Boundary Value Problems for Infinite Metric Graphs: A Study Motivated by Biological Networks

The biological world solves some of its transport problems using extremely complex graphical structures. Examples include the human circulatory and nervous systems, and the root systems and leaf veins of plants. Infinite graph models motivated by such examples are considered. Classical problems like heat flow and wave motion display new aspects on infinite graphs. We describe a rich family of graphs for which these problems have satisfactory solutions. Connections with harmonic functions and possibly probablity will be discussed.