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Introduction to Introduction to Graph TheoryGraph Theory

Theodora WelchManagement & Marketing Department

College of Management SNA Workshop

July 31, 2008(SOURCE: Introduction to graph theory, Lecture notes by Tom Snijders,

http://www.stats.ox.ac.uk/~snijders/D1pm_Graph_theory_basics.pdf)

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Graph Theory • 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

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“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.

x

y

y = f(x)

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“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:

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• 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).

(5, 7)}

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In Graph G:• Vertex set V(G) = {v1 , , vn}

• Edge set E(G) = { e1 , , em}

(5, 7)}

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Graph G: To this directed graph we add elements (3,1), (4,2)

(5, 7)}

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• 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

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• The empty graph (on 5 vertices): N5

• The complete graph (on 5 vertices): K5

Some Special Graphs

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• The star graph (on 6 vertices)

• The cycle graph (on 5 vertices): C5

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• 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)

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• The node-connectivity of a connected graph is the minimum number of vertices that need to be removed to disconnect the graph: k-connected

Connectivity

0 1 2 4

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Degree Centrality

• 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