A simple graph that is isomorphic to its complement is self-complementary. (i) Prove that, if G is self-complementary, then G has 4k or 4k+1 vertices, where k is an integer. (ii) Find all self-complementary graphs with 4 and 5 vertices. (iii) Find a self-complementary graph with 8 vertices.

Q: If a simple graph G is having 15 vertices and 5 edges then its complement graph G will have None of…

A: 15 vertices and 105 edges

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Q: Find all r-regular graphs of order n for a convenient selection of r, n.

A:

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Q: 2n 4. Prove or disprove: for every n-vertex graph G, x(G) ≤ 2w(G) + a (G) (6 + 10.

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Q: (c) Prove that the two graphs below are isomorphic.

A: Given, We will prove these two graphs are isomorphic.

Q: Let Vn be the set of connected graphs having n edges, vertex set [n], and exactly one cycle. Form a…

A: Given Let Vn be the set of connected graphs having n edges, vertex set [n], and exactly one cycle.…

Q: (b) Give an example of a disconnected simple graph on n vertices in which every vertex has degree…

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A:

Q: Show that if G is a simple graph with n vertices (where n is a positive integer) and each vertex has…

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

Q: Evaluate the given Jacobi symbols using the properties of the Jacobi symbol and the generalized LQR:…

A: (a) Given Jacobi symbol is 853861

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Q: (b) Show that G may not necessarily have three cycle subgraphs by giving an example of a graph G,…

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Q: (c) A complete graph with two connected components.

A: