Graph Theory. Directed and undirected graphs make useful mental models for many situations. These objects are loosely defined as follows:


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1 Graph Theory Directed and undirected graphs make useful mental models for many situations. These objects are loosely defined as follows: Definition An undirected graph is a (finite) set of nodes, some of which are connected by edges. A directed graph is a (finite) set of nodes, some of which are connected by arrows. Examples The nodes are the points. Undirected graphs Directed graphs Undirected and directed graphs roughly correspond to symmetric and asymmetric relations, though unlike relations there can be more than one edge or arrow between two nodes (as in the last example above). Try to think of situations that you can model with these graphs. For instance, the first undirected graph above could model the pattern of friendship between four people, or the layout of twoway streets between four corners, or the connections between four concepts. Directed graphs can be used to model all sorts of situations involving networks computer networks, social networks (who can influence who), networks of cause and effect, and so on.
2 When thinking about graphs, it is important to realize that it is only the pattern of connection between the nodes that matters not the actual way that we draw the graph. So for our purposes all of the following pictures would represent the same graph: (The technical term for two objects that have the same structure is isomorphic. All of the graphs above are isomorphic.) Graphs have many interesting properties; we will now study a few of the more basic properties. First, we loosely define some concepts which should be intuitively clear. Definition In an undirected graph, a path is sequence of edges from one node to another. Examples The following pictures show different paths in the same undirected graph: For directed graphs, the idea of a path is similar, except we must follow the direction of the arrows: Definition In a directed graph, a path is sequence of arrows from one node to another such that the head of each of the arrows is at the tail of the next arrow. Examples The following pictures show different paths in the same directed graph:
3 The third example of each of the two sets of examples above shows that it is possible for a path to stop and start at the same node. These special types of path are worth giving a special name: Definition A cycle is a path that starts and stops at the same node. Note that this definition works for both undirected and directed graphs. The notion of a path is useful in pinning down the idea that a graph may or may not have several different pieces. Definition An undirected graph is connected if there is a path between any two nodes. Examples A connected graph A graph which is not connected There is a special type of undirected graph that comes up a lot in information technology called a tree. Definition A tree is a connected undirected graph with no cycles. Examples Trees Not a tree The name tree is very suggestive. Given a connected undirected graph with no cycles we can always draw a picture of it that looks like an upsidedown realworld tree (think of grabbing one node, pulling up on it, and letting the other nodes hang down).
4 is isomorphic to When drawn this way, the top node is usually called the root of the tree. Try to think of some situations that could model with trees. Typical examples include the organizational hierarchy of a corporation (the root might be the CEO), a family tree (the root could be your greatgrandfather), or the file system of a computer (the root would be C:\ on a Windows machine). Here s a simple property of trees: FACT In a tree, the number of nodes is one more than the number of edges. To see why this is fact is true, note that we can build up any tree by starting with a single root node and then adding one edge and one node at a time. The root node has one more node than it has edges (it has 1 node and 0 edges) and each time we add an edge we also add one node, so we increase the number of nodes and the number of edges by the same amount, which means that the difference between these numbers is still 1. Another useful concept is the degree of a node: Definition For an undirected graph, the degree of a node is the number of edges touching it. Example The nodes of the following graph have been labeled with their degrees.
5 FACT The sum of the degrees of the nodes of an undirected graph is twice the number of edges. For example, if we sum the degrees of the graph above, we get = 12 and the graph has half that number namely six edges. This fact is true because each edge connects two nodes, so summing the degrees amounts to adding a 2 for each edge. This leads us to the FACT In an undirected graph, the number of nodes of odd degree is even. For example, the graph above has two nodes of odd degree (one of degree 1 and one of degree 3). This fact follows from the previous fact: the sum of the degrees is an even number (twice the number of edges) and the only way to get an even number when adding up a bunch of numbers is to have an even number of odd ones. (That is, if have an odd number of odd numbers, like = 11 they sum to an odd number.) This fact has the following amusing interpretation: FACT The number of people in the world who have shaken hands with an odd number of people is even. To see this, imagine an enormous undirected graph with a node for every person who has ever lived. If two of these people have shaken hands, connect the corresponding nodes with an edge. A person that has shaken hands with an odd number of people corresponds to a node of odd degree, and we know that there is an even number of such nodes. To finish this section, let s turn our attention to directed graphs. Definition In a directed graph, a source is a node such that arrows touching the node point away from the node. A sink is a node such that all arrows touching the node point into the node. A source A sink
6 FACT A directed graph with no cycles must have a source and a sink. Examples No cycles must have a source and a sink With a cycle there may not be a source or a sink We will see why this fact is true. Suppose we have a directed graph with no cycles; we ll first try to find a sink. Recall that in defining directed graphs we have assumed that the set of nodes is finite. Pick any node to start with if there are no arrows out of the node then we have already found our sink, so suppose that there are arrows (at least one anyway) pointing out of the node. Pick any arrow pointing out of the node and follow it to the next node. Again, this next node must have an arrow pointing out (or the node is a sink) so we can follow another arrow to another new node. We can keep on following arrows like this to new nodes but we can t do it forever since there are only a finite number of nodes. And we can t arrive at a node that we ve visited before because the path of arrows cannot be a cycle, so the only thing that can happen is that we eventually arrive at a sink. To find a source, do this same process of following arrows out of nodes, but follow them backwards (going from head to tail). Again, since there are no cycles and only a finite number of nodes we eventually have to arrive at a source.
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