Things to note about the structure of the initial graph.

- a cycle of length 5
- have 1 and 2 length line attached to the same vertex.

(A) No cycle of length 5

(C) line graph of length 1 and 2 attached to a different vertex.

(D) No cycle of length 5

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26 votes

3

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42 votes

Best answer

For these type of questions find which are not isomorphic.

The graph in option (A) has a $3$ length cycle whereas the original graph does not have a $3$ length cycle

The graph in option (C) has vertex with degree $3$ whereas the original graph does not have a vertex with degree $3$

The graph in option (D) has a $4$ length cycle whereas the original graph does not have a $4$ length cycle

**so option B must be correct.**

sir can we do in this way

in given graph we have one 4 degree vertex delete that vertex obviously now we will have to delete all the adjacent edges to that 4 degree vertex,repeat thesame process in the (b) option.

after that check that both the graph are same or not ..here both graph are same therfore they are isomorphic!!

in given graph we have one 4 degree vertex delete that vertex obviously now we will have to delete all the adjacent edges to that 4 degree vertex,repeat thesame process in the (b) option.

after that check that both the graph are same or not ..here both graph are same therfore they are isomorphic!!

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A Quicker Solution is

* **Degree Sequence of Option A and C** are **not as same** as the Given Graph. So can be eliminated

* In **Option D vertex with degree 4** is adjacent to **vertices with degrees {2,2,2,2}** whereas in the **given graph **it is (vertex with degree 4) adjacent to vertices **with degrees {1,2,2,2 }** so option D can also be eliminated.

* Option B satisfies all necessary conditions and also also has a one to one corresspondence with the given graph.

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7 votes

Ans: **(B)**

**Explanation: ***Degree sequence of the given graph 4-2-2-2-2-2-1-1.*

*Degree sequene for other options: *

* A) 4-3 failed*

* C) 3- failed*

* For B) and D) degree sequence matches with given 4-2-2-2-2-2-1-1. Comparing B) , D) and original graph: original graph has 5-cycle. But D) have only 4-cycle. *

* B) matches all the criterias for isomorphic graph.*

5 votes

if graphs have to be isomorphic then they must follow the three condition

(1) no of points must be same

(2) no of edges also must be same

(3) degree sequence must be same

if these three conditon is following simultaneously then graph will be isomorpphic

option b is correct

(1) no of points must be same

(2) no of edges also must be same

(3) degree sequence must be same

if these three conditon is following simultaneously then graph will be isomorpphic

option b is correct

**Isomorphic graph: **Two graphs are isomorphic if there is one-to-one correspondence between them which preserves the** Incidence relation**.

if graphs have to be isomorphic then they must follow the 4 condition

(1) no of vertices must be same

(2) no of edges also must be same

(3) degree sequence must be same

(4)Incidence relation must be preserves

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2 votes

For Isomorphic :i)No of edges must be same ii)No of vertices must be same iii)And Pendant vertex same too

all options have equal number of edges and vertices but the thing which differ it from right one is pendant vertex

In given graph a node which have pendant vertex is 4 and 1 vertex has 2 nodes and another is one so option B contains same so option B is right one

all options have equal number of edges and vertices but the thing which differ it from right one is pendant vertex

In given graph a node which have pendant vertex is 4 and 1 vertex has 2 nodes and another is one so option B contains same so option B is right one