On the Ratio of Shannon Numbers of Graphs
Sharareh Alipour, Amin Gohari, Mehrshad Taziki
TL;DR
The paper introduces the relative fractional independence number α^*(G|H) to bound the ratio of graph invariants under strong graph products, unifying two concrete constructions for Γ(G,H) into a single linear-programming framework. It proves the two approaches are dual forms and shows α^*(G|H) frequently bounds ratios of invariants such as the Shannon capacity, independence number, and Lovász-type numbers, with sharp results for vertex-transitive and Cayley graphs. The authors provide explicit upper bounds on the Shannon-capacity ratio for pairs of Cayley graphs and derive new lower bounds (and exact Haemers numbers) for certain Johnson graphs, along with an extension of the No-Homomorphism Lemma via α^*(G|H). They develop the Expand operations to characterize when α^*(G|H) ≤ 1 implies structural inclusion G∈Expand(H) for cycles and perfect graphs, and pose open problems about computational aspects, equality conditions, and connections to fractional Haemers numbers. Overall, the work yields a versatile tool for bounding capacity ratios and guiding structural graph inequalities with potential impact on zero-error information theory and graph invariants.
Abstract
Let $Γ$ be a function that maps two arbitrary graphs $G$ and $H$ to a non-negative real number such that $$α(G^{\boxtimes n})\leq α(H^{\boxtimes n})Γ(G,H)^n$$ where $n$ is any natural number and $G^{\boxtimes n}$ is the strong product of $G$ with itself $n$ times. We establish the equivalence of two different approaches for finding such a function $Γ$. The common solution obtained through either approach is termed ``the relative fractional independence number of a graph $G$ with respect to another graph $H$". We show this function by $α^*(G|H)$ and discuss some of its properties. In particular, we show that $α^*(G|H)\geq \frac{X(G)}{X(H)} \geq \frac{1}{α^*(H|G)},$ where $X(G)$ can be the independence number, the Shannon capacity, the fractional independence number, the Lovász number, or the Schrijver's or Szegedy's variants of the Lovász number of a graph $G$. This inequality is the first explicit non-trivial upper bound on the ratio of the invariants of two arbitrary graphs, as mentioned earlier, which can also be used to obtain upper or lower bounds for these invariants. As explicit applications, we present new upper bounds for the ratio of the Shannon capacity of two Cayley graphs and compute new lower bounds on the Shannon capacity of certain Johnson graphs (yielding the exact value of their Haemers number). Moreover, we show that $α^*(G|H)$ can be used to present a stronger version of the well-known No-Homomorphism Lemma.