An entropy based proof of the Moore bound for irregular graphs

We provide proofs of the following theorems by considering the entropy of random walks: Theorem 1.(Alon, Hoory and Linial) Let G be an undirected simple graph with n vertices, girth g, minimum degree at least 2 and average degree d: Odd girth: If g=2r+1,then n \geq 1 + d*(\Sum_{i=0}^{r-1}(d-1)^i) Even girth: If g=2r,then n \geq 2*(\Sum_{i=0}^{r-1} (d-1)^i) Theorem 2.(Hoory) Let G = (V_L,V_R,E) be a bipartite graph of girth g = 2r, with n_L = |V_L| and n_R = |V_R|, minimum degree at least 2 and the left and right average degrees d_L and d_R. Then, n_L \geq \Sum_{i=0}^{r-1}(d_R-1)^{i/2}(d_L-1)^{i/2} n_R \geq \Sum_{i=0}^{r-1}(d_L-1)^{i/2}(d_R-1)^{i/2}

💡 Research Summary

The paper presents a novel proof of the Moore bound for irregular (non‑regular) graphs by exploiting the entropy of random walks. The Moore bound gives a lower bound on the number of vertices a graph must have given its average degree and girth (the length of the shortest cycle). Traditional proofs rely on combinatorial expansion trees or algebraic arguments and become cumbersome when the graph is irregular, i.e., when vertex degrees vary. The authors replace these techniques with an information‑theoretic approach: they consider a simple random walk on the graph, define the entropy of the distribution over all length‑(k) walk paths, and relate this entropy to the number of distinct vertices reachable within (k) steps.

The key observation is that if the girth is at least (g), then for the first (\lfloor g/2\rfloor) steps the walk cannot encounter a cycle; the walk’s trajectory is exactly the same as that in a tree that expands from the starting vertex. In a tree whose root has average degree (d) and whose subsequent levels expand by a factor of ((d-1)), the number of distinct vertices at distance at most (r) from the root is \