zbMATH — the first resource for mathematics

On bipartite \(Q\)-polynomial distance-regular graphs with diameter 9, 10, or 11. (English) Zbl 1391.05174
Summary: Let \(\Gamma\) denote a bipartite distance-regular graph with diameter \(D\). In [Graphs Comb. 20, No. 1, 47–57 (2004; Zbl 1054.05101)] J. S. Caughman IV showed that if \(D \geq 12\), then \(\Gamma\) is \(Q\)-polynomial if and only if one of the following (i)-(iv) holds: (i) \(\Gamma\) is the ordinary \(2D\)-cycle, (ii) \(\Gamma\) is the Hamming cube \(H(D,2)\), (iii) \(\Gamma\) is the antipodal quotient of the Hamming cube \(H(2D,2)\), (iv) the intersection numbers of \(\Gamma\) satisfy \(c_i = (q^i - 1)/(q-1)\), \(b_i = (q^D-q^i)/(q-1)\) \((0 \leq i \leq D)\), where \(q\) is an integer at least \(2\). In this paper we show that the above result is true also for bipartite distance-regular graphs with \(D \in \{9,10,11\}\).

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05E30 Association schemes, strongly regular graphs
Zbl 1054.05101
Full Text: Link
[1] E. Bannai, T. Ito, Algebraic Combinatorics I: Association Schemes, The Benjamin Cummings Lecture Notes Ser. 58, Menlo Park, CA, 1984.
[2] A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer Verlag, Berlin, Heidelberg, 1989. · Zbl 0747.05073
[3] J. S. Caughman, Spectra of bipartite P - and Q-polynomial association schemes, Graphs Combin. 14 (1998), 321-343. · Zbl 0917.05088
[4] J. S. Caughman, The Terwilliger algebras of bipartite P - and Q-polynomial schemes, Discrete Math. 196 (1999), 65-95. · Zbl 0924.05067
[5] J. S. Caughman, Bipartite Q-polynomial quotients of antipodal distance-regular graphs, J. Combin. Theory Ser. B 76 (1999), 291-296. · Zbl 0938.05064
[6] J. S. Caughman, The parameters of bipartite Q-polynomial distance-regular graphs, J. Algebraic Combin. 15 (2002), 223-229. · Zbl 0997.05098
[7] J. S. Caughman, Bipartite Q-polynomial distance-regular graphs, Graphs Combin. 20 (2004), 47-57. · Zbl 1054.05101
[8] B. Curtin, Almost 2-homogeneous bipartite distance-regular graphs, European J. Combin. 21 (2000), 865-876. · Zbl 1002.05069
[9] G. Dickie, Q-polynomial structures for association schemes and distance-regular graphs, Ph.D. Thesis, University of Wisconsin, 1995. · Zbl 0852.05085
[10] E. R. van Dam, J. H. Koolen, and H. Tanaka, Distance-regular graphs, Electron. J. Combin. #DS22 (2016). the electronic journal of combinatorics 25(1) (2018), #P1.52 11 · Zbl 1335.05062
[11] C. D. Godsil, Algebraic combinatorics, Chapman and Hall, New York, 1993.
[12] M. S. Lang, Bipartite distance-regular graphs: The Q -polynomial property and pseudo primitive idempotents, Discrete Math. 331 (2014), 27-35. · Zbl 1297.05264
[13] K. Nomura, Homogeneous graphs and regular near polygons, J. Combin. Theory Ser. B 60 (1994), 63-71. · Zbl 0793.05130
[14] K. Nomura, Spin models on bipartite distance-regular graphs, J. Combin. Theory Ser. B 64 (2) (1994), 300-313. · Zbl 0827.05060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.