On generalized truncations of complete graphs

For a k-regular graph Gamma and a graph Upsilon of order k, a generalized truncation of Gamma by Upsilon is constructed by replacing each vertex of Gamma with a copy of Upsilon. E. Eiben, R. Jajcay and P. Sparl 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.