Simple Undirected Graphs

Abstract

When we consider graphical models for the multivariate complex normal distribution we formulate the models in terms of simple undirected graphs. This chapter presents the concept of simple undirected graphs. As the main purpose is to define and introduce the later needed results, the presentation is at times short and compact. The well known results are stated without proof, but references containing further information are given. Results which are not quite well known are treated in more detail. First of all we define a simple undirected graph and associated basic definitions. Afterwards we consider the concepts separation, decomposition and decomposability of simple undirected graphs. This involves investigation of chordless 4-cycles, running intersection property orderings, the maximum cardinality search algorithm and an algorithm to determine decomposability of a simple undirected graph. Then we move on to definition of collapsibility and we treat the concept a regular edge. We observe that a simple undirected decomposable graph and a decomposable subgraph of it with one edge less differ by a regular edge. Finally we state some decompositions of subgraphs in a simple undirected graph containing a regular edge. We begin by defining a simple undirected graph.