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Advances in the Shannon Capacity of Graphs

Nitay Lavi, Igal Sason

TL;DR

This work advances the study of Shannon capacity by deriving exact values and bounds for the $q$-Kneser and Tadpole graph families, and by constructing an infinite family of connected graphs whose capacity is not attained by any finite strong power of the graph. It develops a polynomial-in-graphs framework, building on Schrijver’s results to identify when the capacity of a graph polynomial equals the polynomial of component capacities, and proves an AM-GM–type inequality linking the capacity of strong products to disjoint unions. It also sharpens our toolkit with exact caps for $q$-Kneser graphs and comprehensive Tadpole-graph analyses, and lays out several avenues for future work including instances where capacity cannot be achieved by any finite power and the broader implications of the new inequality. Overall, the paper provides both concrete capacity results for notable graph families and general methodological advances to facilitate future capacity computations and bounds, while remaining accessible to researchers entering the area.

Abstract

We derive exact values and new bounds for the Shannon capacity of two families of graphs: the $q$-Kneser graphs and the tadpole graphs. We also construct a countably infinite family of connected graphs whose Shannon capacity is not attained by the independence number of any finite strong power. Building on recent work of Schrijver, we establish sufficient conditions under which the Shannon capacity of a polynomial in graphs, formed via disjoint unions and strong products, equals the corresponding polynomial of the individual capacities, thereby reducing the evaluation of such capacities to that of their components. Finally, we prove an inequality relating the Shannon capacities of the strong product of graphs and their disjoint union, which yields streamlined proofs of several known bounds. In addition to contributing to the computation of the Shannon capacity of graphs, this paper is intended to serve as an accessible entry point to those wishing to work in this area.

Advances in the Shannon Capacity of Graphs

TL;DR

This work advances the study of Shannon capacity by deriving exact values and bounds for the -Kneser and Tadpole graph families, and by constructing an infinite family of connected graphs whose capacity is not attained by any finite strong power of the graph. It develops a polynomial-in-graphs framework, building on Schrijver’s results to identify when the capacity of a graph polynomial equals the polynomial of component capacities, and proves an AM-GM–type inequality linking the capacity of strong products to disjoint unions. It also sharpens our toolkit with exact caps for -Kneser graphs and comprehensive Tadpole-graph analyses, and lays out several avenues for future work including instances where capacity cannot be achieved by any finite power and the broader implications of the new inequality. Overall, the paper provides both concrete capacity results for notable graph families and general methodological advances to facilitate future capacity computations and bounds, while remaining accessible to researchers entering the area.

Abstract

We derive exact values and new bounds for the Shannon capacity of two families of graphs: the -Kneser graphs and the tadpole graphs. We also construct a countably infinite family of connected graphs whose Shannon capacity is not attained by the independence number of any finite strong power. Building on recent work of Schrijver, we establish sufficient conditions under which the Shannon capacity of a polynomial in graphs, formed via disjoint unions and strong products, equals the corresponding polynomial of the individual capacities, thereby reducing the evaluation of such capacities to that of their components. Finally, we prove an inequality relating the Shannon capacities of the strong product of graphs and their disjoint union, which yields streamlined proofs of several known bounds. In addition to contributing to the computation of the Shannon capacity of graphs, this paper is intended to serve as an accessible entry point to those wishing to work in this area.

Paper Structure

This paper contains 26 sections, 61 theorems, 183 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic definitions and graph families
  4. Terminology
  5. Graph operations
  6. Basic graph invariants under isomorphism
  7. Fractional invariants of graphs
  8. Graph spectrum
  9. Some structured families of graphs
  10. The Shannon capacity of graphs
  11. The Lovász theta-function of graphs
  12. Concluding preliminaries
  13. The Shannon capacity of polynomials in graphs
  14. The Shannon capacity of Tadpole graphs
  15. When the graph capacity is not attained by the independence number of any finite power?
  16. ...and 11 more sections

Key Result

Theorem 2.1

Let $\mathsf{G}$ and $\mathsf{H}$ be simple graphs. Then,

Figures (3)

  • Figure 1: A $5$-cycle graph and its orthonormal representation (Lovász umbrella).
  • Figure 2: The Tadpole graph $\mathrm{T}({5},{6})$.
  • Figure 3: The DMCs of $\mathrm{T}({k},{\ell})$ (left plot) and $\mathop{\mathrm{\mathsf{C}}}\nolimits_{k+\ell}$ (right plot).

Theorems & Definitions (149)

  • Definition 2.1: Subgraphs and graph connectivity
  • Definition 2.2: Isomorphic graphs
  • Definition 2.3: Complement and self-complementary graphs
  • Example 2.1
  • Definition 2.4: Strong product of graphs
  • Definition 2.5: Disjoint union of graphs
  • Definition 2.6: Independent sets
  • Definition 2.7: Independence number
  • Definition 2.8: Cliques
  • Definition 2.9: Clique number
  • ...and 139 more