On optimality of designs with three distinct eigenvalues
  Let $\D_{v,b,k}$ denote the family of all connected block designs with $v$ treatments and $b$ blocks of size $k$. Let $d\in\D_{v,b,k}$. The replication of a treatment is the number of times it appears in the blocks of $d$. The matrix $C(d)=R(d)-\frac{1}{k}N(d)N(d)^\top$ is called the information matrix of $d$ where $N(d)$ is the incidence matrix of $d$ and $R(d)$ is a diagonal matrix of the replications. Since $d$ is connected, $C(d)$ has $v-1$ nonzero eigenvalues $\mu_1(d),...,\mu_{v-1}(d)$. Let $\D$ be the class of all binary designs of $\D_{v,b,k}$. We prove that if there is a design $d^*\in\D$ such that (i) $C(d^*)$ has three distinct eigenvalues, (ii) $d^*$ minimizes trace of $C(d)^2$ over $d\in\D$, (iii) $d^*$ maximizes the smallest nonzero eigenvalue and the product of the nonzero eigenvalues of $C(d)$ over $d\in\D$, then for all $p>0$, $d^*$ minimizes $(\sum_{i=1}^{v-1}\mu_i(d)^{-p})^{1/p}$ over $d\in\D$. In the context of optimal design theory, this means that if there is a design $d^*\in\D$ such that its information matrix has three distinct eigenvalues satisfying the condition (ii) above and that $d^*$ is E- and D-optimal in $\D$, then $d^*$ is $\Phi_p$-optimal in $\D$ for all $p>0$. As an application, we demonstrate the $\Phi_p$-optimality of certain group divisible designs. Our proof is based on the method of KKT conditions in nonlinear programming. 
M. R. Faghihi, E. Ghorbani, G. B. Khosrovshahi, and S. Tat
math.CO