ULethbridge - Number Theory and Combinatorics Seminar: Walter Carballosa Torres

In this talk I will present a study on the Gromov's hyperbolicity of some product of graphs. The presentation is based on joint works that studied hyperbolicity of the lexicographic product, tensor product and graph join of two graphs, respectively. Product of graphs is a topic that has widely been study due to its applications in networks and regular structures. We characterize the hyperbolicity of these products

Lexicographic product: the lexicographic product graph G1∘G2$G_1\circ G_2$ is hyperbolic if and only if G1$G_1$ is (unless if G1$G_1$ is trivial). In particular, we obtain the sharp inequalities δ(G1)≤δ(G1∘G2)≤δ(G1)+3/2$\delta (G_1)\le \delta (G_1\circ G_2)\le \delta (G_1)+3/2$, if G1$G_1$ is not a trivial graph, and we prove that second inequality is attained only if G1$G_1$ is a tree.

Tensor product: the tensor product graph G1×G2$G_1\times G_2$ is hyperbolic, then one factor is bounded and the other is hyperbolic. Besides, we prove that this necessary condition is also sufficient in many cases. In other cases, we find (not so simple) characterizations of hyperbolic tensor products.