Graphene nanocones and Pascal matrices

TL;DR

This paper explores conjectured identities linking the determinants of adjacency matrices of graphene nanostructures to Pascal matrices and other combinatorial objects, supported by analytic and computational evidence.

Contribution

It introduces new conjectures connecting graphene adjacency determinants with Pascal matrices and combinatorial counting problems, supported by analytic and computational validation.

Findings

Determinant of graphene triangle adjacency matrix equals the characteristic polynomial of the Pascal matrix.

Determinant of trapezium adjacency matrix equals a sub-matrix determinant.

Determinant of the tight binding matrix equals its permanent.

Abstract

I conjecture three identities for the determinant of adjacency matrices of graphene triangles and trapezia with Bloch (and more general) boundary conditions. For triangles, the parametric determinant is equal to the characteristic polynomial of the symmetric Pascal matrix. For trapezia it is equal to the determinant of a sub-matrix. Finally, the determinant of the tight binding matrix equals its permanent. The conjectures are supported by analytic evaluations and Mathematica, for moderate sizes. They establish connections with counting problems of partitions, lozenge tilings of hexagons, dense loops on a cylinder.