ALGEBRAIC COMBINATORICS

no. 6
A spectral version of the Moore problem for bipartite regular graphs
Cioabă, Sebastian M.1; Koolen, Jack H.2; Nozaki, Hiroshi3
1 University of Delaware Department of Mathematical Sciences Ewing Hall Newark, DE 19716-2553, USA
2 School of Mathematical Sciences University of Science and Technology of China Wen-Tsun Wu Key Laboratory of the Chinese Academy of Sciences Hefei, Anhui, China
3 Aichi University of Education 1 Hirosawa, Igaya-cho, Kariya Aichi 448-8542, Japan
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 0≤θ<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 g≥2d-2. For certain values of d, there are infinitely many such graphs of various valencies k. However, for d=11 or d≥15, we prove that there are no bipartite distance-regular graphs with g≥2d-2.

Received:
Revised:
Accepted:
Published online:
DOI: 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.
Cioabă, Sebastian M. 1; Koolen, Jack H. 2; Nozaki, Hiroshi 3

1 University of Delaware Department of Mathematical Sciences Ewing Hall Newark, DE 19716-2553, USA
2 School of Mathematical Sciences University of Science and Technology of China Wen-Tsun Wu Key Laboratory of the Chinese Academy of Sciences Hefei, Anhui, China
3 Aichi University of Education 1 Hirosawa, Igaya-cho, Kariya Aichi 448-8542, Japan
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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/articles/10.5802/alco.71/

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