Kronecker Product Graphs and Counting Walks in Restricted Lattices

TL;DR

This paper derives formulas for counting walks in Kronecker product graphs and computes their spectral distributions, especially for 2D restricted lattices, using Mellin convolution and elliptic integrals.

Contribution

It introduces new formulas for walk counting and spectral analysis in Kronecker product graphs and applies these to specific lattice structures.

Findings

Formulas for walk counts in Kronecker product graphs

Spectral distributions expressed via elliptic integrals

Identification of 2D lattices with Kronecker structure

Abstract

Formulas are derived for counting walks in the Kronecker product of graphs, and the associated spectral distributions are obtained by the Mellin convolution of probability distributions. Two-dimensional restricted lattices admitting the Kronecker product structure are listed, and their spectral distributions are calculated in terms of elliptic integrals.