Overview
This paper examines the Shannon capacity of lexicographic products of finite confusability graphs. The research also investigates the Lovász theta function and the fractional Haemers number in this context. The lexicographic product specifically models zero-error communication, where one graph component describes confusability between distinct classes, and the other component describes confusability within each of those classes.
Research Context
The study builds upon existing knowledge concerning graph invariants relevant to information theory. Specifically, it addresses the Shannon capacity, a fundamental measure in zero-error information theory, alongside the Lovász theta function and the fractional Haemers number. These functions are known for their connections to graph properties and their utility in bounding the Shannon capacity.
Approach
The paper provides elementary, self-contained proofs for three established results. These include the multiplicativity of both the Lovász theta function and the fractional Haemers number when applied to lexicographic products. Additionally, a proof for the equality of the fractional and ordinary Lovász theta functions is presented. The investigation into Shannon capacity involves proving its supermultiplicativity under lexicographic products, considering both possible orders of the product. These ordered products are then compared with the strong product. A key aspect of the methodology involves the explicit construction of a countably infinite family of graphs. For this family, the study demonstrates strict supermultiplicativity of Shannon capacity under the lexicographic product, specifically when a family member is paired with its complement. This construction also reveals that the multiplicative gap can be arbitrarily large. The comparisons made in the paper yield bounds and establish sufficient conditions for the exact determination of Shannon capacities. Furthermore, these comparisons contribute to establishing the incomparability of the resulting upper bounds. The research also determines the Shannon capacities of lexicographic products involving Kneser graphs, their complements, and their $q$-analogues. It identifies that a complete outer factor preserves the inner factor's capacity. Finally, the capacities of iterated lexicographic powers are determined, encompassing powers of self-complementary graphs that exhibit vertex-transitive or strongly regular properties.
Findings
- The Lovász theta function and the fractional Haemers number exhibit multiplicativity under lexicographic products.
- The fractional and ordinary Lovász theta functions are equal.
- Shannon capacity is supermultiplicative under lexicographic products in both possible orders of the graphs.
- Lexicographic products of specific graphs (a countably infinite family and its complement) demonstrate strict supermultiplicativity of Shannon capacity, with an arbitrarily large multiplicative gap.
- Comparisons of ordered lexicographic products with the strong product yield bounds and sufficient conditions for determining Shannon capacities exactly.
- The resulting upper bounds for Shannon capacity are established as incomparable.
- The Shannon capacities for lexicographic products involving Kneser graphs, their complements, and their $q$-analogues have been determined.
- A complete outer factor in a lexicographic product preserves the inner factor's Shannon capacity.
- Capacities for iterated lexicographic powers, including powers of vertex-transitive or strongly regular self-complementary graphs, have been determined.
Why This Matters
The lexicographic product serves as a model for zero-error communication scenarios, specifically distinguishing between confusability between classes and confusability within classes. The findings on Shannon capacity, a key metric for zero-error communication, directly inform the theoretical understanding of such systems.
Potential Applications
The abstract poses an open problem, suggesting avenues for future research building upon the presented findings.