bibkey: molinari2022graphene authors: L. G. Molinari year: 2022 title: “Graphene nanocones and Pascal matrices” doi: 10.48550/arXiv.2206.14428 url: https://arxiv.org/abs/2206.14428v2 claim: “Conjecture 2: the Hückel determinant of a honeycomb trapezium equals the displayed reduced Pascal determinant.” strata_touched:

• D5/S3/Combinatorics/Graph/HoneycombTrapeziumHuckelDeterminant license: citation-only triage: anchor

Graphene nanocones and Pascal matrices

L. G. Molinari, arXiv:2206.14428v2 (2022-08-21), math-ph.

Page 4 defines the horizontal blocks:

A block Tₘ(xₘ,yₘ) is a square matrix of size 2m+1 that describes a row of 2m+1 atoms with boundary parameters xₘ,yₘ:

The displayed $T_0$ is $x_0+y_0$. For $m>0$ the block has ones on the two adjacent diagonals, $y_m$ at $(0,2m)$ and $x_m$ at $(2m,0)$. The vertical blocks are defined on the same page:

The blocks Rₘ are (2m+1)×(2m−1), with unit elements for the vertical edges in the graph, joining atoms in rows m−1 and m:

The displayed $R_1,R_2,R_3$ have ones at $(2j+1,2j)$, $0\le j<m$. The triangle matrix has $T_m$ on the diagonal and $R_m,R_m^T$ on its adjacent block diagonals.

Section 5, page 9, defines the trapezium:

If rows 0,1,…,k−1 are removed from a honeycomb triangle, a trapezium results, with rows of lengths 2k+1, … , 2n+1. The corresponding Hückel matrix Hₖ,ₙ(𝐱,𝐲) is obtained by deleting the first k² rows and columns of Hₙ(𝐱,𝐲). Its size is (n+1)²−k²=(2k+1) + … + (2n+1).

Conjecture 2, verbatim (page 9):

The determinant of the Hückel matrix Hₖ,ₙ(𝐱, 𝐲) of a honeycomb trapezium with rows with 2k+1, 2k+3, … , 2n+1 sites, size (n+1)²−k², is equal to the determinant of the following matrix of size n+1−k:

The displayed matrix has indices $0\le i,j\le n-k$, with rows in decreasing source-row order:

$$M_{ij}=\{\begin{array}{ll}{x}_{n-i}+y_{n-i},& i=j,\\ (-1{)}^{j-i}(\genfrac{}{}{0ex}{}{n-i}{j-i})y_{n-i},& i<j,\\ \left(\genfrac{}{}{0ex}{}{n-j}{i-j}\right)x_{n-j},& i>j.\end{array}$$

The identity is read over any commutative ring with arbitrary boundary weights. Its $k=0$ specialization is the paper’s Conjecture 1 for triangles; Conjecture 2 is the assertion treated here. No assertion about Conjecture 3 on permanents is made.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2206.14428
• URL: https://arxiv.org/abs/2206.14428v2
• PDF: https://arxiv.org/pdf/2206.14428v2, page 4 for the blocks and page 9 for Conjecture 2.
• Original source: https://arxiv.org/src/2206.14428v2, HUCKEL_PASCAL_3.tex, Conjecture 2 and its following determinant display.