A family of edge-transitive Cayley graphs
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Edge-transitive graphs of order a prime or a product of two distinct primes with any positive integer valency, and of square-free order with valency at most 7 have been classified by a series of papers. In this paper, a complete classification is given of edge-transitive Cayley graphs of square-free order with valency less than the smallest prime divisor of the order. This leads to new constructions of infinite families of both arc-regular Cayley graphs and edge-regular Cayley graphs (so half-transitive). Also, as by-products, it is proved that, for any given positive integers \(k,s\ge 1\) and \(m,n\ge 2\), there are infinitely many arc-regular normal circulants of valency 2k and order a product of s primes, and there are infinitely many edge-regular normal metacirculants of valency 2m and order a product of n primes; such arc-regular and edge-regular examples are also specifically constructed.
KeywordsEdge-transitive graph Half-transitive graph Arc-regular graph Edge-regular graph Normal Cayley graph
Mathematics Subject Classification20B15 20B30 05C25
The authors are very grateful to the referee for the helpful comments.
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