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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.32 no.4 Antofagasta Dec. 2013

http://dx.doi.org/10.4067/S0716-09172013000400007 

 

On generalized binomial series and strongly regular graphs

 

Vasco Moço Mano, Luís António de Almeida Vieira
University of Porto, Portugal

Enide Andrade Martins
University of Aveiro, Portugal


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.

Keyword : Strongly regular graph Euclidean Jordan algebra Matrix analysis.


REFERENCES

1 L.W.Beineke,R.J.Wilsonand P. J. Cameron, eds.,Topicsin Algebraic Graph Theory, Cambridge University Press, (2004).         [ Links ]

2 R. C. Bose, Strongly regular graphs, partial geometries and partially balanced designs, Pacific J. Math 13, pp. 389-419, (1963).         [ Links ]

3 R. C. Bose, block designs with two associate classes, J. Am. Statist. Assoc. 47, pp. 151-184, (1952).         [ Links ]

4 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).         [ Links ]

5 D. M. Cardoso and L. A. Vieira, Euclidean Jordan algebras with strongly regular graphs, Journal of Mathematical Sciences 120, pp.881-894, (2004).         [ Links ]

6 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).         [ Links ]

7 J. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford Science Publications, Oxford, (1994).         [ Links ]

8 L. Faybusovich, Euclidean Jordan algebras and interior-point algorithms, J. Positivity 1, pp. 331-357, (1997).         [ Links ]

9 L. Faybusovich, Linear systems in Jordan algebras and primal-dual interior-point algorithms, Journal of Computational and Applied Mathematics 86, pp. 148-175, (1997).         [ Links ]

10 C. Godsil and G. Royle, Algebraic Graph Theory, Springer-Verlag, New York, (2001).         [ Links ]

11 R. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, (1985)        [ Links ]

12 X. L. Hubaut, Strongly regular graphs, Discrete Math. 13, pp. 357-381,(1975).         [ Links ]

13 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).         [ Links ]

14 M. Koecher, The Minnesota Notes on Jordan Algebras and Their Applications, Springer, Berlin, (1999).         [ Links ]

15 J. H. V. Lint and R. M. Wilson, A Course in Combinatorics, Cambridge University Press, Cambridge, (2004).         [ Links ]

16 V. M. Mano, E.A.MartinsandL.A. Vieira, Feasibility conditions on the parameters of a strongly regular graph, Electronic Notes in Discrete Mathematics 38, pp. 607-613, (2011).         [ Links ]

17 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).         [ Links ]

18 H. Massan and E. Neher, Estimation and testing for lattice conditional independence models on Euclidean Jordan algebras, Ann. Statist., 26, pp. 1051-1082, (1998).         [ Links ]

19 A. Neumaier, Strongly regular graphs with smallest eigenvalue -m, Archiv der Mathematik 33, pp. 392-400, (1979).         [ Links ]

20 L. L. Scott Jr., A condition on Higman's parameters, Notices of Amer. Math. Soc. 20 (1973) A-97 (Abstract 721-20-45).         [ Links ]

 

Vasco Moco Mano
Department of Mathematics
Faculty of Sciences
University of Porto
Rua do Campo Alegre
687; 4169-007, Porto, Portugal
e-mail : vascomocomano@gmail.com

 

Enide Andrade Martins
CIDMA - Center for Research and Development in Math. and Appl.
Department of Mathematics
University of Aveiro
3810-193 Aveiro
Portugal
e-mail : enide@ua.pt

 

Luis Antonio de Almeida Vieira
CMUP - Center of Research of Mathematics
Department of Mathematics
Faculty of Sciences
University of Porto
Rua do Campo Alegre
687; 4169-007 Porto
Portugal
e-mail : lvieira@fe.up.pt

1. Enide Andrade Martins is supported in part by FEDER funds through COMPETE Operational Programme Factors of Competitiveness ("Programa Operacional Factores de Competitividade") and by Portuguese funds through the Center for Research and Develpment in Mathematics and Applications and the Portuguese Foundation for Science and Technology (iFCT - Fundaçao paraa Ciência e a Tecnologia") within project PEest-C/MAT/UI4106/ 2011 with COMPETE number FCOMP-01-0124-FEDER-022690 and Project PTDC/MAT/ 112276/2009.

2. Luis Vieira research funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the FCT -Fundaçao paraa Ciência e a Tecnologia under the project PEest-C/MAT/UI0144/2011.

Received : April 2013. Accepted : August 2013.

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