{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:05Z","timestamp":1753893785040,"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 $\\Gamma$ be an antipodal distance-regular graph with diameter $4$ and eigenvalues $\\theta_0&gt;\\theta_1&gt;\\theta_2&gt;\\theta_3&gt;\\theta_4$. Then $\\Gamma$ is tight in the sense of Juri\u0161i\u0107, Koolen, and Terwilliger (J. Algebraic Combin, 2000) whenever $\\Gamma$ is locally strongly regular with nontrivial eigenvalues $p:=\\theta_2$ and $-q:=\\theta_3$. Assume that $\\Gamma$ is tight. Then the intersection numbers of $\\Gamma$ are expressed in terms of $p$, $q$, and $r$, where $r$ is the size of the antipodal classes of $\\Gamma$. We denote $\\Gamma$ by $\\mathrm{AT4}(p,q,r)$ and call this an antipodal tight graph of diameter $4$ with parameters $p,q,r$. In this paper, we give a new feasibility condition for the $\\mathrm{AT4}(p,q,r)$ family. We determine a necessary and sufficient condition for the second subconstituent of $\\mathrm{AT4}(p,q,2)$ to be an antipodal tight graph.Using this condition, we prove that there does not exist $\\mathrm{AT4}(q^3-2q,q,2)$ for $q\\equiv3$ $(\\mathrm{mod}~4)$. We discuss the $\\mathrm{AT4}(p,q,r)$ graphs with $r=(p+q^3)(p+q)^{-1}$.<\/jats:p>","DOI":"10.37236\/11332","type":"journal-article","created":{"date-parts":[[2023,4,7]],"date-time":"2023-04-07T09:01:18Z","timestamp":1680858078000},"source":"Crossref","is-referenced-by-count":0,"title":["A New Feasibility Condition for the AT4 Family"],"prefix":"10.37236","volume":"30","author":[{"given":"Zheng-Jiang","family":"Xia","sequence":"first","affiliation":[]},{"given":"Jae-Ho","family":"Lee","sequence":"additional","affiliation":[]},{"given":"Jack H.","family":"Koolen","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2023,4,7]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v30i2p7\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v30i2p7\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,7]],"date-time":"2023-04-07T09:01:18Z","timestamp":1680858078000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v30i2p7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,7]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,4,7]]}},"URL":"https:\/\/doi.org\/10.37236\/11332","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2023,4,7]]},"article-number":"P2.7"}}