{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:59Z","timestamp":1753893839563,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let ${\\cal D}_{v,b,k}$\u00a0denote the family of all connected block designs\u00a0with $v$ treatments and $b$ blocks of size $k$. Let $d\\in{\\cal D}_{v,b,k}$.\u00a0The replication of a treatment is the number of times it appears in the blocks of $d$.\u00a0The 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.\u00a0Since $d$ is connected, $C(d)$ has $v-1$ nonzero eigenvalues $\\mu_1(d),\\ldots,\\mu_{v-1}(d)$.Let ${\\cal D}$ be the class of all binary designs of ${\\cal D}_{v,b,k}$.\u00a0We prove that if there is a design $d^*\\in{\\cal D}$ such that (i) $C(d^*)$ has three distinct eigenvalues, (ii) $d^*$ minimizes trace of $C(d)^2$ over $d\\in{\\cal D}$, (iii) $d^*$ maximizes the smallest nonzero eigenvalue and the product of the nonzero eigenvalues of $C(d)$ over $d\\in{\\cal D}$,\u00a0then for all $p&gt;0$, $d^*$ minimizes $\\left(\\sum_{i=1}^{v-1}\\mu_i(d)^{-p}\\right)^{1\/p}$ over $d\\in{\\cal D}$.\u00a0In the context of optimal design theory, this means that if there is a design $d^*\\in{\\cal D}$ such that its information matrix has three distinct eigenvalues satisfying the condition (ii) above and that $d^*$ is E- and D-optimal in ${\\cal D}$,\u00a0then $d^*$ is $\\Phi_p$-optimal in ${\\cal D}$ for all $p&gt;0$. As an application, we demonstrate the $\\Phi_p$-optimality of certain group divisible designs.\u00a0Our proof is based on the method of KKT conditions in nonlinear programming.<\/jats:p>","DOI":"10.37236\/2709","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T00:13:13Z","timestamp":1578701593000},"source":"Crossref","is-referenced-by-count":1,"title":["On Optimality of Designs with Three Distinct Eigenvalues"],"prefix":"10.37236","volume":"20","author":[{"given":"M. R.","family":"Faghihi","sequence":"first","affiliation":[]},{"given":"E.","family":"Ghorbani","sequence":"additional","affiliation":[]},{"given":"G. B.","family":"Khosrovshahi","sequence":"additional","affiliation":[]},{"given":"S.","family":"Tat","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2013,4,24]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i2p16\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i2p16\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T06:21:14Z","timestamp":1579242074000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v20i2p16"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,4,24]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2013,4,9]]}},"URL":"https:\/\/doi.org\/10.37236\/2709","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2013,4,24]]},"article-number":"P16"}}