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Iwao Mizukai

Publications and source records attributed to Iwao Mizukai.

4 recordsLinked to original sources
arXiv2026

Local Shellings of Broken-Circuit Complexes via Boundary States and the Chromatic Specialization of the Negami Polynomial

For edge-disjoint graphs meeting along a finite boundary, we construct relative shellings of NBC faces with prescribed boundary connectivity. The construction adds paths in spanning forests to the restriction faces of auxiliary rooted-forest complexes. When every edge of the first graph precedes every edge of the second, these relative shellings combine into a shelling with consecutive connectivity blocks. The resulting nonnegative decomposition of the $h$-polynomial realizes the chromatic specialization of the Negami splitting identity. We also give path formulas for gluing graphic Orlik--Terao Gröbner bases across boundaries of size two or three, and a virtual-clique elimination formula for arbitrary boundaries. An exponential output lower bound at treewidth two distinguishes explicit basis enumeration from coefficient computation at fixed treewidth. Finally, state polytopes and their Cayley lifts describe an obstruction to linear block orders in the original coordinates and a lifted realization of these orders.

math.CO↗
arXiv2026

The Negami Polynomial and Broken-Circuit Stanley--Reisner Rings

We establish a direct connection between Seiya Negami's three-variable graph polynomial $f(G;t,x,y)$ and the Stanley-Reisner ring of the broken-circuit complex of the graphic matroid. For a connected loopless graph $G$, the chromatic specialization $f(G;q,-1,1)=P_G(q)$ together with Whitney's broken-circuit theorem yields \[ h_{\BC(G)}(z)=(-z)^r\left[\frac{f(G;q,-1,1)}{q}\right]_{q=(z-1)/z},\qquad r=|V(G)|-1. \] For brevity, we call this explicit composite map the \emph{Negami--Hilbert correspondence}. The underlying chromatic/characteristic-polynomial-to-broken-circuit-Hilbert-series relation is classical, and no claim of novelty is made for that underlying identity. We place the formulation in the context of works of Negami, Oxley, Whitney, Brylawski--Oxley, Proudfoot--Speyer, Llamas--Martínez-Bernal--Merino, and Berget. We then derive closed or low-degree formulas for cycles $C_n$, wheels $W_n$, complete graphs $K_n$, complete bipartite graphs $K_{m,n}$, strongly regular graphs, large-girth regular graphs, and Ramanujan graphs. Finally, as a first nontrivial construction retaining the full three-variable Negami polynomial, we equip the squarefree edge algebra of the triangle $C_3$ with the graphic-matroid rank filtration and recover the complete polynomial $f(C_3;t,x,y)$ from the bigraded Hilbert polynomial of the associated graded algebra.

math.CO↗
arXiv2024

Ollivier Ricci curvature of Cayley graphs for dihedral groups, generalized quaternion groups, and cyclic groups

Lin, Lu, and Yau formulated the Ricci curvature of edges in simple undirected graphs[2]. Using their formulations, we calculate the Ricci curvatures of Cayley graphs for the dihedral groups, the general quaternion groups, and cyclic groups with some generating sets that are chosen so that their cardinal numbers are less than or equal to four. For the dihedral group and the general quaternion group, we obtained the Ricci curvatures of all edges of the Cayley graph with generator sets consisting of the four elements that are the two generators defining each group and their inverses elements.For the cyclic group (Z/nZ, +), we have the Ricci curvatures of edges of the Cayley graph generating by S_{1, k} = {+1, -1, +k, -k}.

math.CO↗