## 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.)

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:

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

More can be found on a separate page.

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