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 the Haemers number of certain Johnson graphs. 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$$\alpha^*(G|H)=\max_{W}\frac{\alpha(G\boxtimes W)}{\alpha(H\boxtimes W)},$$其中最大值取自所有图$W$，$G\boxtimes W$是$G$与$W$的强积，$\alpha$表示独立数。我们给出了一个用于计算$\alpha^*(G|H)$的非平凡线性规划，并讨论了其若干性质。我们证明$$\alpha^*(G|H)\geq \frac{X(G)}{X(H)},$$其中$X(G)$可以是图$G$的独立数、零差错香农容量、分数独立数、Lovász数，或Schrijver与Szegedy变体下的Lovász数。该不等式是首个针对任意两图不变量比值的显式非平凡上界，如前所述，它也可用于获得这些不变量的上界或下界。作为显式应用，我们给出了两个Cayley图零差错香农容量比值的新上界，并计算了某些Johnson图的Haemers数。此外，我们证明相对分数独立数可用于增强著名的无同态引理。该引理广泛用于证明两图间不存在同态，并用于给出图独立数的上界。我们对无同态引理的扩展在计算上比原版本更易处理。