Definitions
1. Vertex: A point is called a vertex (plural vertices). Sometimes called nodes.
2. Edge: A line is called an edge. Sometimes called arcs.
3. Graph: The whole diagram is called a graph or a network.
4. Degree of a vertex: The number of edges that lead to a vertex is called the
degree.
5. Loop:
An edge that leaves a vertex and wraps back around to the same vertex without
passing through another vertex.
Example p330 text
6. Subgraph
7. Simple graph or network: when pairs of vertices are connected by one edge at
most.
Example p330 text
7. Complete graph
Graph where an edge connects each vertex to all other vertices in the graph.
Example p330 text
Which of the above are also simple?
All of them as pairs of vertices are connected by one edge at most.
8. Bipartite graph:
Red vertices only connect (adjacent to) blue vertices.
This bipartite graph is complete as each vertex in one group is connected
to every vertex in the other group.
9. Directed graph: (or diagraphs)
Only possible to move along the edges in one
direction. Edges are represented by arrows.
Otherwise it is an undirected graph.
10. Arc: Another word for edge.
11. Weighted graph: A graph where the arcs have a numerical value.
12. Planar graph
13. Face
14.
15. Connected graph: graph where it is possible to reach every vertex by moving
along the edges. If this is not possible it is called a disconnected graph or network.
Example p330 text
16. Bridge
A bridge of a connected graph is a graph edge whose removal disconnects the
graph.
17. Eulerian Graphs
An Euler path may be a Euler trail or Semi Euler trail
For an Euler trail to exist all vertices must be of an even degree OR for a Semi Euler
trail to exist there must be exactly 2 vertices of odd degree
18. Hamiltonian graph Semi Hamiltonian graph
Simple graph
Not a simple graph
Pairs of vertices are connected by at most
one edge
Pairs of vertices may be connected
by multiple edges. Loops may also
Complete graph
be present
Not a complete graph
Graph where an edge connects each vertex
to all other vertices in the graph.
why?
Bipartite graph
A bipartite graph is a set of graph vertices
decomposed into two disjoint sets such that
no two graph vertices within the same set
are adjacent.
This is Not a complete bipartite graph
why is this not a bipartite graph?
This is a complete bipartite
graph
Planar graph
(lines cross over each other but can be
redrawn to create a planar graph)
Planar graph
Same graph drawn as a planar
graph
Connected graph
Disconnected graph
Not possible
to reach each
vertex by
moving along
edges.
Euler path (E for Euler ...E for edge)
An Euler path may be a Euler trail or Semi
Euler trail
For an Euler trail to exist all vertices must be
of an even degree OR for a Semi Euler trail
to exist there must be exactly 2 vertices of
odd degree
Make a table and count the degree
of each vertex to prove a Euler path
is not possible.
Make a take and count the degree of each
vertex to prove this.
Hamiltonian cycles and semi H. cycles
reach all vertices once and may not use
every edge. The semi starts and finishes at
a different vertex.
Why is this Non Hamiltonian?