Characterizing the Lovasz theta function via walk generating functions

TL;DR

This paper introduces a novel characterization of the Lovasz theta function using walk-generating functions, leading to new bounds for graph independence numbers and insights into related algebraic graph theory concepts.

Contribution

It establishes a new relationship between the Lovasz theta function and walk-generating functions, enabling natural generalizations of Hoffman bounds to non-regular graphs.

Findings

New characterization of Lovasz theta via walk-generating functions

Generalizations of Hoffman bounds for non-regular graphs

Introduction of the spherical independence number as a relaxation of independence number

Abstract

A new characterization of the Lovasz theta function is provided by relating it to the (weighted) walk-generating function, thus establishing a relationship between two seemingly quite distinct concepts in algebraic graph theory. An application of this new characterization is given by showing how it straightforwardly entails multiple natural generalizations of the Hoffman upper bound (on both the independence number and Lovasz number) to arbitrary non-regular graphs. These new bounds possess properties that make them advantageous to previously derived such generalizations. It will also be shown that the Lovasz theta function equals a natural relaxation of the independence number, here dubbed the spherical independence number -- the determination of which involves producing a vector corresponding to a generalized maximum independent set which might be significant for the maximum independent set problem. Lastly, the derivation of the new characterization involves proving a certain analysis result which may in itself be of interest.