On the p-Ranks of the Adjacency Matrices of Distance-Regular Graphs
René Peeters
DOI: 10.1023/A:1013842904024
Abstract
Let 
be a distance-regular graph with adjacency matrix A. Let I be the identity matrix and J the all-1 matrix. Let p be a prime. We study the p-rank of the matrices A + bJ - cI for integral b, c and the p-rank of corresponding matrices of graphs cospectral with .
be a distance-regular graph with adjacency matrix A. Let I be the identity matrix and J the all-1 matrix. Let p be a prime. We study the p-rank of the matrices A + bJ - cI for integral b, c and the p-rank of corresponding matrices of graphs cospectral with . Using the minimal polynomial of A and the theory of Smith normal forms we first determine which p-ranks of A follow directly from the spectrum and which, in general, do not. For the p-ranks that are not determined by the spectrum (the so-called relevant p-ranks) of A the actual structure of the graph can play a rôle, which means that these p-ranks can be used to distinguish between cospectral graphs.
Pages: 127–149
Keywords: $p$-rank; distance-regular graph; adjacency matrix; minimal polynomial
Full Text: PDF
References
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2. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Ergebnisse der Mathematik 3.18, Springer, Heidelberg, 1989.
3. A.E. Brouwer and C.A. van Eyl, “On the p-rank of the adjacency matrices of strongly regular graphs,” J. Alg. Combin. 1 (1992), 329-346.
4. A.E. Brouwer and H.A. Wilbrink, “Ovoids and fans in the generalized quadrangle Q(4, 2),” Geom. Dedicata 36 (1990), 121-124.
5. E.R. van Dam, “Regular graphs with four eigenvalues,” Linear Algebra Appl. 226-228 (1995), 139-162.
6. C.D. Godsil, “Modular Bose-Mesner Algebras,” Preprint, 1999.
7. W.H. Haemers, “Distance-regularity and the spectrum of graphs,” Linear Algebra Appl. 236 (1996), 265-278.
8. W.H. Haemers and E. Spence, “Graphs cospectral with distance-regular graphs,” Linear and Multilinear Algebra 39 (1995), 91-107.
9. E.S. Lander, Symmetric Designs: An Algebraic Approach, London Math. Soc. Lect. Notes Series 74, Cambridge Univ. Press, Cambridge, 1983.
10. R. Peeters, “Ranks and structure of graphs,” Ph.D. Thesis, Tilburg University, 1995.
11. R. Peeters, “Uniqueness of strongly regular graphs having minimal p-rank,” Linear Algebra Appl. 226-228 (1995), 9-31.
12. R.M. Wilson, “A diagonal form for the incidence matrices of t-subsets vs. k-subsets,” European J. Combin. 11 (1990), 609-615.
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