Basic Terminology
An edge connects two vertices
Two vertices are adjacent if they are connected
An edge is incident with the two vertices it connects
Vertices are the endpoints of the edge connecting them
The degree of a vertex is the number of incident edges
An isolated vertex has degree zero (0)
A pendant vertex has degree one (1)
Every undirected graph has an even number of vertices of odd degree.
The "First Theorem" of Graph Theory
E
E
O
EO
O
O
O
E
O
O
E
E
E
O
In a graph with directed edges the in-degree of a vertex v, denoted by deg-(v), is the number of edges with v as their terminal vertex.
The out-degree of v, denoted by deg+(v), is the number of edges with v as their initial vertex.
A Theorem for Directed Graphs
EvvVvVv
)(deg)(deg
Let G=(V,E) be a graph with directed edges. Then
K1 K2 K3 K4 K5 K6
Complete Graphs Kn
A complete graph is a simple graph with one edge between every pair of vertices.
How many edges are there in a complete graph of n vertices?
First we note that each vertex of Kn has degree n-1.
Using the Handshaking Theorem, we have
2e = deg(v) = n*(n-1),
therefore
e = n*(n-1)/2.
The Connection Machine was a series of supercomputers that grew out of Danny Hillis's research in the early 1980s at MIT on alternatives to the traditional von Neumann architecture of computation. The Connection Machine was originally intended for applications in artificial intelligence and symbolic processing, but later versions found greater success in the field of computational science.
Connection Machine
CM-2 CM-5
http://en.wikipedia.org/wiki/Connection_Machine
K2,3 K3,3
K3,5 K2,6
Some Complete Bipartite Graphs
The "first theorem" of planar graph theory - K3,3 is not planar.
The Arc Reversal Algorithm
The arc-reversal algorithm has applications in computer communications, parallel processing, flow analysis, scheduling and Bayesian Networks.
For an n-node graph we build an nxn array with 1's indicating edges and 0's no edge the main diagonal of the matrix is unused unless a node has an edge connected to itself. If graph is weighted, 1's are replaced with edge weight values
Adjacency Matrix Graph Representation
adjacency matrix
A B C D E F G HA - 1 1 1 1 1 0 0B 1 - 1 0 1 0 0 1C 1 1 - 1 1 0 0 1D 1 0 1 - 0 1 1 1E 1 1 1 0 - 1 1 0F 1 0 0 1 1 - 1 1G 0 0 0 1 1 1 - 1H 0 1 1 1 0 1 1 -
A D F
C H
B E G