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Define isomorphism of graphs. find if the following two graphs are isomorphic. if yes find one to one correspondence between the vertices.

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Define isomorphism of graphs. find if the following two graphs are isomorphic. if yes find one to one correspondence between the vertices.

**Solution:**

Let $G_1$ and $G_2$ be the two given graph then they are said to be isomorphic to each other if following conditions are satisfied.

**a)** No. of vertices of $G_1$ = No of vertices of $G_2$

**b)** No of angles of $G_1$ = No of edges of $G_2$

**c)** Both the graph must have same number of vertices with equal degree.

**1]** No. of vertices of $G_1$ = No of vertices of $G_2$

$V(G_1) = V(G_2)$

8 = 8

**2]** No of edges of $G_1$ = No of edges of $G_2$

$E(G_1) = E(G_2)$

12 = 12

**3]**

Definition of vertices of $G_1$ | Definition of vertices of $G_2$ |
---|---|

def(A) = 3 | def (A) = 3 |

def(B) = 3 | def(B) = 3 |

def (C) = 3 | def (C) = 3 |

def (D) = 3 | def (D) = 3 |

def (E) = 3 | def (E) = 3 |

def (F) = 3 | def (F) = 3 |

def (G) = 3 | def (H) = 3 |

def(H) = 3 | def (H) = 3 |

$\therefore$ Both the graph $G_1$ and $G_2$ isomorphic to each other.

One to one correspondence between $G_1$ and $G_2$

Graph $G_1$ | A | B | C | D | E | F | G | H |
---|---|---|---|---|---|---|---|---|

Graph $G_2$ | A | F | D | G | E | B | C | H |

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