{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,20]],"date-time":"2026-05-20T19:49:33Z","timestamp":1779306573167,"version":"3.51.4"},"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>If X is a geodesic metric space and $x_1,x_2,x_3\\in X$, a geodesic triangle $T=\\{x_1,x_2,x_3\\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $\\delta$-hyperbolic $($in the Gromov sense$)$ if any side of $T$ is contained in a $\\delta$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$. We denote by $\\delta(X)$ the sharp hyperbolicity constant of $X$, i.e., $\\delta(X):=\\inf\\{\\delta\\ge 0: X \\text{ is }\\delta\\text{-hyperbolic}\\}$.  The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. The main aim of this paper is to obtain information about the hyperbolicity constant of the line graph $\\mathcal{L}(G)$ in terms of parameters of the graph $G$. In particular, we prove qualitative results as the following: a graph $G$ is hyperbolic if and only if $\\mathcal{L}(G)$ is hyperbolic; if $\\{G_n\\}$ is a T-decomposition of $G$ ($\\{G_n\\}$ are simple subgraphs of $G$), the line graph $\\mathcal{L}(G)$ is hyperbolic if and only if $\\sup_n \\delta(\\mathcal{L}(G_n))$ is finite. Besides, we obtain quantitative results. Two of them are quantitative versions of our qualitative results. We also prove that $g(G)\/4 \\le \\delta(\\mathcal{L}(G)) \\le c(G)\/4+2$, where $g(G)$ is the girth of $G$ and $c(G)$ is its circumference. We show that $\\delta(\\mathcal{L}(G)) \\ge \\sup \\{L(g):\\, g \\,\\text{ is an isometric cycle in }\\,G\\,\\}\/4$. Furthermore, we characterize the graphs $G$ with $\\delta(\\mathcal{L}(G))  &lt;  1$.<\/jats:p>","DOI":"10.37236\/697","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:40:44Z","timestamp":1578714044000},"source":"Crossref","is-referenced-by-count":26,"title":["On the Hyperbolicity Constant of Line Graphs"],"prefix":"10.37236","volume":"18","author":[{"given":"Walter","family":"Carballosa","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jos\u00e9 M.","family":"Rodr\u00edguez","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jos\u00e9 M.","family":"Sigarreta","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mar\u00eda","family":"Villeta","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2011,10,31]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p210\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p210\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:00:29Z","timestamp":1579302029000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i1p210"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,10,31]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/697","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,10,31]]},"article-number":"P210"}}