{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:41Z","timestamp":1753893821631,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>The Steiner distance of a set of vertices in a graph is the fewest number of edges in any connected subgraph containing those vertices. The order-$k$ Steiner distance hypermatrix of an $n$-vertex graph is the $n \\times \\cdots \\times n$ ($k$ terms) array indexed by vertices, whose entries are the Steiner distances of their corresponding indices. In the case of $k=2$, this reduces to the classical distance matrix of a graph. Graham and Pollak showed in 1971 that the determinant of the distance matrix of a tree only depends on its number $n$ of vertices. Here, we show that the hyperdeterminant of the Steiner distance hypermatrix of a tree vanishes if and only if (a) $n \\geq 3$ and $k$ is odd, (b) $n=1$, or (c) $n=2$ and $k \\equiv 1 \\pmod{6}$. Two proofs are presented of the $n=2$ case - the other situations were handled previously - and we use the argument further to show that the distance spectral radius for $n=2$ is equal to $2^{k-1}-1$. Some related open questions are also discussed.<\/jats:p>","DOI":"10.37236\/12850","type":"journal-article","created":{"date-parts":[[2024,7,12]],"date-time":"2024-07-12T10:03:43Z","timestamp":1720778623000},"source":"Crossref","is-referenced-by-count":0,"title":["Note on the Spectra of Steiner Distance Hypermatrices"],"prefix":"10.37236","volume":"31","author":[{"given":"Joshua","family":"Cooper","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhibin","family":"Du","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2024,7,12]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v31i3p4\/9071","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v31i3p4\/9071","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,12]],"date-time":"2024-07-12T10:03:43Z","timestamp":1720778623000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v31i3p4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,12]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,7,12]]}},"URL":"https:\/\/doi.org\/10.37236\/12850","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2024,7,12]]},"article-number":"P3.4"}}