On generalized truncations of complete graphs

For a k -regular graph Γ and a graph Υ of order k , a generalized truncation of Γ by Υ is constructed by replacing each vertex of Γ with a copy of Υ . E. Eiben, R. Jajcay and P.  S parl introduced a method for constructing vertex-transitive generalized truncations. For convenience, we call a graph obtained by using Eiben et al. ’s method a special generalized truncation . In their paper, Eiben et al.  proposed a problem to classify special generalized truncations of a complete graph K n by a cycle of length n  − 1 . In this paper, we completely solve this problem by demonstrating that with the exception of n  = 6 , every special generalized truncation of a complete graph K n by a cycle of length n  − 1 is a Cayley graph of AGL(1, n ) where n is a prime power. Moreover, the full automorphism groups of all these graphs and the isomorphisms among them are determined.