The Parameters of Bipartite Q-polynomial Distance-Regular Graphs
John S.IV Caughman
IV
DOI: 10.1023/A:1015008423615
Abstract
Let 
denote a bipartite distance-regular graph with diameter D 
3 and valency k 3. Suppose 
0, 1, ..., D is a Q-polynomial ordering of the eigenvalues of . This sequence is known to satisfy the recurrence i - 1 - 
i + i + 1 = 0 (0 > i > D), for some real scalar . Let q denote a complex scalar such that q + q -1 = . Bannai and Ito have conjectured that q is real if the diameter D is sufficiently large.
denote a bipartite distance-regular graph with diameter D 3 and valency k 3. Suppose 0, 1, ..., D is a Q-polynomial ordering of the eigenvalues of . This sequence is known to satisfy the recurrence i - 1 - i + i + 1 = 0 (0 > i > D), for some real scalar . Let q denote a complex scalar such that q + q -1 = . Bannai and Ito have conjectured that q is real if the diameter D is sufficiently large. We settle this conjecture in the bipartite case by showing that q is real if the diameter D 4. Moreover, if D = 3, then q is not real if and only if 1 is the second largest eigenvalue and the pair ( 
, k) is one of the following: (1, 3), (1, 4), (1, 5), (1, 6), (2, 4), or (2, 5). We observe that each of these pairs has a unique realization by a known bipartite distance-regular graph of diameter 3.
4. Moreover, if D = 3, then q is not real if and only if 1 is the second largest eigenvalue and the pair ( , k) is one of the following: (1, 3), (1, 4), (1, 5), (1, 6), (2, 4), or (2, 5). We observe that each of these pairs has a unique realization by a known bipartite distance-regular graph of diameter 3.Pages: 223–229
Keywords: distance-regular graph; bipartite; association scheme; $P$-polynomial; $Q$-polynomial
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References
1. E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings, London, 1984.
2. N. Biggs, Algebraic Graph Theory, Cambridge University Press, Cambridge, 1974.
3. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, 1989.
4. J.S. Caughman, IV, “Spectra of bipartite P- and Q-polynomial association schemes,” Graphs Combin. 14 (1998), 321-343.
5. B. Curtin, “2-homogeneous bipartite distance-regular graphs,” Discrete Math. 187 (1998), 39-70.
6. G. Dickie, “Q-polynomial structures for association schemes and distance-regular graphs,” Ph.D. Thesis, University of Wisconsin, 1995.
7. F.S. Roberts, Applied Combinatorics, Prentice-Hall, New Jersey, 1984.
8. J.H. van Lint and R.M. Wilson, A Course in Combinatorics, Cambridge University Press, Cambridge, 1992.
2. N. Biggs, Algebraic Graph Theory, Cambridge University Press, Cambridge, 1974.
3. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, 1989.
4. J.S. Caughman, IV, “Spectra of bipartite P- and Q-polynomial association schemes,” Graphs Combin. 14 (1998), 321-343.
5. B. Curtin, “2-homogeneous bipartite distance-regular graphs,” Discrete Math. 187 (1998), 39-70.
6. G. Dickie, “Q-polynomial structures for association schemes and distance-regular graphs,” Ph.D. Thesis, University of Wisconsin, 1995.
7. F.S. Roberts, Applied Combinatorics, Prentice-Hall, New Jersey, 1984.
8. J.H. van Lint and R.M. Wilson, A Course in Combinatorics, Cambridge University Press, Cambridge, 1992.
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