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For $k\ge 1$, the $k$-independence number $\alpha_k$ of a graph is themaximum number of vertices that are mutually at distance greater than $k$. Thewell-known inertia and ratio bounds for the (1-)independence number$\alpha(=\alpha_1)$ of a graph, due to Cvetkovi\'c and Hoffman, respectively,were generalized recently for every value of $k$. We show that, for graphs withenough regularity, the polynomials involved in such generalizations are closelyrelated and give exact values for $\alpha_k$, showing a new relationshipbetween the inertia and ratio type bounds. Additionally, we investigate theexistence and properties of the extremal case of sets of vertices that aremutually at maximum distance for walk-regular graphs. Finally, we obtain newsharp inertia and ratio type bounds for partially walk-regular graphs by usingthe predistance polynomials.

On inertia and ratio type bounds for the $k$-independence number of a graph and their relationship