Parametric restrictions on quasi-symmetric designs. (January 2022)

• Record Type: Journal Article
• Authors: Bagchi, Bhaskar
• Is Part Of: European journal of combinatorics. Volume 99(2022)
• Publication Date: 2022-01
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2021.103434
• Languages: English
• ISSNs: 0195-6698

Abstract

Abstract: In this paper, we attach several new invariants to connected strongly regular graphs (excepting conference graphs on non-square number of vertices): one invariant called the discriminant, and a p-adic invariant corresponding to each prime number p. We prove parametric restrictions on quasi-symmetric 2-designs with a given connected block graph G and a given defect (absolute difference of the two intersection numbers) solely in terms of the defect and the parameters of G, including these new invariants. This is a natural analogue of Schutzenberger's Theorem and the Shrikhande–Chowla–Ryser theorem. This theorem is effective when these graph invariants can be explicitly computed. We do this for complete multipartite graphs, co-triangular graphs, symplectic non-orthogonality graphs (over the field of order 2) and the Steiner graphs, yielding explicit restrictions on the parameters of quasi-symmetric 2-designs whose block graphs belong to any of these four classes.