{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:58Z","timestamp":1753893838559,"version":"3.41.2"},"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>Let $p$ be an odd\u00a0 prime, $q=p^e$, $e \\geq 1$, and $\\mathbb{F} = \\mathbb{F}_q$ denote the finite field of $q$ elements.\u00a0 Let $f: \\mathbb{F}^2\\to \\mathbb{F}$ and\u00a0 $g: \\mathbb{F}^3\\to \\mathbb{F}$\u00a0 be functions, and\u00a0 let $P$ and $L$ be two copies of the 3-dimensional vector space $\\mathbb{F}^3$. Consider a bipartite graph $\\Gamma_\\mathbb{F} (f, g)$ with vertex partitions $P$ and $L$ and with edges defined as follows: for every $(p)=(p_1,p_2,p_3)\\in P$ and every $[l]= [l_1,l_2,l_3]\\in L$, $\\{(p), [l]\\} = (p)[l]$ is an edge in $\\Gamma_\\mathbb{F} (f, g)$ if\u00a0$$p_2+l_2 =f(p_1,l_1) \\;\\;\\;\\text{and}\\;\\;\\; p_3 + l_3 = g(p_1,p_2,l_1).$$The following question\u00a0 appeared in Nassau: Given $\\Gamma_\\mathbb{F} (f, g)$,\u00a0 is it always possible to find a function $h:\\mathbb{F}^2\\to \\mathbb{F}$ such that the graph $\\Gamma_\\mathbb{F} (f, h)$\u00a0 with the same vertex set as $\\Gamma_\\mathbb{F} (f, g)$ and with edges $(p)[l]$\u00a0 defined in a similar way\u00a0 by the system $$p_2+l_2 =f(p_1,l_1) \\;\\;\\;\\text{and}\\;\\;\\; p_3 + l_3 = h(p_1,l_1),$$\u00a0is isomorphic to $\\Gamma_\\mathbb{F} (f, g)$ for infinitely many $q$?\u00a0 In this paper we show that the\u00a0 answer to the question is negative and the graphs $\\Gamma_{\\mathbb{F}_p}(p_1\\ell_1, p_1\\ell_1p_2(p_1 + p_2 + p_1p_2))$ provide such an example for $p \\equiv 1 \\pmod{3}$. Our argument is based on proving that the automorphism group of these graphs has order $p$, which is the smallest possible order of the automorphism group of graphs of the form $\\Gamma_{\\mathbb{F}}(f, g)$.<\/jats:p>","DOI":"10.37236\/10707","type":"journal-article","created":{"date-parts":[[2022,3,11]],"date-time":"2022-03-11T06:28:50Z","timestamp":1646980130000},"source":"Crossref","is-referenced-by-count":0,"title":["A New Family of Algebraically Defined Graphs with Small Automorphism Group"],"prefix":"10.37236","volume":"29","author":[{"given":"Felix","family":"Lazebnik","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vladislav","family":"Taranchuk","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2022,3,11]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v29i1p43\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v29i1p43\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,3,11]],"date-time":"2022-03-11T06:28:50Z","timestamp":1646980130000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v29i1p43"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,11]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,1,27]]}},"URL":"https:\/\/doi.org\/10.37236\/10707","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2022,3,11]]},"article-number":"P1.43"}}