A spectral version of the Moore problem for bipartite regular graphs
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Let b(k,θ) be the maximum order of a connected bipartite k-regular graph whose second largest eigenvalue is at most θ. In this paper, we obtain a general upper bound for b(k,θ) for any 0≤θ< 2√(k-1). Our bound gives the exact value of b(k,θ) whenever there exists a bipartite distance-regular graph of degree k, second largest eigenvalue θ, diameter d and girth g such that g≥ 2d-2. For certain values of d and g, there are infinitely many such graphs. However, for d=11 or d≥ 15, we prove that there are no bipartite distance-regular graphs with g≥ 2d-2.
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