Singular Distance Powers of Circuits

Abstract

In this work 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.