On generalized binomial series and strongly regular graphs
DOI:
https://doi.org/10.4067/S0716-09172013000400007Keywords:
Strongly regular graph Euclidean Jordan algebra Matrix analysis.Abstract
We consider a strongly regular graph, G, and associate a three dimensional Euclidean Jordan algebra, V, to its adjacency matrix A. Then, by considering binomial series of Hadamard powers of the idem-potents of the unique complete system of orthogonal idempotents of V associated to A, we establish feasibility conditions for the existence of strongly regular graphs.References
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[2] R. C. Bose, Strongly regular graphs, partial geometries and partially balanced designs, Pacific J. Math 13, pp. 389-419, (1963). block designs with two associate classes, J. Am. Statist. Assoc. 47, pp. 151-184, (1952).
[3] A. E. Brouwer and J. H. van Lint, Strongly regular graphs and partial geometries, Enumeration and Design (D. M. Jackson and S. A. Vanstone, eds.), Academic Press, (1982).
[4] D. M. Cardoso and L. A. Vieira, Euclidean Jordan algebras with strongly regular graphs, Journal of Mathematical Sciences 120, pp. 881-894, (2004).
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[7] L. Faybusovich, Euclidean Jordan algebras and interior-point algorithms, J. Positivity 1, pp. 331-357, (1997).
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[10] R. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, (1985). Cambridge, (1991).
[11] X. L. Hubaut, Strongly regular graphs, Discrete Math. 13, pp. 357-381, (1975).
[12] P. Jordan, J. V. Neuman, and E. Wigner, On an algebraic generalization of the quantum mechanical formalism, Annals of Mathematics 35, pp. 29-64, (1934).
[13] M. Koecher, The Minnesota Notes on Jordan Algebras and Their Applications, Springer, Berlin, (1999).
[14] J. H. V. Lint and R. M. Wilson, A Course in Combinatorics, Cambridge University Press, Cambridge, (2004).
[15] V. M. Mano, E. A. Martins and L. A. Vieira, Feasibility conditions on the parameters of a strongly regular graph, Electronic Notes in Discrete Mathematics 38, pp. 607-613, (2011).
[16] V. M. Mano and L. Vieira, Admissibility conditions and asymptotic behavior of strongly regular graphs, International Journal of Mathematical Models and Methods in Applied Sciences, Issue 6, Vol 5, pp. 1027-1034, (2011).
[17] H. Massan and E. Neher, Estimation and testing for lattice conditional independence models on Euclidean Jordan algebras, Ann. Statist., 26, pp. 1051-1082, (1998).
[18] A. Neumaier, Strongly regular graphs with smallest eigenvalue -m, Archiv der Mathematik 33, pp. 392-400, (1979).
[19] L. L. Scott Jr., A condition on Higman’s parameters, Notices of Amer. Math. Soc. 20 (1973) A-97 (Abstract 721-20-45).
[2] R. C. Bose, Strongly regular graphs, partial geometries and partially balanced designs, Pacific J. Math 13, pp. 389-419, (1963). block designs with two associate classes, J. Am. Statist. Assoc. 47, pp. 151-184, (1952).
[3] A. E. Brouwer and J. H. van Lint, Strongly regular graphs and partial geometries, Enumeration and Design (D. M. Jackson and S. A. Vanstone, eds.), Academic Press, (1982).
[4] D. M. Cardoso and L. A. Vieira, Euclidean Jordan algebras with strongly regular graphs, Journal of Mathematical Sciences 120, pp. 881-894, (2004).
[5] Ph. Delsarte, J. M. Goethals and J. J. Seidel, Bounds for system of lines and Jacobi polynomials, Philips Res. Rep. 30, pp. 91-105, (1975).
[6] J. Faraut and A. Kor´anyi, Analysis on Symmetric Cones, Oxford Science Publications, Oxford, (1994).
[7] L. Faybusovich, Euclidean Jordan algebras and interior-point algorithms, J. Positivity 1, pp. 331-357, (1997).
[8] L. Faybusovich, Linear systems in Jordan algebras and primal-dual interior-point algorithms, Journal of Computational and Applied Mathematics 86, pp. 148-175, (1997).
[9] C. Godsil and G. Royle, Algebraic Graph Theory, Springer-Verlag, New York, (2001).
[10] R. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, (1985). Cambridge, (1991).
[11] X. L. Hubaut, Strongly regular graphs, Discrete Math. 13, pp. 357-381, (1975).
[12] P. Jordan, J. V. Neuman, and E. Wigner, On an algebraic generalization of the quantum mechanical formalism, Annals of Mathematics 35, pp. 29-64, (1934).
[13] M. Koecher, The Minnesota Notes on Jordan Algebras and Their Applications, Springer, Berlin, (1999).
[14] J. H. V. Lint and R. M. Wilson, A Course in Combinatorics, Cambridge University Press, Cambridge, (2004).
[15] V. M. Mano, E. A. Martins and L. A. Vieira, Feasibility conditions on the parameters of a strongly regular graph, Electronic Notes in Discrete Mathematics 38, pp. 607-613, (2011).
[16] V. M. Mano and L. Vieira, Admissibility conditions and asymptotic behavior of strongly regular graphs, International Journal of Mathematical Models and Methods in Applied Sciences, Issue 6, Vol 5, pp. 1027-1034, (2011).
[17] H. Massan and E. Neher, Estimation and testing for lattice conditional independence models on Euclidean Jordan algebras, Ann. Statist., 26, pp. 1051-1082, (1998).
[18] A. Neumaier, Strongly regular graphs with smallest eigenvalue -m, Archiv der Mathematik 33, pp. 392-400, (1979).
[19] L. L. Scott Jr., A condition on Higman’s parameters, Notices of Amer. Math. Soc. 20 (1973) A-97 (Abstract 721-20-45).
How to Cite
[1]
V. Moco Mano, E. Andrade Martins, and L. A. de Almeida Vieira, “On generalized binomial series and strongly regular graphs”, Proyecciones (Antofagasta, On line), vol. 32, no. 4, pp. 393-408, 1.
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