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⩾…⩾λn its eigenvalues. Then ∑nj=2λij=nti−di, for i=0, 1, …, g−1, where ti 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 nti−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|, |λn|. 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.