Abstract
Characterizing regular covers of symmetric graphs is one of the fundamental topics in the field of algebraic graph theory, and is often a key step for approaching general symmetric graphs. Complete graphs, which are typical symmetric graphs, naturally appear in the study of many symmetric graphs as normal quotient graphs. In this paper, a characterization of edge-transitive cyclic covers of complete graphs with prime power order is given by using the techniques of finite group theory and the related properties of coset graphs. Certain previous results are generalized and some new families of examples are founded.
Highlights
Throughout the paper, by a graph, we mean a connected, undirected and simple graph with a valency of at least three.For a graph Γ, denote its vertex set, edge set and arc set by V Γ, EΓ and AΓ respectively
A characterization of edge-transitive cyclic covers of complete graphs with prime power order is given by using the techniques of finite group theory and the related properties of coset graphs
For an automorphism group X ≤ Aut Γ, Γ is called X-vertex-transitive, X-edge-transitive or X-arc-transitive, if X is transitive on V Γ, EΓ or AΓ, respectively; Γ is called X-locally-primitive if the vertex
Summary
Throughout the paper, by a graph, we mean a connected, undirected and simple graph with a valency of at least three. Numerous relative results have been obtained in the literature, see [2] [3] [4] [5] and reference therein for edge- or arc-transitive cyclic or abelian covers of graphs with small order. 2-arc-transitive cyclic, 2p - and 3p -covers with p a prime of complete graphs are determined in [7] [8]. The main purpose of this paper is to characterize edge-transitive cyclic covers of complete graphs with prime power order. A locally-primitive cyclic cover of a complete graph K pn with p an odd prime is isomorphic to K pn , pn − pnK2. Let Γ be a locally-primitive m -cover of the complete graph K p2 with p a prime.
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