On generalized binomial series and strongly regular graphs


  • Vasco Moco Mano University of Porto.
  • Enide Andrade Martins University of Aveiro.
  • Luis Antonio de Almeida Vieira University of Porto.




Strongly regular graph Euclidean Jordan algebra Matrix analysis.


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.

Author Biographies

Vasco Moco Mano, University of Porto.

Department of Mathematics, Faculty of Sciences. Rua do Campo Alegre 687; 4169-007, Porto.

Enide Andrade Martins, University of Aveiro.

CIDMA - Center for Research and Development in Math.and Appl. Department of Mathematics 3810-193 Aveiro.

Luis Antonio de Almeida Vieira, University of Porto.

CMUP - Center of Research of Mathematics Department of Mathematics, Faculty of Sciences.   Rua do Campo Alegre 687; 4169-007 Porto.


[1] L. W. Beineke, R. J. Wilson and P. J. Cameron, eds., Topics in Algebraic Graph Theory, Cambridge University Press, (2004).

[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

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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