Euclidean Jordan Algebras with Strongly Regular Graphs

• Type: Article in International Scientific Journal
• Date: 2004

Authors

Domingos Moreira Cardoso (Author) Other; Luís Almeida Vieira (Author) FEUP

Journal

Title: Journal of Mathematical Sciences

Vol. 120 No. 1

Pages: 881-894

ISSN: 1072-3374

Publisher: Springer Nature

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Scientific classification

FOS: Natural sciences > Mathematics

CORDIS: Physical sciences > Mathematics

Other information

Authenticus ID: P-00H-J7Q

DOI: 10.1023/b:joth.0000013553.99598.cb

Abstract (EN): Homogeneous coherent symmetric algebras (also known as symmetric association schemes [1]) are particular cases of cellular algebras [9] and Euclidean Jordan algebras [8]. Cellular algebras appear in the study of symmetries and isomorphisms of graphs by using matrix-stabilization procedures and in the framework of the automorphism groups of graphs [4, 7]. Among several readable papers about Euclidean Jordan algebras, we refer to [3, 8]. Previous works on strongly regular graphs (see, e.g., [5, 9]) were developed with the aim of obtaining the minimal coherent algebra which includes a particular set of matrices and then its symmetric homogeneous coherent three-dimensional subalgebras, all of them defining strongly regular graphs [6]. In this paper, from a particular element of a basis B = {A1,... ,Am} of a Euclidean Jordan algebra Vn (where A1 = In is the identity matrix of order n), with as many distinct eigenvalues as the dimension of the algebra, we construct another basis B
of idempotents by exploiting the algebraic and combinatorial properties of Vn. From B
, we easily obtain the character table of Vn. Then, similarly to [2] (but in the different context of association schemes), fusing all the matrices of B but A1 and Am, a strongly regular graph is obtained for even n. In addition, for particular even values of n, other strongly regular graphs included in the Euclidean Jordan algebra Vn are deduced. The notation will be introduced throughout the remaining sections, which are organized as follows. In Sec. 2, we present the basic notions on Euclidean Jordan algebras and the Euclidean Jordan algebra Vn, which has specific combinatorial properties. In Sec. 3, we introduce a procedure for determining the character table of Vn. In Sec. 4, some strongly regular graphs whose adjacency matrices belong to Vn are deduced and the corresponding spectra determined. Finally, in Sec. 5, a numerical example is described.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 14