A continuous analogue of the girth problem

Let A be the adjacency matrix of a d-regular graph of order n and girth g and d = λ₁ ≥ ⋯ ≥ λ_n its eigenvalues. Then ∑ⁿ_j=2 λⁱ_j = nt_i - dⁱ, for i = 0, 1, ⋯, g - 1, where t_i is the number of closed walks of length i on the d-regular infinite tree. Here we consider distributions on the real line, whose ith moment is also nt_i - dⁱ for all i = 0, 1, ⋯, g - 1. We investigate distributional analogues of several extremal graph problems involving the parameters n, d, g, and Λ = max λ₂, λ_n. Surprisingly, perhaps, many similarities hold between the graphical and the distributional situations. Specifically, we show in the case of distributions that the least possible n, given d, g is exactly the (trivial graph-theoretic) Moore bound. We also ask how small Λ can be, given d, g, and n, and improve the best known bound for graphs whose girth exceeds their diameter.