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 d̄. Odd girth: If g = 2r + 1, then n ≥ 1 + d̄ r−1 ∑ i=0 (d̄− 1)i. Even girth: If g = 2r, then n ≥ 2 r−1 ∑ i=0 (d̄− 1)i. Theorem 2.(Hoory) Let G = (VL, VR, E) be a bipartite graph of girth g = 2r, with nL = |VL| and nR = |VR|, minimum degree at least 2 and the left and right average degrees dL and dR. Then, nL ≥ r−1 ∑ i=0 (dR − 1) ⌈ i 2 (dL − 1) ⌊ i 2 ⌋ ,