Hello I don\'t understand this example Please explain it in details Determine wh

Question

Hello

I don't understand this example

Please explain it in details

Determine which of the graphs represented below are isomorphic a and b are not isomorphic The vertices D(a) and F(b) Have the same degree but not the same adjacent degree a and d are not isomorphic The vertices D(a) and A(d) Have the same degree but not the same adjacent degree Determine which of the graphs represented below are isomotphic. a and c are isomorphic Ah rightarrow F.B rightarrow E, C rightarrow D, D rightarrow G, E rightarrow C,F rightarrow A, G rightarrow B1 b and d are not isomorphic The vertices B (d) have no similar vertix in b c is not isomorphic to b or d

Explanation / Answer

I hope u know what isomorphism is...so lets proceed further look at fig a ... consider vertex D, it is the simplest one as its degree is 2. first find a vertex in fig b whose degree is 2... it is F and C..since it is symmetric, so chose any of the two for comparison. D of fig a vs F of fig b both have same degree....i.e. 2 Now look at the vertices adjacent to them... for fig a, the vertices adjacent to D are :C and A C has degree 4, while A has degree 3 for fig b, the vertices adjacent to F are : E and A both E and A are of 3 degrees. Hence D is not equivalent to F or even C ( F and C are symmetric) Also, there is no other 2 degree vertex in fig b. so D (a) has no equivalence in fig b Hence fig a and fig b are not isomorphic Hope it is clear now... :) Consider fig a and fig d now: again starting by 2 degree vertices: fig a has two 2 degree vertices (D and B) fig d also has two of these (A and C) we have already analyzed fig a, so lets compare it with vertex A of fig d: the vertices adjacent to A (in fig d) are : F and B both of these have 3 degrees, fine??? but we found earlier that, the degress of vertices adjacent to D (in fig a) had degrees 3 and 4 respectively. It should be clear that again A and C( in fig d) are symmetric and there are no other 2 degree vertices. Hence fig a and fig d are also not isomorphic :) ------------------------------------------------------------------------- now look at a and c compare vertex D(a) with G(b) their degrees are 2 in each case... and the degrees of their adjacent vertices are also same... for fig a; degree of C = 4 and A = 3 for fig c; degree of D = 4 and F = 3 so u can see that D->G; C->D; A->F and so on... all other vertices have their related vertex.... Hence these 2 are isomorphic.... --------------------------------- Now comes fig b and d look at vertex B of fig d its degree is 3 and degree of its adjacent vertices are 2 and 2 is there any such vertex in fig b??? No... So the are not isomorphic... ----------------------------------- c is isomorphic to a a is not isomorphic to b and d so c is also not isomorphic to b and d :) Cheers :)

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