Odd-cycle defects in the Alon-Friedland bound

Let $G$ be a finite simple graph. We study perfect matchings through two complementary viewpoints: reductions to bipartite permanent terms and the directed cycle covers counted by the ordinary adjacency permanent. The main identity is an explicit odd-cycle-indexed form of the classical cycle-cover expansion: it separates $\text{perfmat}(G{)}^2$, the even-cycle-cover contribution coming from superposing two perfect matchings, from the contribution of cycle covers containing odd cycles. This gives a nonnegative odd-cycle defect ${\delta }_{\mathrm{o}\mathrm{d}\mathrm{d}}(G)$. Combining the identity with the Bregman–Minc inequality yields a structural refinement of the Alon–Friedland degree-sequence bound. We prove product, positivity, and fractional-perfect-matching interpretations for the defect; show that for $K_{2n}$ the defect asymptotically accounts for almost the entire Bregman–Minc target; and derive from this a derangement identity. We also study near equality in the Alon–Friedland bound: we classify all one-edge perturbations of the extremal graphs, record the uniform obstruction coming from $K_4$, and formulate a sharp bounded-degree gap problem whose two natural candidate extremal graphs cross between maximum degrees $9$ and $10$.