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The spectral excess theorem states that, in a regular graph G, the average excess, which is the mean of the numbers of vertices at maximum distance from a vertex, is bounded above by the spectral excess (a number that is computed by using the adjacency spectrum of G), and G is distance-regular if and only if equality holds.
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2014
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E.R. van Dam, J.H. Koolen, and H. Tanaka, Distance-regular graphs, manuscript (2014), available online at https://sites.google.com/site/edwinrvandam/home/papers/drg.pdf
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M.A. Fiol, S. Gago, and E. Garriga, A simple proof of the spectral excess theorem for distance-regular graphs, Linear Algebra Appl
2010
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G.-S. Lee, C.-W. Weng, A characterization of bipartite distance-regular graphs, Linear Algebra Appl
2014
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