(a) Draw a picture of a simple (undirected) graph G whose adjacency matrix is 0 1 0 0 1 1 0 10 0 A = | 0 1 0 1 0 0 0 10 1 1 0 0 10 (b) Does there exist a graph with degree sequence (2,1, 2, 2, 2) ? If ,draw a picture of one such graph. If not, briefly explain. so,
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- graph and optimization leçon! question 4(b) Suppose G is a simple connected graph with 12 vertices and 16 edges. Suppose 4 of its vertices are degree 1, and 3 of its vertices are degree 2. Prove that G is planar. (Hint: Kuratowski) (c) Let G be any simple connected planar graph with n vertices and e edges. Suppose there are exactly y vertices of degree 2. Assume that n - y > 3. Prove that e < 3n - y - 6. (Hint: Explain why the degree-2 vertices can be erased, and how to take care of any resulting loops or multiple edges.) (d) Suppose that a connected simple graph G' has exactly 10 vertices of degree 4, 8 vertices of degree 5, and all other vertices have degree 7. Find the maximum possible number of degree-7 vertices G could have, so that G would still be planar.3. (b) The degree of every vertex of a graph G is one of three consecutive integers. If, for each of the three consecutive integers x, the graph G contains exactly x vertices of degree x, prove that two-thirds of the vertices of G have odd degree. (c) Construct a simple graph with 12 vertices satisfying the property described in part (b).
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- a. Draw all non-isomorphic graphs with 3 edges on 6 vertices.b. Draw all non-isomorphic graphs with 4 edges on 5 vertices.c. Draw all connected non-isomorphic graphs with degree sequence 5,5,2,2,2,2,2,2,2,2.d. Draw all non-isomorphic trees with degree sequence 3,3,2,2,2,1,1,1,1.Using graph C16. Let G be a graph with n vertices , t of which have degree K and the others have degree K+1 ,prove that t = (K+1)n-2e , e is number of edges in G .