Abstract
There exist several generalizations of the classical Dirichlet convolution, for instance the so-called A-convolutions analyzed by Narkiewicz. We shall connect the concept of A-convolutions satisfying a weak form of regularity and Ramanujan sums with the spectrum of integral circulant graphs. These generalized Cayley graphs, having circulant adjacency matrix and integral eigenvalues, comprise a great amount of arithmetical features. By use of our concept we obtain a multiplicative decomposition of the so-called energy of integral circulant graphs with multiplicative divisor sets. This will be fundamental for the study of open problems, in particular concerning the detection of integral circulant graphs with maximal or minimal energy.
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