The optimal bound on the 3-independence number obtainable from a polynomial-type method

A $k$-independent set in a connected graph is a set of vertices such that any two vertices in the set are at distance greater than $k$ in the graph. The $k$-independence number of a graph, denoted ${\alpha }_k$, is the size of a largest $k$-independent set in the graph. Recent results have made use of polynomials that depend on the spectrum of the graph to bound the $k$-independence number. They are optimized for the cases $k=1,2$. There are polynomials that give good (and sometimes) optimal results for general $k$, including case $k=3$. In this paper, we provide the best possible bound that can be obtained by choosing a polynomial for case $k=3$ and apply this bound to well-known families of graphs including the Hamming graph.