Applications in Mathematics

Subsection 4.2.1 Basic Definitions

The way we avoid ambiguities in mathematics is to provide concrete and rigorous definitions. Crafting good definitions is not easy, but it is incredibly important. The definition is the agreed upon starting point from which all truths in mathematics proceed. Is there a graph with no edges? We have to look at the definition to see if this is possible.

We want our definition to be precise and unambiguous, but it also must agree with our intuition for the objects we are studying. It needs to be useful: we could define a graph to be a six legged mammal, but that would not let us solve any problems about bridges. Instead, here is the (now) standard definition of a graph.

Strange. Nowhere in the definition is there talk of dots or lines. From the definition, a graph could be

Here we have a graph with four vertices (the letters \(a, b, c, d\)) and four edges (the pairs \(\{a,b\}, \{a,c\}, \{b,c\}, \{b,d\}, \{c,d\})\)).

Looking at sets and sets of 2-element sets is difficult to process. That is why we often draw a representation of these sets. We put a dot down for each vertex, and connect two dots with a line precisely when those two vertices are one of the 2-element subsets in our set of edges. Thus one way to draw the graph described above is this:

However we could also have drawn the graph differently. For example either of these:

The exact placement of the vertices is unimportant. Both drawings above represent the same graph.

Back to some basic graph theory definitions. Notice that all the graphs we have drawn above have the property that no pair of vertices is connected more than once, and no vertex is connected to itself. Graphs like these are sometimes called simple, although we will just call them graphs. This is because our definition for a graph says that the edges form a set of 2-element subsets of the vertices. Remember that it doesn't make sense to say a set contains an element more than once. So no pair of vertices can be connected by an edge more than once. Also, since each edge must be a set containing two vertices, we cannot have a single vertex connected to itself by an edge.

That said, there are times we want to consider double (or more) edges and single edge loops. For example, the “graph” we drew for the Bridges of Königsberg problem had double edges because there really are two bridges connecting a particular island to the near shore. We will call these objects multigraphs. This is a good name: a multiset is a set in which we are allowed to include a single element multiple times.

The graphs above are also connected: you can get from any vertex to any other vertex by following some path of edges. A graph that is not connected can be thought of as two separate graphs drawn close together. For example, the following graph is NOT connected because there is no path from \(a\) to \(b\text{:}\)

Most of the time, it makes sense to treat non-connected graphs as separate graphs (think of the above graph as two squares), so unless otherwise stated, we will assume all our graphs are connected.

Vertices in a graph do not always have edges between them. If we add all possible edges, then the resulting graph is called complete. That is, a graph is complete if every pair of vertices is connected by an edge. Since a graph is determined completely by which vertices are adjacent to which other vertices, there is only one complete graph with a given number of vertices. We give these a special name: \(K_n\) is the complete graph on \(n\) vertices.

Each vertex in \(K_n\) is adjacent to \(n-1\) other vertices. We call the number of edges emanating from a given vertex the degree of that vertex. So every vertex in \(K_n\) has degree \(n-1\text{.}\) How many edges does \(K_n\) have? One might think the answer should be \(n(n-1)\text{,}\) since we count \(n-1\) edges \(n\) times (once for each vertex). However, each edge is incident to 2 vertices, so we counted every edge exactly twice. Thus there are \(n(n-1)/2\) edges in \(K_n\text{.}\) Alternatively, we can say there are \({n \choose 2}\) edges, since to draw an edge we must choose 2 of the \(n\) vertices.

Subsection 4.2.2 Graphs and Degrees

In general, if we know the degrees of all the vertices in a graph, we can find the number of edges. The sum of the degrees of all vertices will always be twice the number of edges, since each edge adds to the degree of two vertices. Notice this means that the sum of the degrees of all vertices in any graph must be even!

What other observations can we make about the degrees of the vertices in a graph? If we list out the degree of each vertex in a graph, this is the degree sequence of a graph.

Example 4.2.3

Find the degree sequence of the graph below.

Solution

We typically list the degrees in descending numerical order, so the degree sequence for this graph is \(3,3,2,2\text{.}\)

Of course not every sequence of numbers will be the degree sequence for a graph. For instance, since we know the sum of all degrees must be even we know that the sequence \(3,3,3\) is not a valid degree sequence. If a sequence of numbers does represent the degree sequence of a simple graph then we call that sequence graphic.