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Let G = (V, E) be a simple graph with |V| Z 2. The complement graph G of G' is the simple graph whose vertex set is V and whose edge set is the set

These 3 questions are related to graph theory. Need solutions

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Let G = (V, E) be a simple graph with |V| Z 2. The complement graph G of G’ is the simple
graph whose vertex set is V and whose edge set is the set V2 \ E , namely the pairs of vertices
of V that are not edges in G’. State whether the following statements are Correct or Incorrect and provide a proof. (A) If G is disconnected, then C‘ is connected. (B) If G is connected, then C? is disconnected.

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Suppose that G is a simple graph with 272 nodes, for n 2 1, and 110 triangles (ie, 110 cycles
of length 3). Prove that G has at most n2 edges.

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Show that any simple graph G = (V, E) with 6(G) > % - (|V| — 2) is connected. Note that
6(G) is defined as the minimum degree of the vertices of the graph G, namely 6(G‘) 2:
min{d(v) : v E V}. (Hint: Use a proof by contradiction. Consider two vertices in different connected components
and consider the sets of the neighbors of each of the vertices; you may find the inclusion— exclusion principle for two sets helpful.)

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