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\begin{document}$$x_1,x_2,x_3\in X$$\end{document}, a geodesic triangleT={x1,x2,x3}\documentclass[12pt]{minimal}
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\begin{document}$$T=\{x_1,x_2,x_3\}$$\end{document} is the union of the three geodesics [x1x2]\documentclass[12pt]{minimal}
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\begin{document}$$[x_1x_2]$$\end{document}, [x2x3]\documentclass[12pt]{minimal}
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\begin{document}$$[x_2x_3]$$\end{document} and [x3x1]\documentclass[12pt]{minimal}
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\begin{document}$$[x_3x_1]$$\end{document} in X. The space X is δ\documentclass[12pt]{minimal}
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\begin{document}$$\delta $$\end{document}-hyperbolic (in the Gromov sense) if any side of T is contained in a δ\documentclass[12pt]{minimal}
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\begin{document}$$\delta $$\end{document}-neighborhood of the union of the two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by δ(X)\documentclass[12pt]{minimal}
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\begin{document}$$\delta (X)$$\end{document} the sharp hyperbolicity constant of X, i.e. δ(X)=inf{δ≥0:Xisδ-hyperbolic}\documentclass[12pt]{minimal}
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\begin{document}$$\delta (X)=\inf \{\delta \ge 0: \, X \, \text { is }\delta \text {-hyperbolic}\}$$\end{document}. In this paper, we characterize the lexicographic product of two graphs G1∘G2\documentclass[12pt]{minimal}
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\begin{document}$$G_1\circ G_2$$\end{document} which are hyperbolic, in terms of G1\documentclass[12pt]{minimal}
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\begin{document}$$G_1$$\end{document} and G2\documentclass[12pt]{minimal}
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\begin{document}$$G_2$$\end{document}: the lexicographic product graph G1∘G2\documentclass[12pt]{minimal}
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\begin{document}$$G_1\circ G_2$$\end{document} is hyperbolic if and only if G1\documentclass[12pt]{minimal}
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\begin{document}$$G_1$$\end{document} is hyperbolic, unless if G1\documentclass[12pt]{minimal}
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\begin{document}$$G_1$$\end{document} is a trivial graph (the graph with a single vertex); if G1\documentclass[12pt]{minimal}
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\begin{document}$$G_1$$\end{document} is trivial, then G1∘G2\documentclass[12pt]{minimal}
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\begin{document}$$G_1\circ G_2$$\end{document} is hyperbolic if and only if G2\documentclass[12pt]{minimal}
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\begin{document}$$G_2$$\end{document} is hyperbolic. In particular, we obtain the sharp inequalities δ(G1)≤δ(G1∘G2)≤δ(G1)+3/2\documentclass[12pt]{minimal}
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\begin{document}$$\delta (G_1)\le \delta (G_1\circ G_2) \le \delta (G_1) + 3/2$$\end{document} if G1\documentclass[12pt]{minimal}
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\begin{document}$$G_1$$\end{document} is not a trivial graph, and we characterize the graphs for which the second inequality is attained.
