On the number of maximum independent sets in Doob graphs

TL;DR

This paper investigates the growth rate of the number of maximum independent sets in Doob graphs, revealing an exponential growth pattern and establishing a connection to distance-2 MDS codes in Hamming graphs.

Contribution

It proves the asymptotic growth of maximum independent sets in Doob graphs and constructs an injective map linking these sets to MDS codes in Hamming graphs.

Findings

Number of maximum independent sets grows as 2^{2m+n-1}(1+o(1))

Established a correspondence between independent sets in Doob graphs and MDS codes

Provided bounds on the enumeration of maximum independent sets

Abstract

The Doob graph $D(m,n)$ is a distance-regular graph with the same parameters as the Hamming graph $H(2m+n,4)$. The maximum independent sets in the Doob graphs are analogs of the distance-$2$ MDS codes in the Hamming graphs. We prove that the logarithm of the number of the maximum independent sets in $D(m,n)$ grows as $2^{2m+n-1}(1+o(1))$. The main tool for the upper estimation is constructing an injective map from the class of maximum independent sets in $D(m,n)$ to the class of distance-$2$ MDS codes in $H(2m+n,4)$.