# A new class of polynomials from to the spectrum of a graph, and its application to bound the $k$-independence number

## TL;DR

This paper introduces a new class of polynomials derived from a graph's spectrum, enabling tight bounds on the $k$-independence number of $k$-partially walk-regular graphs using spectral techniques.

## Contribution

It presents a novel family of polynomials from graph spectra and applies them with interlacing methods to bound the $k$-independence number, extending spectral graph theory.

## Key findings

- Derived tight spectral bounds for $k$-independence number
- Identified that odd graphs $O_{ ext{ell}}$ lack 1-perfect codes
- Provided examples where bounds are tight

## Abstract

The $k$-independence number of a graph is the maximum size of a set of vertices at pairwise distance greater than $k$. A graph is called $k$-partially walk-regular if the number of closed walks of a given length $l\le k$, rooted at a vertex $v$, only depends on $l$. In particular, a distance-regular graph is also $k$-partially walk-regular for any $k$. In this note, we introduce a new family of polynomials obtained from the spectrum of a graph. These polynomials, together with the interlacing technique, allow us to give tight spectral bounds on the $k$-independence number of a $k$-partially walk-regular graph. Together with some examples where the bounds are tight, we also show that the odd graph $O_{\ell}$ with $\ell$ odd has no $1$-perfect code.