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Designs over edge-regular, co-edge-regular and amply regular graphs are investigated. Using linear algebra, we obtain lower bounds in certain inequalities involving the parameters of the designs. Some results on designs meeting the bounds are obtained. These designs are over connected regular graphs with least eigenvalue $$-2$$ , have the minimal number of blocks and do not appear in an earlier work. Partial classification such designs over strongly regular graphs with least eigenvalue $$-2$$ is given.

Triangle-free quasi-symmetric 2- designs with intersection numbers ; and are investigated. Possibility of triangle-free quasi-symmetric designs with or is ruled out. It is also shown that, for a fixed x and a fixed ratio , there are only finitely many triangle-free quasi-symmetric designs. © 2012 Wiley Periodicals, Inc. J Combin Designs 00: 1-6, 2012

Triangle-free quasi-symmetric 2- (v, k, λ) designs with intersection numbers x, y; 0 1, are investigated. It is proved that λ⩾2y − x − 3. As a consequence it is seen that for fixed λ, there are finitely many triangle-free quasi-symmetric designs. It is also proved that: k⩽y(y − x) + x. Copyright © 2011 Wiley Periodicals, Inc. J Combin Designs 19:422-426, 2011

The following results for proper quasi-symmetric designs with non-zero intersection numbers x,y and λ > 1 are proved. (1) Let D be a quasi-symmetric design with z = y − x and v ≥ 2k. If x ≥ 1 + z + z3 then λ

The following result is proved: Let D be a quasi-symmetric 3-design with intersection numbers x, y(0≤x