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Graph-02. Adjacency list. Graph representation. Adjacency matrix. Graph representation. When sparse use adjacency list When dense use adjacency matrix. Graph representation. Isomorphic Graph.

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Graph representation

Adjacency list

Graph representation


Graph representation1

Adjacency matrix

Graph representation


Graph representation2

When sparse use adjacency list

When dense use adjacency matrix

Graph representation


Isomorphic graph
Isomorphic Graph

  • The simple graphs G1 = (V1,E1) and G2 = (V2, E2) are isomorphic if there is a one-to-one and onto function f from V1 to V2 with the property that a and b are adjacent in G1 if and only if f(a) and f(b) are adjacent in G2 , for all a and b in V1 . Such a function f is called an isomorphism.


Properties
Properties

  • must have same number of vertices

  • must have the same number of edges

  • the degrees of the vertices must be the same.



Path

  • Path is a sequence of edges that begins at a vertex of a graph and travels from vertex to vertex along edges of the graph.

  • The path is a circuitif it begins and ends at the same vertex, that is, if u = v and has length greater than zero.

  • A path or circuit is simple if it does not contain the same edge more than once.


Connected graph
Connected Graph

  • An undirected graph is called connected if there is a path between every pair of distinct vertices of the graph.

  • A directed graph is strongly connected if there is a path from a to b and from b to a whenever a and b are vertices in the graph.


Theorem
THEOREM

  • Let G be a graph with adjacency matrix A with respect to the ordering V1, V2, . . Vn (with directed or undirected edges, with multiple edges and loops allowed). The number of different paths of length r from Vi to Vj , where r is a positive integer, equals the (i, j)th entry of N.


Euler circuit
Euler Circuit

  • An Euler circuit in a graph G is a simple circuit containing every edge of G. An Euler path in G is a simple path containing every edge of G .

    Solution:

    The graph G1 has an Euler circuit, for example, a , e, c, d, e, b, a. Neither of the graphs G2 or G3 has an Euler circuit.

    However, G3 has an Euler path, namely, a , c, d, e, b, d, a , b. G2 does not have an Euler path.


Solution:

The graph H2 has an Euler circuit, for example, a , g, c, b, g, e, d, f, a. Neither H1 nor H3has an Euler circuit.

H3has an Euler path, namely, c, a , b, c, d, b but H1does not.


Theorem1
THEOREM

  • A connected multigraph with at least two vertices has an Euler circuit if and only if each of its vertices has even degree.

  • A connected multigraphhas an Euler path but not an Euler circuit if and only if it has exactly two vertices of odd degree.


  • G1contains exactly two vertices of odd degree, namely, b and d. Hence, it has an Euler path that must have b and d as its endpoints. One such Euler path is d, a , b, c, d, b.

  • Similarly,G2has exactly two vertices of odd degree, namely, b and d. So it has an Euler path that must have b and d as endpoints. One such Euler path is b, a , g, j, e, d, c, g, b, c, j, d.


Hamilton path
Hamilton Path

  • A simple path in a graph G that passes through every vertex exactly once is called a Hamilton path and a simple circuit in a graph G that passes through every vertex exactly once is called a Hamilton circuit.


  • G1 has a Hamilton circuit: a , b, c, d, e, a.

  • There is no Hamilton circuit in G2 (this can be seen by noting that any circuit containing every vertex must contain the edge {a , b} twice),

  • but G2 does have a Hamilton path, namely, a , b, c, d.

  • G3 has neither a Hamilton circuit nor a Hamilton path, because any path containing all vertices must contain one of the edges {a, b}, {e, f}, and {c, d} more than once.



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