Abstract
Let G be a finite group and Cay( G, S) the Cayley graph of G with respect to S. A subset S is called a CI-subset if, for any T ⊂ G, Cay( G, S) ≅ Cay( G, T) implies S α = T for some α ∈ Aut( G). In this paper, we investigate the finite groups G in which every subset S with size at most m and (S⋋ = G is a CI-subset where m is a positive integer. As a corollary, we classify symmetric graphs of order p 3 and of valency 2 p where p is a prime.
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