Posets associated with subspaces in a \(d\)-bounded distance-regular graph. (English) Zbl 1216.05164
Summary: Let \(\Gamma= (X,R)\) denote a \(d\)-bounded distance-regular graph with diameter \(d\geq 3\). A regular strongly closed subgraph of \(\Gamma\) is said to be a subspace of \(\Gamma\). For \(x\in X\), let \(P(x)\) be the set of all subspaces of \(\Gamma\) containing \(x\). For each \(i= 1,2,\dots, d- 1\), let \(\Delta_0\) be a fixed subspace with diameter \(d- i\) in \(P(x)\), and let
\[
{\mathcal L}(d, i)= \{\Delta\in P(x)\mid\Delta+ \Delta_0= \Gamma, d(\Delta)= d(\Delta\cap\Delta_0)+ i\}\cup\{\emptyset\}.
\]
If we define the partial order on \({\mathcal L}(d,i)\) by ordinary inclusion (resp. reverse inclusion), then \({\mathcal L}(d,i)\) is a finite poser, denoted by \({\mathcal L}_0(d,i)\) (resp.\({\mathcal L}_R(d,i)\)). In the present paper we show that both \({\mathcal L}_0(d,i)\) and \({\mathcal L}_R(d,i)\) are atomic, and compute their characteristic polynomials.
References:
| [1] | Aigner, M., Combinatorial Theory (1979), Springer-Verlag: Springer-Verlag Berlin · Zbl 0415.05001 |
| [2] | Birgkhoff, G., Lattice Theory (1967), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0153.02501 |
| [3] | Brouwer, A. E.; Cohen, A. M.; Neumaier, A., Distance-Regular Graphs (1989), Springer-Verlag: Springer-Verlag New York · Zbl 0747.05073 |
| [4] | Gao, S.; Guo, J.; Liu, W., Lattices generated by strongly closed subgraphs in \(d\)-bounded distance-regular graphs, European J. Combin., 28, 1800-1813 (2007) · Zbl 1121.05127 |
| [5] | Gao, S.; Guo, J.; Zhang, B.; Fu, L., Subspaces in \(d\)-bounded distance-regular graphs and their applications, European J. Combin., 29, 592-600 (2008) · Zbl 1146.05020 |
| [6] | Gao, Y., Lattices generated by orbits of subspaces under finite singular unitary group and its characteristic polynomials, Linear Algebra Appl., 368, 243-268 (2003) · Zbl 1025.20038 |
| [7] | Gao, Y.; You, H., Lattices generated by orbits of subspaces under finite singular classical groups and its characteristic polynomials, Comm. Algebra, 31, 2927-2950 (2003) · Zbl 1034.51001 |
| [8] | Guo, J.; Gao, S., Lattices generated by join of strongly closed subgraphs in \(d\)-bounded distance-regular graphs, Discrete Math., 308, 1921-1929 (2008) · Zbl 1151.05052 |
| [9] | Guo, J.; Gao, S.; Wang, K., Lattices generated by subspaces in \(d\)-bounded distance-regular graphs, Discrete Math., 308, 5260-5264 (2008) · Zbl 1177.05131 |
| [10] | Guo, J., Lattices associated with finite vector spaces and finite affine spaces, Ars Combin., 88, 47-53 (2008) · Zbl 1224.51005 |
| [11] | Guo, J.; Li, Z.; Wang, K., Lattices associated with totally isotropic subspaces in classical spaces, Linear Algebra Appl., 431, 1088-1095 (2009) · Zbl 1171.51003 |
| [12] | Guo, J.; Nan, J., Lattices generated by orbits of flats under finite affine-symplectic groups, Linear Algebra Appl., 431, 536-542 (2009) · Zbl 1184.51004 |
| [13] | Huo, Y.; Liu, Y.; Wan, Z., Lattices generated by transitive sets of subspaces under finite classical groups I, Comm. Algebra, 20, 1123-1144 (1992) · Zbl 0763.51002 |
| [14] | Huo, Y.; Liu, Y.; Wan, Z., Lattices generated by transitive sets of subspaces under finite classical groups II, the orthogonal case of odd characteristic, Comm. Algebra, 20, 2685-2727 (1993) · Zbl 0821.20030 |
| [15] | Huo, Y.; Liu, Y.; Wan, Z., Lattices generated by transitive sets of subspaces under finite classical groups, the orthogonal case of even characteristic III, Comm. Algebra, 21, 2351-2393 (1993) · Zbl 0820.51007 |
| [16] | Huo, Y.; Wan, Z., On the geomericity of lattices generated by orbits of subspaces under finite classical groups, J. Algebra, 243, 339-359 (2001) · Zbl 1024.51004 |
| [17] | J. Nan, J. Guo, Lattices generated by two orbits of subspaces under finite singular classical groups, Comm. Algebra (in press); J. Nan, J. Guo, Lattices generated by two orbits of subspaces under finite singular classical groups, Comm. Algebra (in press) · Zbl 1215.51002 |
| [18] | Suzuki, H., On strongly closed subgraphs of highly regular graphs, European J. Combin., 16, 197-220 (1995) · Zbl 0821.05020 |
| [19] | Wang, K.; Feng, Y., Lattices generated by orbits of flats under finite affine groups, Comm. Algebra, 34, 1691-1697 (2006) · Zbl 1123.51007 |
| [20] | Wang, K.; Guo, J., Lattices generated by orbits of totally isotropic flats under finite affine-classical groups, Finite Fields Appl., 14, 571-578 (2008) · Zbl 1158.51003 |
| [21] | Wang, K.; Guo, J., Lattices generated by two orbits of subspaces under finite classical groups, Finite Fields Appl., 15, 236-245 (2009) · Zbl 1169.51010 |
| [22] | Wang, K.; Li, Z., Lattices associated with vector space over a finite field, Linear Algebra Appl., 429, 439-446 (2008) · Zbl 1161.05305 |
| [23] | Weng, C., D-bounded distance-regular graphs, European J. Combin., 18, 211-229 (1997) · Zbl 0869.05025 |
| [24] | Weng, C., Classical distance-regular graphs of negative type, J. Combin. Theory Ser. B, 76, 93-116 (1999) · Zbl 0938.05067 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
