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)=

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)=

Application of Adjacency Matrices in Graph Theory Graph Isomorphism –Same adjacency matrix = isomorphic –Different adjacency matrix = may not be isomorphic –Ex: –Rearrange A(G)-

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=

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

Application of Adjacency Matrices in Network Graph Analysis Social Network Graph Directed Graph

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 G=H=

A(G) A(G)²A(G)³

A(H) A(H)²A(H)³

Application of Adjacency Matrices in Network Graph Analysis Significance –Marketers –Social Network Websites

Sources Chartrand, Gary, and Gary Chartrand. Introductory Graph Theory. New York: Dover, Hanneman, Robert A., and Mark Riddle. Introduction to Social Network Methods. Riverside: University of California, Web. 28 Apr Farmer, Jesse. "Graph Theory: Part III (Facebook)." 20bits. Web. 28 Apr "Graph." Wolfram MathWorld: The Web's Most Extensive Mathematics Resource. Wolfram Research, Inc., Web. 28 Apr Gross, Jonathan L., and Jay Yellen. Graph Theory and Its Applications. Boca Raton: Chapman & Hall/CRC, "Linear Algebra." Wolfram MathWorld: The Web's Most Extensive Mathematics Resource. Wolfram Research, Inc., Web. 28 Apr West, Douglas Brent. Introduction to Graph Theory. Upper Saddle River, NJ: Prentice Hall, 1996.