Large regular bipartite graphs with median eigenvalue 1
β Scribed by Guo, Krystal; Mohar, Bojan
- Book ID
- 122179896
- Publisher
- Elsevier Science
- Year
- 2014
- Tongue
- English
- Weight
- 256 KB
- Volume
- 449
- Category
- Article
- ISSN
- 0024-3795
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π SIMILAR VOLUMES
We show that, if a bipartite distance-regular graph of valency k has an eigenvalue of multiplicity k, then it becomes 2-homogeneous. Combined with a result on bipartite 2-homogeneous distance-regular graphs by K. Nomura, we have a classification of such graphs.
## Abstract Let __B(G)__ be the edge set of a bipartite subgraph of a graph __G__ with the maximum number of edges. Let __b~k~__ = inf{|__B(G)__|/|__E(G)__β__G__ is a cubic graph with girth at least __k__}. We will prove that lim~k β β~ __b~k~__ β₯ 6/7.
We show that any k-regular bipartite graph with 2n vertices has at least \ (k&1) k&1 k k&2 + n perfect matchings (1-factors). Equivalently, this is a lower bound on the permanent of any nonnegative integer n\_n matrix with each row and column sum equal to k. For any k, the base (k&1) k&1 Γk k&2 is l