We study supersingular \(\ell\) -isogeny graphs as finite regular multigraphs through the lens of non-backtracking spectral graph theory. Using the Hashimoto operator, we show that traces of its powers count cyclically non-backtracking closed isogeny chains, and we derive an exact Möbius inversion formula for the associated primitive cycle numbers. This leads to a finite primitive cycle certificate for supersingular isogeny graphs, invariant under graph isomorphism and computable from the oriented-edge data or, equivalently, from the Ihara–Bass determinant formula. The resulting framework provides a rigorous discrete and algorithmic invariant for measuring primitive cycle structure in arithmetic regular graphs.
