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A spectral version of the Moore problem for bipartite regular graphs. (English) Zbl 1428.05187
Summary: Let $$b(k,\theta)$$ be the maximum order of a connected bipartite $$k$$-regular graph whose second largest eigenvalue is at most $$\theta$$. In this paper, we obtain a general upper bound for $$b(k,\theta)$$ for any $$0\leq \theta < 2 \sqrt{k-1}$$. Our bound gives the exact value of $$b(k,\theta)$$ whenever there exists a bipartite distance-regular graph of degree $$k$$, second largest eigenvalue $$\theta$$, diameter $$d$$ and girth $$g$$ such that $$g\geq 2d-2$$. For certain values of $$d$$, there are infinitely many such graphs of various valencies $$k$$. However, for $$d = 11$$ or $$d\geq 15$$, we prove that there are no bipartite distance-regular graphs with $$g\geq 2d-2$$.
##### MSC:
 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C40 Connectivity 05C12 Distance in graphs 05C35 Extremal problems in graph theory 05E30 Association schemes, strongly regular graphs 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 90C05 Linear programming 94B65 Bounds on codes
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