1. Introduction to Graph Theory
Theodora Welch
Management & Marketing Department
College of Management
SNA Workshop
July 31, 2008
(SOURCE: Introduction to graph theory, Lecture notes by Tom Snijders,

2. Graph Theory Presentation coverage Mathematical theory of networks
Powerful tool for modeling & analyzing networks
Graphs are important & effective mathematical objects for modeling relationships and structures
Presentation coverage
Graph-theoretic definitions & notation
Some special graphs
Node-connectivity
Degree centrality
are important and effective mathematical constructs for modeling relationships and structural information.
Graphs are used in many different problems, including
In these problems the graph itself is processed by some algorithm
Graph theory serves as a powerful tool for modeling
Learn the basic properties of graph theory
Graphs themselves can be analyzed (processed) by algorithms & software packages
Laura: Matrices & Statistics in SNA
Rich: Using UCINET6

3. “Graphs”
Often, when the word graph is used in an applied mathematical setting, we think of:
In this situation, the word graph is short for graph of a function.
In this module, we will study mathematical objects called graphs. Often, when the word graph is used in a mathematical setting, people think of something resembling:
The kind of graphs that we will study are sometimes called combinatorial graphs to distinguish them from the graphs described above. Combinatorial graphs can sometimes be represented pictorially as networks of dots (called vertices) connected by lines (called edges). Examples include:

4. “Combinatorial Graphs”
The kind of graphs that we are interested in are sometimes called combinatorial graphs
Combinatorial graphs can be represented pictorially as networks of nodes (vertices) connected by lines (edges).
Example:
A graph in this sense is a mathematical object that captures the notion of connection.
We used graph theory (combinatorial graphs) to represent this mathematical object composed of points (vertices) connected by lines (edges)

5. Definition: A undirected Graph G consists of a set of vertices & a set of edges, where each edge joins an unordered pair of vertices. The set of vertices of G is denoted by V(G) & the set of edges is denoted by E(G).
Let’s take a closer look at graphs and formalize our discussion, we’ll need to introduce
some definitions and terminology.
A graph G is a pair of sets V and E
The elements of V are the vertices (a.k.a. nodes or
Points or dots ) of G.
The elements of E are the edges of G.
The function between the two sets sends an edge to the pair of vertices that are its called endpoints,
This function is generally a RELATIONSHIP
And may send more than one edge between a given pair of vertices.
Connections generally come in two forms,
those that are non-directional
and those that have an implicit direction
To distinguish these two cases we need to define two different kinds of graphs: undirected graphs and digraphs or directed graphs.
Throughout this presentation I will refer to undirected graphs and distinguish some aspects of digraphs
Pictures of graphs showing all of the edges and vertices are great ways to represent a graph. It is immediately clear which vertices are joined by edges, and you can often recognize the graph from its picture.
The use of pictorial images in social network analysis are critical both in helping investigators to understand network data and to communicate that understanding to others.
For example:
It is easy to see that this graph has 7 vertices (which here are numbered)
and 8 edges (which are not): simply count the pairs in the edge set.
Every vertex except one meets 2 vertices (note that vertex 5 meets 4 vertices).
In an undirected graph the number of edges that meet a vertex is the vertex Degree
Thus for vertices 1, 2, 3, 4, 6, & 7 the Degree is 2
For vertex 5 it is 4
(5, 7)}

6. (5, 7)} In Graph G: Vertex set V(G) = {v1 ,  , vn}
Edge set E(G) = { e1 ,  , em}
In this slide I denote the Vertex and Edge set elements formally
Let’s take a pretty abstract description of a graph G:
 We are given a vertex set V(G) = {v1 ,  , vn}
 We are given an edge set E(G) = { e1 ,  , em}
If this graph were instead a directed graph – the edges would have directionality
Imagine edges with arrows or even edges with self-loops;
In this imagined directed graph the elements in set E would be “ordered”: thus pair (a,b) or (1, 2) would be distinct from pair (b,a) or (2-1)
(5, 7)}

7. Graph G: To this directed graph we add elements (3,1), (4,2)
In this slide I denote the Vertex and Edge set elements formally
Let’s take a pretty abstract description of a graph G:
 We are given a vertex set V(G) = {v1 ,  , vn}
 We are given an edge set E(G) = { e1 ,  , em}
If this graph were instead a directed graph – the edges would have directionality
Imagine edges with arrows or even edges with self-loops;
In this imagined directed graph the elements in set E would be “ordered”: thus pair (a,b) or (1, 2) would be distinct from pair (b,a) or (2-1)
(5, 7)}

8. Definition: Two vertices are adjacent if there is an edge joining them
Definition: Two vertices are adjacent if there is an edge joining them. Vertices are said to be incident to the edge.
Adjacency-matrix representation of a Graph G= (V,E) is a |V| x |V| matrix A = (aij) such that aij = 1 if (i, j) and 0 otherwise
Here is another important definition
Definition Two vertices are said to be adjacent if there is an edge joining them. The vertices are said to be incident to the edge.
The ideas of adjacency and incidence can be used to come up with different ways of representing a graph.
These will become useful once we get to Matrix representation and operations
What I would like you to note here is that V x V is a SQUARE Matrix – this is our NETWORK \-- what we’re generally interested in

9. Some Special Graphs The empty graph (on 5 vertices): N5
The complete graph (on 5 vertices): K5
Here I introduce several families of graphs

10. The star graph (on 6 vertices)
The cycle graph (on 5 vertices): C5

11. A bipartite graph (vertex set partitioned into 2 subsets; are no edges linking vertices in the same set): K m n
K 3, 2 (top)
A complete bipartite graph (all possible edges are present):
K 5, 1 (middle)
K 3, 2 (bottom)
Complete bipartite graphs are de-noted Kmn where m and n are the sizes of the sets in the partition.
In the afternoon panel I will talk a bit about the data structure and bipartite graphs my current research.

12. Connectivity
The node-connectivity of a connected graph is the minimum number of vertices that need to be removed to disconnect the graph: k-connected
There is edge-connectivity as well….

13. Degree Centrality da /(n-1)
Degree centrality of a node a: CD (a) = d a
Normalized degree centrality of node a:
da /(n-1)
Example: Node x”: degree centrality = 4; normalized degree centrality = 4/6 = 0.67
There are other centrality measures :
Closeness centrality
Betweeness centrality
Maximum centrality
Graph-level measure of centralization
Eigenvector centrality