On the spectral distributions of distance-k graph of free product graphs

TL;DR

This paper investigates the spectral distribution of distance-$k$ graphs derived from regular trees and free product graphs, revealing their convergence properties and asymptotic behavior related to semicircular distributions.

Contribution

It provides explicit spectral distribution calculations for distance-$k$ graphs of regular trees and establishes their convergence and asymptotic distributions in free product graphs.

Findings

Distance-$k$ graph distributions of regular trees are explicitly calculated.

Distance-$k$ graphs of regular graphs converge to the tree distribution.

Asymptotic distributions of free product graphs relate to Chebyshev polynomials and semicircular variables.

Abstract

We calculate the distribution with respect to the vacuum state of the distance-$k$ graph of a $d$-regular tree. From this result we show that the distance-$k$ graph of a $d$-regular graphs converges to the distribution of the distance-$k$ graph of a regular tree. Finally, we prove that, properly normalized, the asymptotic distributions of distance-$k$ graphs of the $d$-fold free product graph, as $d$ tends to infinity, is given by the distribution of $P_k(s)$, where $s$ is a semicircle random variable and $P_k$ is the $k$-th Chebychev polynomial.