Let $\Gamma$ be a function that maps two arbitrary graphs $G$ and $H$ to a non-negative real number such that $$\alpha(G^{\boxtimes n})\leq \alpha(H^{\boxtimes n})\Gamma(G,H)^n$$ 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 $\Gamma$. 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 $\alpha^*(G|H)$ and discuss some of its properties. In particular, 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 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 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：设Γ为将任意两个图G和H映射到非负实数的函数，使得对于任意自然数n满足$$\alpha(G^{\boxtimes n})\leq \alpha(H^{\boxtimes n})\Gamma(G,H)^n$$，其中$G^{\boxtimes n}$表示图G与自身进行n次强积运算的结果。本文建立了寻找此类函数Γ的两种不同方法的等价性。通过任一方法得到的共同解被称为“图G相对于图H的相对分数独立数”，记作$\alpha^*(G|H)$。我们讨论了该函数的部分性质，特别证明了不等式$\alpha^*(G|H)\geq \frac{X(G)}{X(H)} \geq \frac{1}{\alpha^*(H|G)}$成立，其中$X(G)$可取为图G的独立数、香农容量、分数独立数、洛瓦兹数，或洛瓦兹数的Schrijver变体与Szegedy变体。该不等式首次为前述任意两个图的不变量之比提供了显式的非平凡上界，同时也可用于推导这些不变量的上下界。作为具体应用，我们给出了两个凯莱图的香农容量之比的新的上界，并计算了某些约翰逊图香农容量的新下界（由此得到其Haemers数的精确值）。此外，我们证明$\alpha^*(G|H)$可用于推导经典“无同态引理”的强化版本。