Asymptotic Spectral Distributions of Distance $k$-Graphs of Cartesian Product Graphs

TL;DR

This paper derives the asymptotic spectral distribution of distance k-graphs of Cartesian powers of a finite graph, revealing a connection to Hermite polynomials through combinatorics and quantum probability.

Contribution

It provides an explicit description of the spectral distribution limit for these graphs, linking graph spectra to Hermite polynomials with a novel combinatorial and quantum probabilistic approach.

Findings

Spectral distribution converges to a limit described by Hermite polynomials.

Explicit formulas for the eigenvalue distribution of distance k-graphs.

Method combines asymptotic combinatorics and quantum probability theory.

Abstract

Let $G$ be a finite connected graph on two or more vertices and $G^{[N,k]}$ the distance $k$-graph of the $N$-fold Cartesian power of $G$. For a fixed $k\ge1$, we obtain explicitly the large $N$ limit of the spectral distribution (the eigenvalue distribution of the adjacency matrix) of $G^{[N,k]}$. The limit distribution is described in terms of the Hermite polynomials. The proof is based on asymptotic combinatorics along with quantum probability theory.