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Some properties of unitary Cayley graphs. (English) Zbl 1121.05059

Summary: The unitary Cayley graph \(X_n\) has vertex set \(Z_n=\{0,1, \dots ,n-1\}\). Vertices \(a,~b\) are adjacent, if gcd\((a-b,n)=1\). For \(X_n\), the chromatic number, the clique number, the independence number, the diameter and the vertex connectivity are determined. We decide on the perfectness of \(X_n\) and show that all nonzero eigenvalues of \(X_n\) are integers dividing the value \(\varphi(n)\) of the Euler function.

MSC:

05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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