Arc transitive covering digraphs and their eigenvalues

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Abstract
We prove that every finite regular digraph has an arc‐transitive covering digraph (whose arcs are equivalent under automorphisms) and every finite regular graph has a 2‐arc‐transitive covering graph. As a corollary, we sharpen C. D. Godsil's results on eigenvalues and minimum polynomials of vertex‐transitive graphs and digraphs. Using Godsil's results, we prove, that given an integral matrix A there exists an arc‐transitive digraph X such that the minimum polynomial of A divides that of X. It follows that there exist arc‐transitive digraphs with nondiagonalizable adjacency matrices, answering a problem by P. J. Cameron. For symmetric matrices A, we construct a 2‐arc‐transitive graphs X.

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