In the early 2000s Louis Bowen initiated a new era of entropy by introducing the f-invariant, and asymptotic entropy notion in regular trees, defying a decades-old belief that this is impossible.

At the heart of his theory is an edge-vertex entropy inequality. This inequality has been the starting point of many recent results, including the Gamarnik-Sudan theorem on local algorithms and the Backhausz-Szegedy theorem about eigenvector delocalization.

What are the analogous inequalities in other Cayley graphs and finite graphs? Our work with Csoka and Harangi tries to answer this.