Let G be a regular, connected planar graph of order 24. If every vertex of G has order 3, how many regions are in a planar representation of G? Explain your answer.
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- Let u and v be distinct vertices in a connected graph G. There may be several connected subgraphs of G containing u and v. What is the minimum size of a connected subgraph of G containing u and v? Explain your answer.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 simple graph with 11 vertices, each of degree 5 or 6. Prove that G has at least 7 vertices of degree 8 or at least 6 vertices of degree 7. Do not use the planar equation e <= 3v - 6.
- Help me with b and cLet G be a planar graph. Use Euler's Formula to prove that G must have a vertex of degree 5 or less. Hint: what if it didn't?Consider the graph G : O D Which of the graphs below is isomorphic to G? O A. B. H H F X V U Indicate the isomorphism by listing the vertices separated by commas in the order they correspond to A,B,C,D,E,F,G, respectively.
- Consider the graph G: Which of the graphs below is isomorphic to G? A B Indicate the isomorphism by listing the vertices separated by commas in the order they correspond to A,B.C.D.E.F. respectively. r,v.hxyzLet G be a connected graph of order n and size n. Prove that G contains a single cycle.Consider the graph G : Which of the graphs below is isomorphic to G? O A. B. V V U U Z Indicate the isomorphism by listing the vertices separated by commas in the order they correspond to A,B,C,D,E,F,G, respectively. Z, X, U, T, H, V, Y
- Consider the following two graphs G1 = {{a,b, c, d, e, f}, {ab, a f, ae, bc, bf, be, cd, cf, df, ef}}, and G2 = {{v1, v2, v3, v4, V5, V6}, {v1v2, v1V3, V1V4, V1 V5, V2V3, V2V6, V3V4, V3V5, V3 V6, V4V5}}, %3D (a) is the function b c d f a e U5 v2 V4 V3 a graph isomorphism? (b) can you construct a bijection of vertices that is not an isomorphism? explain. (c) provide a drawing that represents this isomorphism class.Let Vn be the set of connected graphs having n edges, vertex set [n], and exactly one cycle. Form a graph Gn whose vertex set is Vn. Include {gn, hn} as an edge of Gn if and only if gn and hn differ by two edges, i.e. you can obtain one from the other by moving a single edge. Tell us anything you can about the graph Gn. For example, (a) How many vertices does it have? (b) Is it regular (i.e. all vertices the same degree)? (c) Is it connected? (d) What is its diameter?Which of the following graphs are isomophic? (Choose two) F F G D D I I А В F F E E G G D H H J I I To submit: For the two that are isomorphic submit the vertex correspondence. For the rest, show why they are not isomorphic to your choice.