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 AM  Vol.9 No.9 , September 2018
Binomial Hadamard Series and Inequalities over the Spectra of a Strongly Regular Graph
Abstract:
Let G be a primitive strongly regular graph of order n and A is adjacency matrix. In this paper we first associate to A a real 3-dimensional Euclidean Jordan algebra  with rank three spanned by In and the natural powers of A that is a subalgebra of the Euclidean Jordan algebra of symmetric matrix of order n. Next we consider a basis  that is a Jordan frame of . Finally, by an algebraic asymptotic analysis of the second spectral decomposition of some Hadamard series associated to A we establish some inequalities over the spectra and over the parameters of a strongly regular graph.
Cite this paper: Vieira, L. (2018) Binomial Hadamard Series and Inequalities over the Spectra of a Strongly Regular Graph. Applied Mathematics, 9, 1055-1071. doi: 10.4236/am.2018.99071.
References

[1]   McCrimmon, K. (2000) A Taste on Jordan Algebras. Springer, New York.

[2]   Faraut, J. and Korányi, A. (1994) Analysis on Symmetric Cones, Oxford Mathematical Monographs. Clarendon Press, Oxford.

[3]   Koecher, M. (1999) The Minnesota Notes on Jordan Algebras and Their Applications. Springer, Berlin.
https://doi.org/10.1007/BFb0096285

[4]   Cardoso, D.M. and Vieira, L.A. (2006) On the Optimal Parameter of a Self-Concordant Barrier over a Symmetric Cone. European Journal of Operational Research, 169, 1148-1157.
https://doi.org/10.1016/j.ejor.2004.11.027

[5]   Schmieta, S.H. and Alizadeh, F. (2001) Associative and Jordan Algebras, and Polynomial time Interior-Point Algorithms for Symmetric Cones. Mathematics of Operations Research, 26, 543-564.
https://doi.org/10.1287/moor.26.3.543.10582

[6]   Faybusovich, L. (1997) Linear Systems in Jordan Algebras and Primal-Dual Interior-Point Algorithms. Journal of Computational and Applied Mathematics, 86, 149-175.
https://doi.org/10.1016/S0377-0427(97)00153-2

[7]   Faybusovich, L. (1997) Euclidean Jordan Algebras and Interior-Point Algorithms. Positivity, 1, 331-357.
https://doi.org/10.1023/A:1009701824047

[8]   Jordan, P., Neuman, J.P. and Wigner, E. (1934) On an Algebraic Generalization of the Quantum Mechanical Formalism. Annals of Mathematics, 35, 29-64.
https://doi.org/10.2307/1968117

[9]   Vieira, L.A. (2018) Asymptotic Properties of the Spectra of a Strongly Regular Graph, Innovation, Engineering and Entrepreneurship. Lecture notes in Electrical Engineering, Book Series, 505, 800-804.

[10]   Cardoso, D.M. and Vieira, L.A. (2004) Euclidean Jordan Algebras with Strongly Regular Graphs. Journal of Mathematical Sciences, 120, 881-894.
https://doi.org/10.1023/B:JOTH.0000013553.99598.cb

[11]   Mano, V.M., Martins, E.A. and Vieira, L.A. (2011) On Generalized Binomial Series and Strongly Regular Graphs. Proyecciones Journal of Mathematics, 4, 393-408.

[12]   Mano, V.M. and Vieira, L.A. (2011) Admissibility Conditions and Asymptotic Behaviour of Strongly Regular Graph. International Journal of Mathematical Models and Methods in Applied Sciences, 5, 1027-1033.

[13]   Mano, V.M. and Vieira, L.A. (2014) Alternating Schur Series and Necessary Conditions for the Existence of Strongly Graphs. International Journal of Mathematical Models and Methods in Applied Sciences Methods, 8, 256-261.

[14]   Vieira, L.A. (2009) Euclidean Jordan Algebras and Inequalities on the Parameters of a Strongly Regular Graph. AIP Conference Proceedings, 1168, 995-998.

[15]   Vieira, L.A. and Mano, V.M. (2015) Generalized Krein Parameters of a Strongly Regular Graph. Applied Mathematics, 6, 37-45.
https://doi.org/10.4236/am.2015.61005

[16]   Massam, H. and Neher, E. (1998) Estimation and Testing for Lattice Condicional Independence Models on Euclidean Jordan Algebras. Annals of Statistics, 26, 1051-1082.
https://doi.org/10.1214/aos/1024691088

[17]   Gowda, M.S. and Tao, J. (2009) Some Inertia Theorems in Euclidean Jordan Algebras. Linear Algebra and Its Applications, 430, 19991-2011.
https://doi.org/10.1016/j.laa.2008.11.015

[18]   Snadjder, R., Gowda, M.S. and Moldovan, M.M. (2012) More Results on Schur Complements in Euclidean Jordan Algbras. Journal of Global Optimization, 53, 121-134.
https://doi.org/10.1007/s10898-011-9734-x

[19]   Gowda, M.S. (2011) Some Inequalities Involving Determinants, Eigenvalues, and Schur Complements in Euclidean Jordan Algebras. Positivity, 15, 381-399.
https://doi.org/10.1007/s11117-010-0086-4

[20]   Alizadeh, F. (2012) An Introduction to Formally Real Jordan Algebras and Their Applications in Optimization. In: Anjos, M.F. and Lasserre, J.B., Eds., Handbook on Semidefinite, Conic and Polynomial Optimization, Springer, New York, 297-338.

[21]   Godsil, C. and Royle, G. (1993) Algebraic Graph Theory. Chapman & Hall, New York.

[22]   Scott, L.L. (1973) A Condition on Higman’s Parameters. Notices of the American Mathematical Society, 20, A-97.

[23]   Delsart, P., Goethals, J.M. and Seidel, J.J. (1975) Bounds for Systems of Lines and Jacobi Polynomials. Philips Research Reports, 30, 91-95.

[24]   Horn, A.R. and Johnson, C.R. (1991) Topics in Matrix Analysis. Cambridge University Press, Cambridge.

 
 
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