Undirected graphs and networks

184 General Mathematics
summary
Vertices and edges
• An undirected graph or network consists of vertices and edges.
• The degree of a vertex is the number of edges leading
to or from that vertex. A loop counts as 2 edges.
• In a connected graph it is possible to reach each vertex
from any other vertex. A connected graph must not
have any isolated vertices.
Planar graphs
• A planar graph has no crossover edges.
• A planar graph divides the plane into a
Multiple
edges
number of regions.
• When counting regions, the region around A planar graph
the outside of the graph is counted as 1.
Not a planar graph
• For any connected planar graph: V = 5
Euler’s Law states that: V + R − E = 2. E = 8
The sum of the degree of all the vertices = R = 5
2 × number of edges. V + R − E
There is always an even number of odd-degree = 5 + 5 − 8
vertices.
Eulerian paths and circuits
• A path is a series of vertices connected by edges.
= 2
• A circuit (or cycle) is a path which starts and finishes at the same vertex and no edge
is traversed more than once.
• An Eulerian path is a path which uses each edge in a graph only once, however a
vertex may be repeated.
• An Eulerian circuit is an Eulerian path which starts and finishes at the same vertex.
• An Eulerian path is possible if the number of odd vertices is 0 or 2.
• An Eulerian circuit is possible if each of the vertices has even degrees.
Hamiltonian paths and circuits
• A Hamiltonian path passes through each vertex only once. It is not necessary to use
all of the edges.
• A Hamiltonian circuit is a Hamiltonian path that starts and finishes at the same vertex.
• To find the shortest path in a network, choose the edge of least distance and move
along the network until the destination is reached. This usually requires a trial and
error approach.
Trees
• A tree is a connected graph without any circuits, loops or multiple edges and
contains only one region.
• A spanning tree is a tree that includes all the vertices in the graph.
• A minimal spanning tree is a spanning tree with the minimum length (or cost, or time).
• To find a minimal spanning tree:
1. select the edge with the minimum value. If there is more than one such edge,
choose any one of them.
2. select the next smallest edge, provided it does not create a cycle.
3. repeat step 2 until all the vertices have been included.
• The vertices and edges in a tree are related by the equation V − E = 1.
Loop
Edge
Vertex