Relationship between Graph Theory and Linear Algebra By Shannon Jones.
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Transcript of Relationship between Graph Theory and Linear Algebra By Shannon Jones.
Relationship between Graph Theory and Linear Algebra
By
Shannon Jones
Outline
• Overview of Graph Theory
• Linear Algebra in Graph Theory
• Application of Adjacency Matrices in Graph Theory
• Application of Adjacency Matrices in Network Graph Analysis
Overview
• Graph Theory– Vertices V(G)– Edges E(G)
Linear Algebra in Graph Theory
• Linear Algebra – study of linear sets of equations and their
transformation properties. – Matrices– Isomorphism
Linear Algebra in Graph Theory
Matrices of a Graph– Matrix
– Adjacency Matrix
Linear Algebra in Graph Theory
• Adjacency Matrix- The adjacency matrix for a simple graph G, denoted A(G), is defined as the symmetric matrix whose rows and columns are both indexed by identical ordering of V(G), such that A(G)[u,v] = 1 if u and v are adjacent, otherwise A(G)[u,v]= 0.
• Ex: G= A(G)=
w
v
u
x
u v w xu 0 1 1 0v 1 0 1 0w 1 1 0 1x 0 0 1 0
Linear Algebra in Graph Theory
• Adjacency Matrix- The adjacency matrix of a simple digraph D, denoted A(D), is the matrix whose rows and columns are both indexed by identical orderings of V(G), such that A(D)[u,v]= 1 if there is an edge from u to v, otherwise A(D)[u,v]= 0.
• Ex: G= A(G)=
W
v
u
x
u v w xu 0 0 1 0v 1 0 0 0w 0 1 0 1x 0 0 0 0
Application of Adjacency Matrices in Graph Theory
• Graph Isomorphism– Same adjacency matrix = isomorphic– Different adjacency matrix = may not be isomorphic– Ex:
– Rearrange A(G)-
w y x zw 0 1 1 0y 1 0 0 1x 1 0 0 1z 0 1 1 0
a b c da 0 0 1 1b 0 0 1 1c 1 1 0 0d 1 1 0 0
w z x yw 0 0 1 1z 0 0 1 1x 1 1 0 0y 1 1 0 0
Application of Adjacency Matrices in Graph Theory
• Walks – A sequence of alternating vertices and edges
– Let G be a graph with adjacency matrix A(G). The value of element (A(G))^r [u,v] of the rth power of matrix A(G) equals the number of u-v walks of length r (or directed walks of length r for a digraph).
Application of Adjacency Matrices in Graph Theory
• Walks
• Ex: G=
A(G)= A(G)²= A(G)³=
w
v
u
x
u v w xu 0 1 1 0v 1 0 1 0w 1 1 0 1x 0 0 1 0
u v w xu 2 1 1 1v 1 2 1 1w 1 1 3 0x 1 1 0 1
u v w xu 2 3 4 1v 3 2 4 1w 4 4 2 3x 1 1 3 0
Application of Adjacency Matrices in Network Graph
Analysis
• Social Network Graph– Vertices = people– Edges = relationship between two people
• “married to”, “friends with”, “related to”
– Corresponding adjacency matrix
Application of Adjacency Matrices in Network Graph
Analysis
• Social Network Graph
• Degree Centrality
Bob
Caro l
Ted A lice
Bob Carol Ted AliceBob 0 1 1 0
Carol 1 0 1 0Ted 1 1 0 1
Alice 0 0 1 0
Application of Adjacency Matrices in Network Graph
Analysis
• Social Network Graph
• Directed Graph
Bob
Caro l
Ted A lice
Bob Carol Ted AliceBob 0 1 1 0
Carol 0 0 1 0Ted 1 1 0 0
Alice 0 0 1 0
Application of Adjacency Matrices in Network Graph
Analysis
• Social Network Graph Adjacency Matrix– Matrix Operations
• Transpose- rows and columns exchange = the measure of degrees of the reciprocity of ties within the graph
• Inverse- (original)(inverse)= identity• Addition and Subraction
Application of Adjacency Matrices in Network Graph
Analysis
• Social Network Graph Adjacency Matrix– Key Matrix Operation
• Powers of the Adjacency Matrix– number of walks of different lengths between people– connectivity of a person in the graph
Application of Adjacency Matrices in Network Graph
Analysis
• Social Network Graph Adjacency Matrix– Key Matrix Operation
• Powers of the Adjacency Matrix
1
2
3
4 5
6
7
1
2
3
4
5
6 7
8
9
10
G= H=
1
2
3
4 5
6
7
1 2 3 4 5 6 71 0 1 1 1 1 1 12 1 0 0 0 0 0 03 1 0 0 0 0 0 04 1 0 0 0 0 0 05 1 0 0 0 0 0 06 1 0 0 0 0 0 07 1 0 0 0 0 0 0
A(G)
1 2 3 4 5 6 71 6 0 0 0 0 0 02 0 1 1 1 1 1 13 0 1 1 1 1 1 14 0 1 1 1 1 1 15 0 1 1 1 1 1 16 0 1 1 1 1 1 17 0 1 1 1 1 1 1
A(G)²1 2 3 4 5 6 7
1 0 6 6 6 6 6 62 6 0 0 0 0 0 03 6 0 0 0 0 0 04 6 0 0 0 0 0 05 6 0 0 0 0 0 06 6 0 0 0 0 0 07 6 0 0 0 0 0 0
A(G)³
1
2
3
4
5
6 7
8
9
10
1 2 3 4 5 6 7 8 9 101 0 1 0 0 1 0 0 1 0 02 1 0 1 1 0 0 0 0 0 03 0 1 0 0 0 0 0 0 0 04 0 1 0 0 0 0 0 0 0 05 1 0 0 0 0 1 1 0 0 06 0 0 0 0 1 0 0 0 0 07 0 0 0 0 1 0 0 0 0 08 1 0 0 0 0 0 0 0 1 19 0 0 0 0 0 0 0 1 0 010 0 0 0 0 0 0 0 1 0 0
1 2 3 4 5 6 7 8 9 101 3 0 1 1 0 1 1 0 1 12 0 3 1 0 1 0 0 1 0 03 1 0 1 1 0 0 0 0 0 04 1 0 1 1 0 0 0 0 0 05 0 1 0 0 3 0 0 1 0 06 1 0 0 0 0 1 1 0 0 07 1 0 0 0 0 1 1 0 0 08 0 1 0 0 1 0 0 3 0 09 1 0 0 0 0 0 0 0 1 110 1 0 0 0 0 0 0 0 1 1
1 2 3 4 5 6 7 8 9 101 0 5 0 0 5 0 0 5 0 02 5 1 3 3 0 1 1 0 1 13 0 3 0 0 1 0 0 1 0 04 0 3 0 0 1 0 0 1 0 05 5 0 1 1 0 3 3 0 1 16 0 1 0 0 3 0 0 1 0 07 0 1 0 0 3 0 0 1 0 08 5 0 1 1 0 1 1 0 1 19 0 1 0 0 1 0 0 1 0 010 0 1 0 0 1 0 0 1 0 0
A(H)
A(H)² A(H)³
Application of Adjacency Matrices in Network Graph
Analysis
• Significance– Marketers– Social Network Websites
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Hanneman, Robert A., and Mark Riddle. Introduction to Social Network Methods. Riverside:
University of California, 2005. Web. 28 Apr. 2011.
<http://faculty.ucr.edu/~hanneman/nettext/index.html>.
Farmer, Jesse. "Graph Theory: Part III (Facebook)." 20bits. Web. 28 Apr. 2011.
<http://20bits.com/articles/graph-theory-part-iii-facebook/>.
"Graph." Wolfram MathWorld: The Web's Most Extensive Mathematics Resource. Wolfram
Research, Inc., 1999. Web. 28 Apr. 2011. <http://mathworld.wolfram.com/Graph.html>.
Gross, Jonathan L., and Jay Yellen. Graph Theory and Its Applications. Boca Raton:
Chapman & Hall/CRC, 2006.
"Linear Algebra." Wolfram MathWorld: The Web's Most Extensive Mathematics Resource. Wolfram
Research, Inc., 1999. Web. 28 Apr.
2011. <http://mathworld.wolfram.com/LinearAlgebra.html>.
West, Douglas Brent. Introduction to Graph Theory. Upper Saddle River, NJ: Prentice Hall, 1996.