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Intersection Matrices Revisited
Author(s) -
Ghareghani N.,
Ghorbani E.,
MohammadNoori M.
Publication year - 2012
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.21308
Subject(s) - intersection (aeronautics) , mathematics , combinatorics , rank (graph theory) , eigenvalues and eigenvectors , operator (biology) , adjacency matrix , scheme (mathematics) , discrete mathematics , algebra over a field , pure mathematics , mathematical analysis , engineering , gene , aerospace engineering , graph , biochemistry , chemistry , physics , repressor , quantum mechanics , transcription factor
Several intersection matrices of s ‐subsets versus k ‐subsets of a v ‐set are introduced in the literature. We study these matrices systematically through counting arguments and generating function techniques. A number of new or known identities appear as natural consequences of this viewpoint; especially, use of the derivative operator d / d z and some related operators reveals some connections between intersection matrices and the “combinatorics of creation‐annihilation.” As application, the eigenvalues of several intersection matrices including some generalizations of the adjacency matrices of the Johnson scheme are derived; two new bases for the Bose–Mesner algebra of the Johnson scheme are introduced and the associated intersection numbers are obtained as well. Finally, we determine the rank of some intersection matrices. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 383–397, 2012