We investigate the effect of the well-known Mycielski construction on the Shannon capacity of graphs and on one of its most prominent upper bounds, the (complementary) Lov\'asz theta number. We prove that if the Shannon capacity of a graph, the distinguishability graph of a noisy channel, is attained by some finite power, then its Mycielskian has strictly larger Shannon capacity than the graph itself. For the complementary Lov\'asz theta function we show that its value on the Mycielskian of a graph is completely determined by its value on the original graph, a phenomenon similar to the one discovered for the fractional chromatic number by Larsen, Propp and Ullman. We also consider the possibility of generalizing our results on the Sperner capacity of directed graphs and on the generalized Mycielsky construction. Possible connections with what Zuiddam calls the asymptotic spectrum of graphs are discussed as well.

翻译：我们研究了著名的米切尔斯基构造对图香农容量及其一个最主要上界——（互补）洛瓦斯θ数的影响。我们证明：如果某个图的香农容量（即噪声信道的可区分性图）可由有限次幂达到，则该图的米切尔斯基图具有严格大于原图的香农容量。对于互补洛瓦斯θ函数，我们证明其在米切尔斯基图上的值完全由原图上的值决定，这一现象类似于Larsen、Propp和Ullman关于分数色数的发现。我们还考虑了将结果推广至有向图的Sperner容量以及广义米切尔斯基构造的可能性，并讨论了与Zuiddam所称的图的渐近谱的可能联系。