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.

在本文中，我们将几个新的不变量附加到连接的强正则图（除了顶点数为非平方的会议图）：一个称为判别式的不变量，以及对应于每个素数 p 的 p-adic 不变量。我们证明了具有给定连通块图的准对称 2-设计的参数限制$G$ 和一个给定的缺陷（两个交点数的绝对差）仅在缺陷和参数方面 $G$，包括这些新的不变量。这是 Schutzenberger 定理和 Shrikhande-Chowla-Ryser 定理的自然模拟。当可以显式计算这些图不变量时，该定理是有效的。我们对完整的多部图、共三角图、辛非正交图（在 2 阶域上）和 Steiner 图执行此操作，对块图属于任何这四个班级。