AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 68(1) (2017), Pages 62–70 Characterization of quasi-symmetric designs with eigenvalues of their block graphs Shubhada M. Nyayate Department of Mathematics, Dnyanasadhana College Thane-400 604 India nyayate.shubhada@gmail.com Rajendra M. Pawale ∗ Department of Mathematics, University of Mumbai Vidyanagari, Santacruze (East), Mumbai 400 098 India rmpawale@yahoo.co.in Mohan S. Shrikhande Mathematics Department, Central Michigan University Mount Pleasant, MI, 48859 U.S.A. Mohan.Shrikhande@cmich.edu Abstract A quasi-symmetric design (QSD) is a (v,k,λ) design with two intersection numbers x, y , where 0 ≤ x<y<k. The block graph of a QSD is a strongly regular graph (SRG), whereas the converse is not true. Using Neumaier’s classification of SRGs related to the smallest eigenvalue, a complete parametric classification of QSDs whose block graph is an SRG with smallest eigenvalue -3, or second largest eigenvalue 2, is obtained. 1 Introduction Let X be a finite set of v elements called points, and β be a set of k-element subsets of X called blocks, such that each pair of points occurs in λ blocks. Then the pair * Corresponding author.