Walks, Paths, and Circuits Graph Theory Walks, Paths, and Circuits

Definitions Walk: sequence of vertices Path: sequence of distinct vertices Length of a path: number of edges in a path Circuit: path that starts and ends at the same vertex, also known as a cycle

Definitions, con’t Euler circuit: a circuit that contains every edge and every vertex of the graph If graph has an Euler circuit then every vertex has even degree

Adjacency Matrices for Graphs Adjacency matrix: an n x n matrix for a graph with n vertices where each entry is the number of edges from each vertex to all the others

Example 1 Suppose A, B and C are three cities. Every day there are 2 nonstop flights from A to B, 3 from B to C and 1 from A to C. There are 2 nonstop flights from B to A, 2 from C to B, and 1 from C to A. Write the adjacency matrix for this information.

Example 2 Draw a picture of a directed graph that has the following adjacency matrix:

Example 3 Find the number of walks of length 2 from v2 to v1 in this graph. Find the number of walks of length 3 from v1 to v3 in this graph.