10. Let G = (V, E) be a loop-free connected planar graph. If G is isomorphic to its dual and|V | = n, what is |E|?
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10. Let G = (V, E) be a loop-free connected planar graph. If G is isomorphic to its dual and
|V | = n, what is |E|?
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- Use symmetry to sketch the graph of xy2=1.a If a graph is symmetric with respect to the x-axis and (a,b) is on the graph, then (,) is also on the graph. b If a graph is symmetric with respect to the y-axis and (a,b) is on the graph, then (,) is also on the graph. c If a graph is symmetric about the origin and (a,b) is on the graph, then (,) is also on the graph.Q. 6: State and prove closed Graph theorem.
- Explain thoroughly!!(Discrete Math-Graph Theory) Let G=(V,E₁∪E₂) be a simple graph such that G₁=(V,E₁) and G₂=(V,E₂) are both forests. Prove or disprove: G must be a planar graph.Consider the graph G with • V(G) = {2,3,6} • e(G) = {a, b, c, d, e, f, g} •E(G) = {(a, [2,2]), (b, [3,3]), (c, [6,6]), (d, [2,6]), (e, [6,2]), (f, [3,6]), (g, [6,3])} Using edge connectivity, we can define the relation R = {(2, 2), (3, 3), (6, 6), (2, 6), (6, 2), (3, 6), (6,3)} Which of the following statements are true? [More than one statement may be true.] R is reflexive. R is symmetric. R is transitive. R is antisymmetric.
- 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?A graph X is called self-complementary if it isomorphic to its complement. For a self-complementary graph X = (V, E), let |V| = y , then (a) find |E| (b) prove that X is connected.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 G be a graph with vertex set V(G) = {v1, v2, V3, V4, V5, V6, V7} and edge set E(G) = {v1v2, V2V3, VZV4, V4V5, V4V1, V3V5, V6V1, V6V2, V6V4, V7V2, V7V3, V7V4} Let H be a graph with vertex set V (H)= {u1, U2, U3, U4, U5, U6, U7} and edge set E(H)={u1u2, U1U5, U2U3, U2U4, UQU5, U2U7, UZU6, UZU7, U4U5, UĄU6, U5U6, U6U7} Are the graphs G and H isomorphic? If they are, then give a bijection f : V (G) V(H) that certifies this, and if they are not, explain why they are not.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.= Let G be a graph and e E E(G). Let H be the graph with V(H) = = V(G) and E(H) = E(G)\{e}. Then e is a bridge of G if H has a greater number of connected components than G. (a) Let G be the simple graph with V(G) = {u, v, w, x, y, z) and E(G) z} {uy, vx, vz, wx, xz}. For each e € E(G), state whether e is a bridge of G. Justify your answer. = (b) Assume that G is connected and that e is a bridge of G with endpoints u and v. Show that H has exactly two connected components H₁ and H₂ with u € V (H₁) and v € V(H₂). To this end, you may want to consider an arbitrary vertex wE V (G) and use a u-w-path in G to construct a u-w-path or a v-w-path in H. (c) Show that e is a bridge of G if and only if it is not contained in a cycle of G.
