{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T07:54:55Z","timestamp":1770710095407,"version":"3.49.0"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>For $i\\geq 0$, the $i$-cube $Q_i$ is the graph on $2^i$ vertices representing $0\/1$ tuples of length $i$, where two vertices are adjacent whenever the tuples differ in exactly one position. (In particular, $Q_0 = K_1$.) Let $\\alpha_i(G)$ be the number of induced $i$-cubes of a graph $G$. Then the cube polynomial $c(G,x)$ of $G$ is introduced as $\\sum_{i\\geq 0} \\alpha_i(G) x^i$. It is shown that any function $f$ with two related, natural properties, is up to the factor $f(Q_0,x)$ the cube polynomial. The derivation $\\partial\\, G$ of a median graph $G$ is introduced and it is proved that the cube polynomial is the only function $f$ with the property $f'(G,x)= f(\\partial\\, G, x)$ provided that $f(G,0)=|V(G)|$. As the main application of the new concept, several relations that widely generalize previous such results for median graphs are proved. For instance, it is shown that for any $s\\geq 0$ we have $c^{(s)}(G,x+1) = \\sum_{i\\geq s}\\, {{c^{(i)}(G,x)}\\over {(i-s)!}}\\,,$ where certain derivatives of the cube polynomial coincide with well-known invariants of median graphs.<\/jats:p>","DOI":"10.37236\/1696","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T02:30:02Z","timestamp":1578709802000},"source":"Crossref","is-referenced-by-count":18,"title":["The Cube Polynomial and its Derivatives: the Case of Median Graphs"],"prefix":"10.37236","volume":"10","author":[{"given":"Bo\u0161tjan","family":"Bre\u0161ar","sequence":"first","affiliation":[]},{"given":"Sandi","family":"Klav\u017ear","sequence":"additional","affiliation":[]},{"given":"Riste","family":"\u0160krekovski","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2003,1,10]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v10i1r3\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v10i1r3\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T05:10:21Z","timestamp":1579324221000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v10i1r3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,1,10]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2003,1,6]]}},"URL":"https:\/\/doi.org\/10.37236\/1696","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,1,10]]},"article-number":"R3"}}