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 $X$. If $X$ is hyperbolic, we denote by $\delta (X)$ the sharp hyperbolicity constant of $X$, i.e., $\delta (X)=\inf\{\delta\geq 0: \, X \, \text{ is $\delta$-hyperbolic}\,\}\,.$ In this paper we characterize the strong product of two graphs $G_1\boxtimes G_2$ which are hyperbolic, in terms of $G_1$ and $G_2$: the strong product graph $G_1\boxtimes G_2$ is hyperbolic if and only if one of the factors is hyperbolic and the other one is bounded. We also prove some sharp relations between $\delta (G_1\boxtimes G_2)$, $\delta (G_1)$, $\delta (G_2)$ and the diameters of $G_1$ and $G_2$ (and we find families of graphs for which the inequalities are attained). Furthermore, we obtain the exact values of the hyperbolicity constant for many strong product graphs.

If X is a geodesic metric space and x 1 , x 2 , x 3 ∈ X, a geodesic triangle T = {x 1 , x 2 , x 3 } is the union of the three geodesics [x 1 x 2 ], [x 2 x 3 ] and [x 3 x 1 ] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. 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. In the context of graphs, to remove and to contract an edge of a graph are natural transformations. The main aim in this work is to obtain quantitative information about the distortion of the hyperbolicity constant of the graph G \ e (respectively, G/e ) obtained from the graph G by deleting (respectively, contracting) an arbitrary edge e from it. This work provides information about the hyperbolicity constant of minor graphs.

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If X is a geodesic metric space and x 1 , x 2 , x 3 ∈ X, a geodesic triangle T = {x 1 , x 2 , x 3 } is the union of the three geodesics [x 1 x 2 ], [x 2 x 3 ] and [x 3 x 1 ] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-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) the sharp hyperbolicity constant of X, i.e., δ(X) = inf{δ ≥ 0 : X is δ-hyperbolic }. Some previous works characterize the hyperbolic product graphs (for the Cartesian, strong, join, corona and lexicographic products) in terms of properties of the factor graphs. However, the problem with the direct product is more complicated. In this paper, we prove that if the direct product G 1 × G 2 is hyperbolic, then one factor is hyperbolic and the other one is bounded. Also, we prove that this necessary condition is, in fact, a characterization in many cases. In other cases, we find characterizations which are not so simple. Furthermore, we obtain formulae or good bounds for the hyperbolicity constant of the direct product of some important graphs.