Punctured adjacency-degree algebras of Cartesian products
Abstract
For a connected regular graph G and a vertex a, we study the algebra generated by the adjacency and degree matrices of G-a and its cyclic module P_a generated by the all-ones vector. Our main theorem determines dim P_a for Cartesian products whose factors have equitable distance partitions at the chosen roots. A normalized logarithmic derivative of the local spectral generating function partitions the factors into boundary classes. We identify the boundary-return space exactly and express dim P_a as a sum of affine ranks on additive spectral fibres. For a distance-regular factor with distinct spectrum Θ, this gives dim P_a(F^{\square m}) = |mΘ| - 1. For products of powers of two distinct complete graphs, we evaluate the fibre formula in closed form. We also determine the full punctured algebras of all Hamming graphs: equality with the compressed Terwilliger algebra holds precisely in dimensions at most four for the hypercube and at most two for larger alphabets. For distance-regular graphs, adjacency moments alone determine the intersection array, with an explicit finite reconstruction. Finally, Cartesian stabilizer formulas separate metric loss from orbit splitting; on Doob graphs their distance-graded defect recovers the number of Shrikhande factors.
Related
- Note on adjacency spectra for twice punctured hypercubes
- A Unified Spectral Framework for Weighted Adjacency-Degree Matrices of Dendrimer and Bethe Trees
- Deep Points of Cluster Algebras II: Punctured Marked Surfaces
- Degree Adjacency Diametrical Matrix of Graphs Spectral Properties and Energy Bounds
- Uniform adjacency-degree probes from differential rigidity
BibTeX
@misc{lu2026punctured,
title = {{Punctured adjacency-degree algebras of Cartesian products}},
author = {Zhipeng Lu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06747},
eprint = {2609.06747},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06747},
}
RIS
TY - UNPB
TI - Punctured adjacency-degree algebras of Cartesian products
AU - Zhipeng Lu
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06747
UR - https://arxiv.org/abs/2609.06747
AB - For a connected regular graph G and a vertex a, we study the algebra generated by the adjacency and degree matrices of G-a and its cyclic module P_a generated by the all-ones vector. Our main theorem determines dim P_a for Cartesian products whose factors have equitable distance partitions at the chosen roots. A normalized logarithmic derivative of the local spectral generating function partitions the factors into boundary classes. We identify the boundary-return space exactly and express dim P_a as a sum of affine ranks on additive spectral fibres. For a distance-regular factor with distinct spectrum Θ, this gives dim P_a(F^{\square m}) = |mΘ| - 1. For products of powers of two distinct complete graphs, we evaluate the fibre formula in closed form. We also determine the full punctured algebras of all Hamming graphs: equality with the compressed Terwilliger algebra holds precisely in dimensions at most four for the hypercube and at most two for larger alphabets. For distance-regular graphs, adjacency moments alone determine the intersection array, with an explicit finite reconstruction. Finally, Cartesian stabilizer formulas separate metric loss from orbit splitting; on Doob graphs their distance-graded defect recovers the number of Shrikhande factors.
ER -