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The Shannon capacity of graph powers

Aida Abiad, Cristina Dalfó, Miquel Àngel Fiol

TL;DR

This paper studies upper bounds on the zero-error capacity $\Theta(G^k)$ of the $k$-th power $G^k$ of a graph $G$ using spectral methods. It introduces two algebraic bounds: a ratio-type bound based on minor polynomials which yields $\Theta(G^k) \le \mathrm{tr} f_k(\boldsymbol{A})$, and a rank-type bound based on Shannon polynomials which yields $\Theta(G^k) \le \min_{p \in \mathbb{R}_k[x], p(\boldsymbol{A})_{ii} \neq 0} \mathrm{rank}(p(\boldsymbol{A}))$. The paper also shows that for $k$-partially walk-regular graphs the Lovász theta number satisfies $\vartheta(G^k) \le \mathrm{tr} f_k(\boldsymbol{A})$, providing a computable route to exact values when the bounds are tight, and demonstrates these notions via computational experiments. The results illustrate that the two algebraic bounds are often sharp, and that the ratio-type bound can rival the Lovász theta bound while being easier to compute via linear programming. By combining spectral theory with optimization, the work advances understanding of the Shannon capacity of graph powers and yields practical tools for estimating $\Theta(G^k)$.

Abstract

For a graph $G$, its $k$-th graph power $G^k$ is constructed by placing an edge between two vertices if they are within distance $k$. We consider the problem of deriving upper bounds on the Shannon capacity of graph powers by using spectral graph theory and linear optimization methods. First, we use the so-called ratio-type bound to provide an alternative and spectral proof of a result by Lovász [IEEE Trans. Inform. Theory 1979], which states that, for a regular graph, the Hoffman ratio bound on the independence number is also an upper bound on the Lovász theta number and, hence, also on the Shannon capacity. In fact, we show that Lovász' result holds in the more general context of graph powers. Secondly, we derive another bound on the Shannon capacity of graph powers, the so-called rank-type bound, which depends on a new family of polynomials that can be computed by running a simple algorithm. Lastly, we provide several computational experiments that demonstrate the sharpness of the two proposed algebraic bounds. As a byproduct, when these two new algebraic bounds are tight, they can be used to easily derive the exact values of the Lovász theta number (which relies on solving an SDP) and the Shannon capacity (which is not known to be computable) of the corresponding graph power.

The Shannon capacity of graph powers

TL;DR

This paper studies upper bounds on the zero-error capacity of the -th power of a graph using spectral methods. It introduces two algebraic bounds: a ratio-type bound based on minor polynomials which yields , and a rank-type bound based on Shannon polynomials which yields . The paper also shows that for -partially walk-regular graphs the Lovász theta number satisfies , providing a computable route to exact values when the bounds are tight, and demonstrates these notions via computational experiments. The results illustrate that the two algebraic bounds are often sharp, and that the ratio-type bound can rival the Lovász theta bound while being easier to compute via linear programming. By combining spectral theory with optimization, the work advances understanding of the Shannon capacity of graph powers and yields practical tools for estimating .

Abstract

For a graph , its -th graph power is constructed by placing an edge between two vertices if they are within distance . We consider the problem of deriving upper bounds on the Shannon capacity of graph powers by using spectral graph theory and linear optimization methods. First, we use the so-called ratio-type bound to provide an alternative and spectral proof of a result by Lovász [IEEE Trans. Inform. Theory 1979], which states that, for a regular graph, the Hoffman ratio bound on the independence number is also an upper bound on the Lovász theta number and, hence, also on the Shannon capacity. In fact, we show that Lovász' result holds in the more general context of graph powers. Secondly, we derive another bound on the Shannon capacity of graph powers, the so-called rank-type bound, which depends on a new family of polynomials that can be computed by running a simple algorithm. Lastly, we provide several computational experiments that demonstrate the sharpness of the two proposed algebraic bounds. As a byproduct, when these two new algebraic bounds are tight, they can be used to easily derive the exact values of the Lovász theta number (which relies on solving an SDP) and the Shannon capacity (which is not known to be computable) of the corresponding graph power.
Paper Structure (7 sections, 11 theorems, 28 equations, 4 figures, 10 tables, 1 algorithm)

This paper contains 7 sections, 11 theorems, 28 equations, 4 figures, 10 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bounds using the minor polynomials
  4. Ratio-type bound
  5. Lovász theta number
  6. Bounds using the Shannon polynomials
  7. Concluding remarks

Key Result

Theorem 1.1

Let $G=(V,E)$ be a regular graph with $n$ vertices and adjacency eigenvalues $\lambda_1 \geq \cdots \geq \lambda_n$. Let $[2, n] = \{2, 3,\ldots, n\}$. Given a polynomial $p \in \mathbb{R}_k[x]$, consider the parameters $W(p) = \max_{u\in V} {(p(A))_{uu}}$ and $\lambda(p) = \min_ {i\in[2,n]}{p(\lamb

Figures (4)

  • Figure 1: Two $3$-minor polynomials for the Coxeter graph in Example \ref{['ex:f3+g3']}.
  • Figure 2: The cube $G=Q_3$ and its 2-power graph $G^{2}=Q_3^{2}$, which is a $(8,6;4,6)$-strongly regular graph with spectrum $\mathop{\mathrm{sp}}\nolimits Q_3^{2}=\{6,0^4,-2^3\}$.
  • Figure 3: The $\{1,2,4\}$-minor polynomials of the Coxeter graph.
  • Figure 4: The Shannon polynomials $s_i(x)$ of the Coxeter graph for $i=1,2,3,4$ (Example \ref{['ex:coxeter-shannon']}).

Theorems & Definitions (23)

  • Theorem 1.1: Ratio-type bound acf19
  • Example 3.1
  • Theorem 3.2: f20
  • Corollary 3.3
  • proof
  • Example 3.4
  • Corollary 3.5
  • Example 3.6
  • Example 3.7
  • Proposition 3.8
  • ...and 13 more