A spectral version of the Moore problem for bipartite regular graphs
Algebraic Combinatorics, Volume 2 (2019) no. 6, pp. 1219-1238.

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 0θ<2k-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 g2d-2. For certain values of d, there are infinitely many such graphs of various valencies k. However, for d=11 or d15, we prove that there are no bipartite distance-regular graphs with g2d-2.

Received: 2018-05-02
Revised: 2019-03-04
Accepted: 2019-03-04
Published online: 2019-12-04
DOI: https://doi.org/10.5802/alco.71
Classification: 05B25,  05C35,  05C50,  05E30,  42C05,  94B65
Keywords: second eigenvalue, bipartite regular graph, bipartite distance-regular graph, expander, linear programming bound.
@article{ALCO_2019__2_6_1219_0,
     author = {Cioab\u a, Sebastian M. and Koolen, Jack H. and Nozaki, Hiroshi},
     title = {A spectral version of the Moore problem for bipartite regular graphs},
     journal = {Algebraic Combinatorics},
     pages = {1219--1238},
     publisher = {MathOA foundation},
     volume = {2},
     number = {6},
     year = {2019},
     doi = {10.5802/alco.71},
     mrnumber = {4049844},
     zbl = {1428.05187},
     language = {en},
     url = {alco.centre-mersenne.org/item/ALCO_2019__2_6_1219_0/}
}
Cioabă, Sebastian M.; Koolen, Jack H.; Nozaki, Hiroshi. A spectral version of the Moore problem for bipartite regular graphs. Algebraic Combinatorics, Volume 2 (2019) no. 6, pp. 1219-1238. doi : 10.5802/alco.71. https://alco.centre-mersenne.org/item/ALCO_2019__2_6_1219_0/

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