{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:49Z","timestamp":1753893829355,"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>For every $h\\in\u00a0 \\mathbb{N}$, a graph $G$ with the vertex set $V(G)$ and the\u00a0 edge set $E(G)$ is said to be $h$-magic if there exists a labeling $l : E(G) \\rightarrow\\mathbb{Z}_h \\setminus \\{0\\}$\u00a0 such that the induced vertex labeling $s : V (G) \\rightarrow \\mathbb{Z}_h$, defined\u00a0 by $s(v) =\\sum_{uv \\in E(G)} l(uv)$ is a constant map. When this constant is zero, we say that $G$ admits a zero-sum $h$-magic labeling. The null set of a graph $G$, denoted by $N(G)$, is the set of all natural numbers $h \\in \\mathbb{ N} $ such that $G$ admits a zero-sum $h$-magic labeling.\u00a0 In 2012, the null sets of 3-regular graphs were determined. In this paper we show that if $G$ is an $r$-regular graph, then for even $r$ ($r &gt; 2$), $N(G)=\\mathbb{N}$ and for odd $r$ ($r\\neq5$),\u00a0 $\\mathbb{N} \\setminus \\{2,4\\}\\subseteq N(G)$. Moreover,\u00a0 we prove that if $r$ is odd and $G$ is a $2$-edge connected $r$-regular graph ($r\\neq 5$), then $ N(G)=\\mathbb{N} \\setminus \\{2\\}$. Also, we show that if $G$ is a $2$-edge connected bipartite graph, then $\\mathbb{N} \\setminus \\{2,3,4,5\\}\\subseteq N(G)$.<\/jats:p>","DOI":"10.37236\/3654","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T01:05:20Z","timestamp":1578704720000},"source":"Crossref","is-referenced-by-count":6,"title":["Zero-Sum Magic Labelings and Null Sets of Regular Graphs"],"prefix":"10.37236","volume":"21","author":[{"given":"Saieed","family":"Akbari","sequence":"first","affiliation":[]},{"given":"Farhad","family":"Rahmati","sequence":"additional","affiliation":[]},{"given":"Sanaz","family":"Zare","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2014,5,2]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v21i2p17\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v21i2p17\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T10:59:56Z","timestamp":1579258796000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v21i2p17"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,5,2]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2014,3,31]]}},"URL":"https:\/\/doi.org\/10.37236\/3654","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2014,5,2]]},"article-number":"P2.17"}}