A gcd-graph is a Cayley graph over a finite abelian group defined by greatest common divisors. Such graphs are known to have integral spectrum. A non-complete extended ▫$p$▫-sum, or NEPS in short, is ...well-known general graph product. We show that the class of gcd-graphs and the class of NEPS of complete graphs coincide. Thus, a relation between the algebraically defined Cayley graphs and the combinatorially defined NEPS of complete graphs is established. We use this link to show that gcd-graphs have a particularly simple eigenspace structure, to be precise, that every eigenspace of the adjacency matrix of a gcd-graph has a basis with entries ▫$-1,0,1$▫ only.
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