## ULethbridge - Number Theory and Combinatorics Seminar: Walter Carballosa Torres

- Date: 11/17/2021
- Time: 13:30

University of Lethbridge

Hyperbolicity of Some Product of Graphs

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∘G2G1∘G2 is hyperbolic if and only if G1G1 is (unless if G1G1 is trivial). In particular, we obtain the sharp inequalities δ(G1)≤δ(G1∘G2)≤δ(G1)+3/2δ(G1)≤δ(G1∘G2)≤δ(G1)+3/2, if G1G1 is not a trivial graph, and we prove that second inequality is attained only if G1G1 is a tree.

Tensor product: the tensor product graph G1×G2G1×G2 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.

Graph join: the graph join G1+G2G1+G2 is always hyperbolic since it's bounded. Moreover, we prove that for every two graphs G1G1 and G2G2 the hyperbolicity constant of its graph join δ(G1+G2)δ(G1+G2) belongs to {0,34,1,54,32}{0,34,1,54,32} characterizing them in each case. Furthermore, we obtain sharp bounds, and even formulas in many cases, for the hyperbolicity constant.

This talk is held in-person and online.

Time: 1:30pm Pacific / 2:30pm MT

Location: SA6006

Online: check website for meeting link