## Simple Graphs Practice

(The instructions for using the applet are available on a separate page and can also be read under the first tab directly in the applet.)

### This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

 What if applet does not run?

(This applet was created to accompany Excursions in Modern Mathematics, Seventh Edition, by Peter Tannenbaum © Pearson Education. Reproduced with permission. An earlier version of the applet is still available online at https://www.cut-the-knot.org/Curriculum/Combinatorics/SimpleGraphs.shtml.)

The sole purpose of the applet is to help accustom a student to the basic concepts of graph theory. A formal definition of graph is a combination of two sets V and E, where elements of V are termed vertices, while the elements of E are edges, and each consists of a pair of vertices from V. The two vertices in an edge may be equal, in which case the edge is called a loop. The number of times a vertex is included in the edges of a graph is called its degree. A vertex is incident to an edge if it's one of the two vertices in the pair (which is the edge.) An edge is incident to each of its constituent vertices. A loop incident to a vertex endows the vertex with two degrees.

A path in a graph between two vertices is a sequence of edges (a path), of which one is incident to the first vertex and another to the second, and which are incident in pairs to the same vertex. A path between two vertices is said to join them. If the two ends of the path coincide, the latter is called a circuit. A graph is connected if any two vertices can be joined by a path.

A vertex of odd degree is said to be odd; otherwise it's even.

A few very general properties of the graphs may be observed using the applet:

1. The number of odd vertices is always even.
2. For a connected graph, the number of odd vertices is either 0 or 2.
3. The total degree of a graph equals double the number of edges.

More can be found on a separate page.