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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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##### References:
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