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Singular distance powers of circuits. (English) Zbl 1172.05040
Summary: A precise condition for the singularity of a circuit distance power \(C_n^{(d)}\) is derived. Namely, either \(n\) and \(d\) are not relatively prime or the order of 2 in \(d+1\) is strictly smaller than in \(n\). It is also shown that the simple eigenvalues of circuit distance powers are contained in \(\{-2,0,2d\}\), generalizing a well-known result for circuits. Further, the nullity of \(C_n^{(d)}\) is calculated.
MSC:
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
15A18 Eigenvalues, singular values, and eigenvectors
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