We define the relative fractional independence number of a graph $G$ with respect to another graph $H$, as $$\alpha^*(G|H)=\max_{W}\frac{\alpha(G\boxtimes W)}{\alpha(H\boxtimes W)},$$ where the maximum is taken over all graphs $W$, $G\boxtimes W$ is the strong product of $G$ and $W$, and $\alpha$ denotes the independence number. We give a non-trivial linear program to compute $\alpha^*(G|H)$ and discuss some of its properties. We show that $\alpha^*(G|H)\geq \frac{X(G)}{X(H)} \geq \frac{1}{\alpha^*(H|G)},$ where $X(G)$ can be the independence number, the zero-error Shannon capacity, the fractional independence number, the Lov\'{a}sz number, or the Schrijver's or Szegedy's variants of the Lov\'{a}sz 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 zero-error 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 $\alpha^*(G|H)$ can be used to present a stronger version of the well-known No-Homomorphism Lemma.

翻译：我们定义图$G$相对于另一个图$H$的相对分数独立数为：$$\alpha^*(G|H)=\max_{W}\frac{\alpha(G\boxtimes W)}{\alpha(H\boxtimes W)},$$ 其中最大值取遍所有图$W$，$G\boxtimes W$为$G$与$W$的强乘积，$\alpha$表示独立数。我们给出了一个非平凡线性规划来计算$\alpha^*(G|H)$，并讨论了其若干性质。我们证明$\alpha^*(G|H)\geq \frac{X(G)}{X(H)} \geq \frac{1}{\alpha^*(H|G)},$ 其中$X(G)$可以表示图$G$的独立数、零误差Shannon容量、分数独立数、Lovász数，或Lovász数的Schrijver变体与Szegedy变体。该不等式是前述首个明确的非平凡上界，用于约束两个任意图的不变量之比，同时也可用于获取这些不变量的上界或下界。作为具体应用，我们给出了两个Cayley图零误差Shannon容量之比的新的上界，并计算了某些Johnson图Shannon容量的新下界（由此得到了其Haemers数的精确值）。此外，我们证明$\alpha^*(G|H)$可用于给出著名的无同态引理的一个更强版本。