The relationship between the spectrum and the automorphism group of a graph is probed with the aid of the theory of finite group representations. Three related topics are explored: l) graphs with non-derogatory adjacency matrix, 2) point-symmetric graphs, and 3) an algorithm for constructing the automorphism group of a prime, point-symmetric graph. First, we give an upper bound on the order of the automorphism group of a graph with non-derogatory adjacency matrix; and show, in a special case, that the degree of each irreducible factor of the minimal polynomial has a natural interpretation in terms of the automorphism group. Second, we prove that the degree of the minimal polynomial of a point-symmetric graph is bounded above by the number of orbits of the stabilizer of any given element. For point-symmetric graphs with a prime number of points, we exhibit a formula linking the degree of the minimal polynomial with the order of the group. Finally, we give a simple algorithm for constructing the automorphism group of a point-symmetric graph with a prime number of points.
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