Metric Problems in Triple Loop Graphs and Digraphs Associated to an Hexagonal Tessellation of the Plane

. The use of plane tessellations by hexagons facilitates the study of a family of triple loop digraphs and a family of triple loop graphs and enables us to give the minimum diameter, mean distance and vulnerability results for every order. 1 Introduction. Most problems arising in the study of networks can be formulated in terms of graphs. One of them, the construction of efficient and reliable networks with a limited number of links, leads to the search of graphs that, with a fixed degree, have minimal diameter for a given order or maximum order for a given diameter. It also leads to the study of their vulnerability i.e. to find how the diameter increases when one or more vertices (or arcs) are deleted. It is possible to associate plane tessellations to certain graphs [1]. This association enables a geometrical, and in general simpler, approach to the above mentioned search. In this paper we deal with the study of some graphs related to hexagonal tessellations of the plane: a family o..