AbstractIn this paper, we present a new connection between representation theory of noncommutative hypersurfaces and combinatorics. Let S be a graded ($$\pm 1$$ ± 1 )-skew polynomial algebra in n variables of degree 1 and $$f =x_1^2 + \cdots +x_n^2 \in S$$ f = x 1 2 + ⋯ + x n 2 ∈ S . We prove that the stable category $$\mathsf {\underline{CM}}^{\mathbb Z}(S/(f))$$ CM ̲ Z ( S / ( f ) ) of graded maximal Cohen–Macaulay module over S/(f) can be completely computed using the four graphical operations. As a consequence, $$\mathsf {\underline{CM}}^{\mathbb Z}(S/(f))$$ CM ̲ Z ( S / ( f ) ) is equivalent to the derived category $$\mathsf {D}^{\mathsf {b}}({\mathsf {mod}}\,k^{2^r})$$ D b ( mod k 2 r ) , and this r is obtained as the nullity of a certain matrix over $${\mathbb F}_2$$ F 2 . Using the properties of Stanley–Reisner ideals, we also show that the number of irreducible components of the point scheme of S that are isomorphic to $${\mathbb P}^1$$ P 1 is less than or equal to $$\left( {\begin{array}{c}r+1\\ 2\end{array}}\right) $$ r + 1 2 .