1. Representing Graphs Wade Trappe

2. Lecture Overview Introduction Some Terminology –Paths Adjacency Matrix

3. Paths and Cycles, pg. 1 Definition: Let v 0 and v n be vertices in a graph. A path from v 0 to v n of length n is an alternating sequence of n+1 vertices and n edges beginning with v 0 and ending with v n, where e i starts at v i-1 and ends at v i. (v 0, e 1, v 1, e 2, v 2, …, v n-1, e n, v n ) Example: (1, e 1, 2, e 2, 3, e 3, 4, e 4, 2) 2 1 3 5 4 6 7e1e1 e3e3 e2e2 e5e5 e7e7 e6e6 e8e8 e4e4 Length 4 path

4. Paths and Cycles, pg. 2 Definition: A connected graph is a graph in which we can get from any vertex to any other vertex via a path. Formally, a graph G is connected if given any vertices v and w in G, there is a path from v to w. 2 1 3 5 4 6 7e1e1 e3e3 e2e2 e5e5 e7e7 e6e6 e4e4 Not Connected e8e8 Connected

5. Paths and Cycles, pg. 3 Obviously, a not-connected graph consists of several pieces. These pieces are subgraphs, or components. Definition: Let G=(V,E) be a graph. We call (V’, E’) a subgraph of G if: For every edge, if e’ is starts at v’ and ends at w’, then Note: You don’t have to include edges! A single vertex is a subgraph! The definition of a subgraph just requires that if you do select an edge, then you must have both ends of it!

6. Paths and Cycles, pg. 4 There are many choices for subgraphs! 1 A Subgraph 2 Another Subgraph 2 1 e1e1 G 2 1 A subgraph Without edges 2 1 e1e1 The entire Graph is a Subgraph!

7. Paths and Cycles, pg. 5 Definition: Let v and w be vertices in a graph. –A simple path from v to w is a path from v to w with no repeated vertices. –A cycle is a path of nonzero length from v to v with no repeated edges –A simple cycle is a cycle from v to v in which, except for the beginning and end, there are no repeated vertices. Example: We will look again at the original graph as an example.

8. Paths and Cycles, pg. 6 2 1 3 5 4 6 7e1e1 e3e3 e2e2 e5e5 e7e7 e6e6 e8e8 e4e4 (6,5,2,4) is a simple path It is not a cycle! (2,6,5,2,4,3,2) is not a simple path It is a cycle, but is not a simple cycle.

9. Konigsberg Bridges, pg. 1 Consider two islands lying in the Pregel River in Konigsberg, connected to each other and the river banks. The challenge: start at A,B,C or D and walk over each bridge exactly once then return the start. B D A D BC A D

10. Konigsberg Bridges, pg. 2 The bridge can be modeled as a graph, where edges are bridges. The Konigsberg Bridge problem is now: Find a cycle in the graph that includes all of the edges and all of the vertices. A cycle in a graph that includes all of the edges and all of the vertices of G is called an Euler Cycle. Note: There are an odd number of edges incident on vertex A… so, if you leave A and come back to A, you use an even amount of vertices… hence, you can’t solve the Konigsberg Bridge problem. The degree of a vertex v, denoted d(v) is the number of edges incident on v. (A loop counts twice).

11. About the Degree… Sum of all the degrees in a graph is an even #: If G is a graph with m edges and vertices {v 1, …, v n } then  d(v i ) = 2m. Proof: When we sum over the degrees of all the vertices, we count each edge (v i, v j ) twice… Once when we count it as (v i,v j ) in the degree of v i and then once again when we count it as (v j, v i ) in the degree of v i.

12. Representation of Graphs, pg. 1 Rather than always draw pictures, it is generally useful to have a data-structured representation for graphs. This is where the adjacency matrix A comes in. The row i, column j entry describes the amount of edges that touch node i and node j. a b ed c

13. Representation of Graphs, pg. 2 The powers of A count the number of paths of various lengths: –The (i,j)-th entry in the matrix A n is the number of paths from i to j of length n. Example: a  d  a and a  b  a gives the 2 a  a paths.