CHAPTER 1

Operators on Graphs. Quantum graphs

In this chapter, we introduce the main players of the quantum graph

theory: metric graphs and differential operators on them. A graph

consists of a set of points (vertices) and a set of segments (edges) con-

necting some of the vertices (Fig. 1).

More notions and results concerning graph theory can be found in

Section 1.1) and Appendix A. Most mathematicians are already famil-

iar with combinatorial graphs (which we survey briefly in Section 1.1),

where the vertices are the main players and the edges merely indicate

some relations between them. In a metric graph, in contrast, atten-

tion is focused on the edges. Metric graphs are introduced in Section

1.3. Quantum graphs are essentially metric graphs equipped with dif-

ferential operators. Such operators (Hamiltonians) are considered in

Section 1.4. The main operator under consideration acts as the second

derivative along the edges with “appropriate” conditions at junctions

(vertices). These conditions generalize the boundary conditions for

ODEs. Here, a lot of attention is devoted to describing what are the

“appropriate” conditions. Considering the quantum graph from the

point of view of waves propagating along edges and scattering at ver-

tices and other more advanced (but fundamental) topics are deferred

to Chapter 2.

Figure 1. A graph

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http://dx.doi.org/10.1090/surv/186/01