The second eigenvalue of regular graphs of given girth
Journal of Combinatorial Theory Series B
Spectra of regular graphs and hypergraphs and orthogonal polynomials
European Journal of Combinatorics
On the spectrum, the growth, and the diameter of a graph
Journal of Combinatorial Theory Series B
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Let A be the adjacency matrix of a d-regular graph of order n and girth g and d = λ1 ≥ ... ≥ λn its eigenvalues. Then Σj=2n λji=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 ≙=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 ≙ can be, given d, g, and n, and improve the best known bound for graphs whose girth exceeds their diameter.