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

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