algorithms. A complete rotation in a Cayley graph on a group G with connection set S is an automorphism of G fixing S setwise and permuting the elements of S cyclically. It is known that if the fixed-point set of such a complete rotation is an independent set and not a vertex-cut, then the gossiping time of the Cayley graph (under a certain model) attains the smallest possible value. In this paper we classify all first-kind Frobenius circulant graphs that admit complete rotations, and describe a means to construct them. This result can be stated as a necessary and sufficient condition for a first-kind Frobenius circulant to be 2-cell embeddable on a closed orientable surface as a balanced regular Cayley map. We construct a family of non-Frobenius circulants admitting complete rotations such that the corresponding fixed-point sets are independent and not vertex-cuts. We also give an infinite family of counterexamples to the conjecture that the fixed-point set of every complete rotation of a Cayley graph is not a vertex-cut.