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{\bf Walter Carballosa, Jos\'e M. Rodr{\'\i}guez, Jos\'e M. Sigarreta and Mar{\'\i}a Villeta}
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{\bf On the Hyperbolicity Constant of Line Graphs}
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If X is a geodesic metric space and $x_1,x_2,x_3\in X$, a {\it
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 $\delta$-\emph{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 $X$, \emph{i.e.},
$\delta(X):=\inf\{\delta\ge 0: \, X \, \text{ is $\delta$-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 $\delta(\mathcal{L}(G)) < 1$.
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