Theoretical Progress on Infinite Graphs and Their Average Degree: Applicability to the European Road Transport Network

Abstract

There are many problems in Graph Theory for finite graphs relating the number of vertices and the number of edges and, therefore, related to the average degree for finite graphs. However, when dealing with real-life problems involving networks, it is often useful to model the situation by using infinite graphs, which can represent extendable systems. In this paper, we will generalize the concept of average degree for infinite graphs in a family of graphs that we call average-measurable. Besides, this new definition allows the generalization of the universal formulae for evaluation of percolation thresholds.

Resource Allocation Approach for Optimal Routing in IoT Wireless Mesh Networks

“…Since the topology of a set of nodes and connections between them is convenient to consider as a graph, it is customary to use graph models. The most commonly used models are directed graphs [9], hypergraphs [10], infinite graphs [11] with Kőnig's lemma and theorem [12], hypernets, etc.…”