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 $$h_{\text{BC}(G)}(z)=(-z{)}^r{\left[\frac{f(G;q,-1,1)}{q}\right]}_{q=(z-1)\mathrm{/}z},r=\mathrm{\mid }V(G)\mathrm{\mid }-1.$$ 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.