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An automorphism of a graph G = (VE) is any isomorphism o G and G.i.e any bijection f : V ~V such that {u.v} € E if and only if {f(u). f(e)} € E. A graph...

Question

An automorphism of a graph G = (VE) is any isomorphism o G and G.i.e any bijection f : V ~V such that {u.v} € E if and only if {f(u). f(e)} € E. A graph is called asymmetric if its only automor- phism is the identity mapping (each Ferter I5 mapped t0 itself)_ (a) Find an example of an asyetric graph with at least 2 vertices (b) Show that no asymmetric graph G exists with 1 < IV(G) <5. Show that graph G with n vertices is asymmetric (sce Exercise 3) if and only if n! distinct graphs On

An automorphism of a graph G = (VE) is any isomorphism o G and G.i.e any bijection f : V ~V such that {u.v} € E if and only if {f(u). f(e)} € E. A graph is called asymmetric if its only automor- phism is the identity mapping (each Ferter I5 mapped t0 itself)_ (a) Find an example of an asyetric graph with at least 2 vertices (b) Show that no asymmetric graph G exists with 1 < IV(G) <5. Show that graph G with n vertices is asymmetric (sce Exercise 3) if and only if n! distinct graphs On the set V(G) are isomorphic to G_



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Suppose that $G$ and $H$ are isomorphic simple graphs. Show that their complementary graphs $\overline{G}$ and $\overline{H}$ are also isomorphic.


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