Beyond Bass Collapse: New Irregular Edge-Space Invariants in Ihara Theory

Abstract: Let (G) be a finite simple graph and let (T) be its Hashimoto operator on the directed-edge space. We show that edge reversal induces a canonical symmetric/antisymmetric splitting under which (T) acquires an explicit (2\times 2) block form. The diagonal blocks are (\tfrac12 L(G)) and (-\tfrac12 A(G)), where (L(G)) is the line-graph adjacency and (A(G)) is the antisymmetric line-graph adjacency, while the off-diagonal block is the mixed incidence product (M=|D|^\top D). This identifies the ordinary and antisymmetric line-graph sectors as the two canonical diagonal sectors of Hashimoto theory and isolates a mixed sector linking them. A Schur-complement argument then gives a factorization [ \det(I-wT)=\det!\bigl(I-\tfrac w2 L(G)\bigr)\,C_G(w), ] where (C_G(w)) is an explicit correction determinant built from the antisymmetric and mixed sectors. We show that the trivial roots (w=\pm1) localize on canonical edge subspaces, and that for line-graph-cospectral pairs all remaining Ihara separation is forced into the correction sector. Although the raw mixed block (M) depends on edge orientation, its natural gauge-invariant shadows, including (MM^\top), (M^\top M), and (M^\top L^kM), define a canonical matrix package attached to the graph. In the regular case these collapse to adjacency-side data, but in the irregular case they need not. As an application, we exhibit irregular non-isomorphic graphs that are adjacency-cospectral and line-graph-cospectral yet are separated by the correction sector, and we find further examples where the gauge-invariant mixed shadows separate even when the scalar Ihara polynomial does not. This isolates new irregular edge-space invariants in Hashimoto--Ihara theory.