Independence numbers of Johnson-type graphs

TL;DR

This paper investigates the independence numbers of a family of Johnson-type graphs defined on vectors with entries in {-1,0,1}, providing exact values for certain parameters and cases, especially for t=0, t=-1, and negative odd t.

Contribution

It determines the independence numbers of Johnson-type graphs $J_{ ext{±}}(n,k,t)$ for large n in specific cases, including complete solutions for k=3 and certain t values.

Findings

Independence number for t=0 and t=-1 cases with large n.

Complete solution for k=3 in these cases.

Independence number for negative odd t when n is large.

Abstract

We consider a family of distance graphs in $\mathbb{R}^n$ and find its independent numbers in some cases. Define graph $J_{\pm}(n,k,t)$ in the following way: the vertex set consists of all vectors from $\{-1,0,1\}^n$ with $k$ nonzero coordinates; edges connect the pairs of vertices with scalar product $t$. We find the independence number of $J_{\pm}(n,k,t)$ for $n > n_0 (k,t)$ in the cases $t = 0$ and $t = -1$; these cases for $k = 3$ are solved completely. Also the independence number is found for negative odd $t$ and $n > n_0 (k,t)$.