Distribution of cycles in supersingular $\ell$-isogeny graphs

Recent work by Arpin, Chen, Lauter, Scheidler, Stange, and Tran counted the number of cycles of length $r$ in supersingular $\ell$-isogeny graphs. In this paper, we extend this work to count the number of cycles that occur along the spine. We provide formulas for both the number of such cycles, and the average number as $p \to \infty$, with $\ell$ and $r$ fixed. In particular, we show that when $r$ is not a power of $2$, cycles of length $r$ are disproportionately likely to occur along the spine. We provide experimental evidence that this result holds in the case that $r$ is a power of $2$ as well.