Asymptotic spectral distributions of distance $k$-graphs of star product graphs

TL;DR

This paper studies the spectral distribution of distance $k$-graphs derived from the $N$-fold star power of a finite connected graph, showing convergence to a Bernoulli distribution as $N$ grows large.

Contribution

It establishes the asymptotic spectral distribution of distance $k$-graphs of star powers, revealing convergence to a Bernoulli distribution, with a proof based on a fourth moment lemma.

Findings

Spectral distribution converges to a Bernoulli distribution as $N$ increases.

The proof utilizes a fourth moment lemma for distribution convergence.

Results apply to distance $k$-graphs of star power graphs.

Abstract

Let $G$ be a finite connected graph and let $G^{[\star N,k]}$ be the distance $k$-graph of the $N$-fold star power of $G$. For a fixed $k\geq1$, we show that the large $N$ limit of the spectral distribution of $G^{[\star N,k]}$ converges to a centered Bernoulli distribution, $1/2\delta_{-1}+1/2\delta_1$. The proof is based in a fourth moment lemma for convergence to a centered Bernoulli distribution.