We define the relative fractional independence number of two graphs, $G$ and $H$, 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 $$\alpha^*(G|H)\geq \frac{X(G)}{X(H)},$$ 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 the relative fractional independence number can be used to present a stronger version of the well-known No-Homomorphism Lemma. The No-Homomorphism Lemma is widely used to show the non-existence of a homomorphism between two graphs and is also used to give an upper bound on the independence number of a graph. Our extension of the No-Homomorphism Lemma is computationally more accessible than its original version.

翻译：我们定义两个图$G$和$H$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$W$，$G\boxtimes W$表示$G$与$W$的强乘积，$\alpha$表示独立数。我们给出了一个计算$\alpha^*(G|H)$的非平凡线性规划，并讨论了其若干性质。我们证明了 $$\alpha^*(G|H)\geq \frac{X(G)}{X(H)},$$ 其中$X(G)$可以是图$G$的独立数、零错误Shannon容量、分数独立数、Lovász数，或Lovász数的Schrijver变体与Szegedy变体。该不等式是首个显式的非平凡上界，用于约束两个任意图的不变量比值，并可进一步用于推导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图零错误Shannon容量比值的新上界，并计算了某些Johnson图Shannon容量的新下界（从而得到其Haemers数的精确值）。此外，我们证明了相对分数独立数可用于推广著名的无同态引理（No-Homomorphism Lemma）。该引理常被用于证明两个图之间不存在同态，并给出图独立数的上界。本文对无同态引理的推广在计算上比原始版本更具可操作性。