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

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 $\frac{1}{2}L(G)$ and $-\frac{1}{2}A(G)$, 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 $M=\mathrm{\mid }D{\mathrm{\mid }}^{\mathrm{\top }}D$. 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 $$\det (I-wT)=\det \text{\hspace{0.17em}⁣}(I-\frac{w}{2}L(G))C_G(w),$$ where $C_G(w)$ is an explicit correction determinant built from the antisymmetric and mixed sectors. We show that the trivial roots $w=\pm 1$ 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 $M{M}^{\mathrm{\top }}$, $M^{\mathrm{\top }}M$, and $M^{\mathrm{\top }}{L}^{k}M$, 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.