A strong Chollet inequality for generalized theta and bicyclic graphs

For a graph Laplacian, the degree-product inequality is stronger than the self formof Chollet’s permanent conjecture. We prove this stronger inequality for every simplegeneralized theta graph and for every principal submatrix of its Laplacian. The mainestimate is quantitative. If the graph consists of r ≥ 2 internally disjoint paths of lengthsℓ1, . . . , ℓr joining two vertices, and C is its normalized Laplacian, thenper(C) − per(C ◦ C) ≥ 22r2 − 48r + 1140r2r∏i=1Fℓi −1 > 0,where F0 = F1 = 1 and Fm = Fm−1 + 14 Fm−2. The proof separates matching contributionsfrom longer permutation cycles. Disjoint classes of matchings give a uniform positivecontribution, while parity bounds the total loss from odd cycles. Established closureproperties then give the strong inequality for graphs whose blocks are bipartite orgeneralized theta graphs, including all connected simple bicyclic graphs. We also obtaina parity-sensitive refinement and an explicit lower bound for the ordinary Chollet defect.Exact integer and rational calculations illustrate the estimates and verify the cycle-coveridentities on a finite family.