Unified bounds for the independence number of graph powers

TL;DR

This paper develops unified, sharp bounds for the $k$-independence number of graphs using semidefinite programming and polynomial techniques, extending existing results and enabling new bounds.

Contribution

It introduces a general framework that unifies and sharpens bounds for the $k$-independence number using advanced mathematical methods.

Findings

Derived sharp bounds for $mbda_k(G)$

Unified various existing bounds into a single framework

Enabled derivation of new bounds for $mbda_k(G)$

Abstract

For a graph $G$, its $k$-th power $G^k$ is constructed by placing an edge between two vertices if they are within distance $k$ of each other. The $k$-independence number $\alpha_k(G)$ is defined as the independence number of $G^k$. By using general semidefinite programming and polynomial methods, we derive sharp bounds for the $k$-independence number of a graph, which extend and unify various existing results. Our work also allows us to easily derive some new bounds for $\alpha_k(G)$.