Hyperbolic Bridged Graphs

In this paper, all graphs are simple and connected, but they are not necessarily finite. As is well known, a (connected) graph G comes equipped with a natural metric on its vertex set V (G), given by defining the distance xy between any pair of vertices x, y ∈ V (G) to be the length of some shortest path or geodesic between x and y. Given a quartet x, y, u, v ∈ V (G), define δ(x, y, u, v) to be the absolute value of the difference between the largest and the second largest of the three sums