Abstract
In this paper we establish inequalities over the parameters and over the spectra of strongly regular graphs in the environment of Euclidean Jordan algebras. We consider a strongly regular graph, G, whose adjacency matrix A has three distinct eigenvalues, and the Euclidean Jordan algebra of real symmetric matrices of order n, \(\text{Sym}(n, \mathbb{R})\) with the vector product and the inner product being the Jordan product and the usual trace of matrices, respectively. We associate a three dimensional real Euclidean Jordan subalgebra \(\mathcal{A}\) of \(\text{Sym}(n, \mathbb{R})\) to A, spanned by the identity matrix and the natural powers of A. Next, we compute the unique complete system of orthogonal idempotents \(\mathcal{B}\) associated to A and we consider particular convergent Hadamard series constructed from the idempotents of \(\mathcal{B}\). Finally, by the analysis of the spectra of the sums of these Hadamard series we establish new conditions for the existence of a strongly regular graph.
Keywords
- Regular Graphs
- Euclidean Jordan Algebra
- Complete Systematic Unity
- Orthogonal Idempotents
- Hadamard Series
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Acknowledgements
Vasco Mano and Enide A. Martins were supported by Portuguese funds through the CIDMA—Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT—Fundação para a Ciência e a Tecnologia”), within project PEst-OE/MAT/UI4106/2014. Enide Martins was also supported by Project PTDC/MAT/112276/2009.
Luís Vieira was partially funded by the European Regional Development Fund Through the program COMPETE and by the Portuguese Government through the FCT—Fundação para a Ciência e a Tecnologia under the project PEst C/MAT/UI0144/2013.
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Mano, V.M., Martins, E.A., Vieira, L.A. (2014). Inequalities on the Parameters of a Strongly Regular Graph. In: Pinto, A., Zilberman, D. (eds) Modeling, Dynamics, Optimization and Bioeconomics I. Springer Proceedings in Mathematics & Statistics, vol 73. Springer, Cham. https://doi.org/10.1007/978-3-319-04849-9_38
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DOI: https://doi.org/10.1007/978-3-319-04849-9_38
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