Given a graph G = (V,E) its complement G is defined to be (V,E) where E = V{2} \ E. That is: a graph on the same vertex-set as G where two vertices are joined in G if and only if they are not joined in G. Prove that if two graphs G₁ & G2 are isomorphic, then their complements G₁ & G2 are isomorphic.

Prove that if  v0 and v1  are distinct vertices of a graph  G  = (V,E) and a path exists in G from v0 to v1 , then there is a simple path in G from v0 to v1 .

Draw graph G and its complement, showing that at least one of G and it's complement, G', is connected.

Graph theory

The complementary graph G' of a simple graph G hasthe same vertices as G. Two vertices are adjacent in G ifand only if they are not adjacent in G. If G is a simple graph with 12 edges and G' has 10 edges, how many vertices does G have?

Let G be a graph of order n that is isomorphic to its complement G. How many edges does G have? Explain your answer. If a graph G has n vertices, all of which but one have odd degree, how many vertices of odd degree are there in G, the complement of G? Prove your answer.

Let G be a connected graph of order n and size n. Prove that G contains a single cycle.

Let G be a graph on at least 3 vertices. Suppose that G has the property that G – v is connected for each vertex v of G. (a) Prove that G must be a connected graph. (b) Prove that every vertex of G has degree at least 2.

The symmetric difference graph of two graphs G. (V. E.) and G₂ (V₁ E₂) on the same vertex Set is defined as G₁ AG₂:= (V, E, DE₂). E₁ DE ₂ = (E₁ \ E₂) U (E₂\E.) : E₁ E₂ = €₁ 0 € ₂ If G₁ and 6₂ are euleran, show that every Vertex in G, D G₂ has even degree FACT: Each vertex of a evlenan graph has even degree. SO: Show G₁ D G₂ is eulerian.