Download - 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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Famous problems: Maximal
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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Famous problems: Maximal
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