[PDF] Adjacency and Incidence Matrices





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Adjacency and Incidence Matrices

1/10

The Incidence Matrix of a Graph

Denition

LetG= (V;E) be a graph whereV=f1;2;:::;ngand

E=fe1;e2;:::;emg. Theincidence matrix of Gis annm

matrixB= (bik), where each row corresponds to a vertex and each column corresponds to an edge such that ifekis an edge betweeniandj, then all elements of columnkare 0 except b ik=bjk= 1.12 3 4e fgB=2 6

641 1 1

1 0 0 0 1 0

0 0 13

7

752/10

The First Theorem of Graph Theory

Theorem

IfGis a multigraph with no loops andmedges, the sum of the degrees of all the vertices ofGis 2m.Corollary The number of odd vertices in a loopless multigraph is even. 3/10

Linear Algebra and Incidence Matrices of Graphs

Recall that the

rank of a matrix is the dimension of its ro w space.Proposition LetGbe a connected graph withnvertices and letBbe the incidence matrix ofG. Then the rank ofBisn1 ifGis bipartite andnotherwise.Example 12 3 4e fgB=2 6

641 1 1

1 0 0 0 1 0

0 0 13

7

754/10

Linear Algebra and Incidence Matrices of Graphs

Recall that the

rank of a matrix is the dimension of its ro w space.Proposition LetGbe a connected graph withnvertices and letBbe the incidence matrix ofG. Then the rank ofBisn1 ifGis bipartite andnotherwise.Example 12 3 4e fghB=2 6

641 1 1 0

1 0 0 1

0 1 0 1

0 0 1 03

7

755/10

The Adjacency Matrix of a Graph

Denition

LetG= (V;E) be a graph with no multiple edges where

V=f1;2;:::;ng. Theadjacency matrix of Gis thenn

matrixA= (aij), whereaij= 1 if there is an edge between vertexiand vertexjandaij= 0 otherwise.Notes

The adjacency matrix of a graph is symmetric.

6/10

Adjacency Matrix Example

12 3 4e fgA=2 6

640 1 1 1

1 0 0 0

1 0 0 0

1 0 0 03

7

757/10

Vertex Degree

Denitions

The degree of a vertex in a graph is the numb erof edges incident on that vertex.

A vertex is

o dd if its degr eeis o dd;otherwise, it i s even .Notes The sum of the elements of rowiof the adjacency matrix of a graph is the degree of vertexi. The sum of the elements of columniof the adjaceny matrix of a graph is the degree of vertexi.8/10

Linear Algebra and Adjacency Matrices of Graphs

Proposition

LetAbe the adjacency matrix of a graph. The (i;i)-entry in A

2is the degree of vertexi.Recall that thetrace of a squa rematrix is the sum of i ts

diagonal entries.Proposition LetGbe a graph witheedges andttriangles. IfAis the adjacency matrix ofG, then (a)trace(A) = 0, (b)trace(A2) = 2e, (c)trace(A3) = 6t.9/10

Acknowledgements

Statements of denitions follow the notation and wording of

Balakrishnan'sIntroductory Discrete Mathematics.

10/10quotesdbs_dbs17.pdfusesText_23
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