Parametrizing Asymmetric Edge-Girth in Cayley Graphs via Affine Homomorphisms: Theory and Structural Obstructions

It is often implicitly assumed that vertex-transitive graphs exhibit uniform edge-girth — that every edge lies at the same distance from the graph's shortest cycles. We show this need not hold, and that the asymmetry can be deliberately parametrized. Using the affine group G = Fₚ ⋊ Fₚ*, we construct a family of Cayley graphs in which three translation generators fix an internal girth of 3, while a fourth, multiplicative generator b of order Q is conjectured to reach isolation depth exactly S (e) =Q once the underlying field is large enough. Extensive computational search (three independent implementations, including native C execution) confirms stabilization at S (e) =Q for Q∈6, 7, 8 across multiple primes. For Q=9, the same search — at volumes well beyond where Q=6, 7, 8 stabilize — consistently caps at S (e) =8. We tested two natural explanations for this gap (a shared-divisor "resonance" with the length-3 translation cycle, and a multiplicative-orbit-diversity criterion) and both are directly refuted by controlled experiments reported here. The Q=9 obstruction is left as an open problem. We also report that varying the number of translation generators (2, 3, or 4) never resolves the obstruction and that additional generators strictly degrade isolation, consistent with higher graph density uniformly shortening alternative paths.