1. Grade 11 AP Mathematics Graph Theory
Definition:
A graph, G, is a set of vertices v(G) = {v1, v2, v3, …, vn} and edges e(G) = {vivj where
1 ≤ i, j ≤ n; i ≠ j}
Vertex set: v(G) = {v1, v2, v3, v4, v5}
Edge set: e(G) = {v1v2, v1v3, v2v3, v2v4, v2v5, v3v4} v1 v5 v4 v3 v2 G

2. Grade 11 AP Mathematics Graph Theory
We will consider only simple graphs with no loops and no multiple edges between vertices. F
Graph F alongside is not a simple graph
We will not attach any direction to the edges (we speak of a digraph in this case) thus, for our purposes, the edges are not directed.
The order of a graph is the number of vertices and the size of a graph is the number of edges. H
Graph H has order 4 and size 5

3. Grade 11 AP Mathematics Graph Theory
Edges are adjacent if they share a common vertex
e.g. edges v1v2 and v2v3 are adjacent since they share a common vertex v2
Vertices are adjacent if they are the endpoints of a particular edge
e.g. vertices v1 and v2 are adjacent since they are the endpoint of edge v1v2
The degree of a vertex is the number of vertices adjacent to it.
e.g. deg (v4) = 2 H v2 v3 v4 v1

4. Grade 11 AP Mathematics Graph Theory
A complete graph (Kn) is a graph in which each vertex is adjacent to every other vertex.
Number of edges in a complete graph =
Graph P is a complete graph K4 since it has 4
vertices and all of its vertices are adjacent to each
of the other three vertices.
Number of edges =
Think of a group of people shaking hands … how many handshakes will you have for 4 people? P v2 v4 v1 v3

5. Grade 11 AP Mathematics Graph Theory
A bipartite graph is a simple graph whose vertices are partitioned into two subsets V1 and V2 such that every edge vivj has i V1 and j V2 i.e. we don’t join vertices in the same subset. v1 v2 v4 v5 v6 v3 Q v1 v2 v4 v5 v6 v3 M
Graph Q is a complete bipartite graph
V1 = {v1, v2, v3, v4}
V2 = {v5, v6}
V1 = {v1, v2, v3, v4}
V2 = {v5, v6}

6. Grade 11 AP Mathematics Graph Theoryv1 v2 v4 v5 v6 v3 Q
A connected graph is a graph such that there is a path from any vertex to any other vertex in the graph.
Graph Q is a connected graph.
By contrast, Graph M is a disconnected graph.
For example, there is no path from v3 to v5. v1 v2 v4 v5 v6 v3 M

7. Grade 11 AP Mathematics Graph Theory
A regular graph is a graph such that all of the vertices have the same degree.
Examples:
2-regular graphs
3-regular graphs (cubics)

8. Grade 11 AP Mathematics Graph Theory
Walks
A walk is a sequence of vertices and edges such that the end vertex of one edge is the start of another edge. A B C D F G H E I

9. Grade 11 AP Mathematics Graph Theory
Paths
A path is a walk such that no edge is travelled more than once. B C A H I E D G F

10. Grade 11 AP Mathematics Graph Theory
Chains
A chain is a path such that no vertex is visited more than once. B C A H I E G F

11. Grade 11 AP Mathematics Graph Theory
A path (and by extension a chain) that starts and ends at the same vertex is called a closed path.
A closed path also called a circuit or a cycle. B C A H I E G F

12. Grade 11 AP Mathematics Graph Theory

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15. Grade 11 AP Mathematics Graph Theory