Graph Matrices

8/6/2019 Graph Matrices

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Matrices

How do we specify a graph?

List the vertices and edges

Specify with computer representation


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Computer representation

adjacency matrix: a |V| x |V| array where each cell i,jcontains theweight of the edge between vi and vj (or 0 for no edge)

adjacency list: a |V| array where each cell icontains a list of allvertices adjacent to vi

incidence matrix: a |V| by |E| array where each cell i,jcontains aweight (or a defined constant HEAD for unweighted graphs) if thevertex i is the head of edge jor a constant TAIL if vertex iis the tail of edge j

c b

a d4

2

6

108

a b c d

a 8 4

b

c 6

d 10 2

a c 8), 4)

b

c b 6)

d c 2), b 10)

1 2 3 4 5

a 8 4

b t t

c 6 t t

d 2 10 t

adjacency

matrix adjacencylist incidencematrix


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Two graphs are isomorphic if a 1-to-1

correspondence between their vertex sets exists

that preserve adjacencies

Determining to two graphs are isomorphic is NP-complete

Famous problems: Isomorphism

a

b

c

d

e

1

2 3

4 5


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Isomorphic GraphsTwo graph G and H are isomorphic if H can be obtained from G by

relabeling the vertices - that is, if there is a one-to-one

correspondence between the vertices of G and those of H, such

that the number of edges joining any pair of vertices in G is equalto the number of edges joining the corresponding pair of vertices in

H.

For example, two unlabeled graphs, such as are isomorphic if

labels can be attached to their vertices so that they become the

same graph.


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Famous problems: Maximal

cliqueA clique of a graph is its complete subgraph ,

A clique not contained in any other clique; the largest

complete subgraph in the graph and then refer to"maximum cliques.

The problem of finding the size of a clique for a given

graph is an NP-complete problem.


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clique A vertex coverof a graph G can be defined as a set Sof vertices of

G such that every edge ofG has at least one of member ofSas anendpoint.

Vertex covers, indicated with red coloring, are shown above for anumber of graphs

The clique number(G) of a graph G is the number of vertices inthe largest clique in G.


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A clique in an undirected graph G = (V, E) is a subset of the vertex set C V, such that for every two vertices inC, there exists an edge connecting the two. This is

equivalent to saying that the subgraph induced by C iscomplete (in some cases, the term clique may also referto the subgraph).

A maximal clique is a clique that cannot be extended by

including one more adjacent vertex, that is, a cliquewhich does not exist exclusively within the vertex set of alarger clique.

A maximum clique is a clique of the largest possible

size in a given graph.

The clique number(G) of a graph G is the number ofvertices in the largest clique in G.


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23 1-vertex cliques (its vertices),

42 2-vertex cliques (its edges),

19 3-vertex cliques (the light blue triangles), and

2 4-vertex cliques (dark blue).

Six of the edges and 11 of the triangles form maximal

cliques.

The two dark blue 4-cliques are both maximum and maximal,

the clique numberof the graph is 4.


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Another example

Determine the no. clique, maximal clique, maximum clique and

clique number.


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clique clique cover: vertex set divided into non-disjoint subsets, each of

which induces a clique

clique partition: a disjoint clique cover

1 2

4 3

Maximal cliques: {1,2,3},{1,3,4}

Vertex cover: {1,3}Clique cover: { {1,2,3}{1,3,4} }

Clique partition: { {1,2,3}{4} }


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Famous problems: Coloring

vertex coloring: labeling the vertices such thatno edge in E has two end-points with the samelabel


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Famous problems: Coloring

chromatic number: The minimum number of colors whichwith the vertices of a graph G may be colored is calledthe chromatic number, denoted


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Famous problems: Bipartite

graphs Bipartite: any graph whose vertices can

be partitioned into two distinct sets so that

every edge has one endpoint in each set.

How colorable is a bipartite graph?

Can you come up with an algorithm to

determine if a graph is bipartite or not?

K4,4


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Petersen graph

The Petersen graph is most commonly

drawn as a pentagon with a pentagram

inside, with five spokes.

Named after Julius Petersen

Vertices 10

Edges 15

Girth 5

Chromatic number 3

Petersen graph

TheP

etersen graph is an undirected graph with 10 vertices and 15 edges.

Girth: The length of the shortest graph cycle in a graph


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Solve:

Exercise: 1.1.2, 1.1.4, 1.1.9, 1.1.11, 1.1.16,

1.1.18, 1.1.19, 1.1.20, 1.1.24, 1.1.25

Introduction to Graph Theory

Second Edition

Douglas B. West

Graph Matrices

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