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Characterizing the Lovasz theta function via walk generating functions

Lasse Harboe Wolff

TL;DR

This work develops two new characterizations of the Lovasz theta function $\vartheta(G)$: it equals the spherical independence number $\alpha_{\mathcal{S}}(G)$ and also admits a walk-generating function representation $\vartheta(G)=\inf_A \min_{x\in[\lambda_{\min}(A)^{-1}, \lambda_{\max}(A)^{-1}]} W_A(x)$. The approach unifies algebraic and combinatorial perspectives by exploiting a duality for reciprocal-function forms and a Lagrangian optimization that connects $\alpha_{\mathcal{S}}(G)$ to $W_A(x)$. Key contributions include natural generalizations of the Hoffman bound to non-regular graphs, a constructive vector $\mathbf{v}$ with $|\mathbf{v}|^2=\vartheta(G)=\alpha_{\mathcal{S}}(G)$ and $\langle \mathbf{v},A\mathbf{v}\rangle=0$, and a duality result for sums of reciprocal terms. These results offer computationally favorable avenues to bound or approximate independence-like quantities and suggest new directions for maximum independent set heuristics grounded in spectral and walk-based methods.

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.

Characterizing the Lovasz theta function via walk generating functions

TL;DR

This work develops two new characterizations of the Lovasz theta function : it equals the spherical independence number and also admits a walk-generating function representation . The approach unifies algebraic and combinatorial perspectives by exploiting a duality for reciprocal-function forms and a Lagrangian optimization that connects to . Key contributions include natural generalizations of the Hoffman bound to non-regular graphs, a constructive vector with and , and a duality result for sums of reciprocal terms. These results offer computationally favorable avenues to bound or approximate independence-like quantities and suggest new directions for maximum independent set heuristics grounded in spectral and walk-based methods.

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.
Paper Structure (9 sections, 18 theorems, 103 equations, 3 figures)

This paper contains 9 sections, 18 theorems, 103 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Proof of the main theorem
  5. Equality between Lovasz number and spherical independence number
  6. Equality between spherical independence number and the expression involving walk-generating functions
  7. Concluding Remarks and outlook
  8. Duality result for the critical points of special reciprocal functions
  9. Proof of sub-multiplicativity for the spherical independence numbers

Key Result

Theorem 1

For any graph $G$, we have where in the rightmost expression above, $A$ ranges over all real symmetric weighted adjacency matrices for $G$.

Figures (3)

  • Figure 1: A drawing of the Golomb graph.
  • Figure 2: A plot of the (unweighted) walk-generating function $W(x)$ for the Golomb graph $G$ as a function of $x$ (See Fig. \ref{['fig:Golomb graph drawing']} for a drawing of the Golomb graph). The blue line is the plot of $W(x)$, while the dashed lines demarcates the interval $(\lambda_n^{-1}, \lambda_1^{-1})$. In this interval, $W(x)$ attains a minimum value of about $4.744$, which then shows $\vartheta(G) \leq 4.744$.
  • Figure 3: A plot of the (unweighted) walk-generating function $W(x)$ of the $17$ vertex path graph $P_{17}$ as a function of $x$. One sees that the critical point at which $W(x)$ attains its largest value indeed lies in the smooth interval containing $0$, i.e. $(\lambda_n^{-1}, \lambda_1^{-1})$. At this value of $x$, $W(x)=9$ which shows $\vartheta(P_{17}) \leq 9$. In fact, we here have $\alpha(P_{17}) = \vartheta(P_{17})=9$.

Theorems & Definitions (39)

  • Definition 1: weighted walk-generating function
  • Remark
  • Definition 2: Spherical independence number
  • Remark
  • Theorem 1
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • Proposition 1
  • ...and 29 more