Home Work 8 (Discrete) 1- Find a graph with degree sequence degn = {3, 3, 3, 3, 2, 2}. Represent this graph by adjacency and incidence matrices. A B E1 E3 C E2 E6 E4 E7 E5 D E8 E F Adjacency Matrices: We order the vertices as A, B, C, D, E, F. The adjacency matrix representing this graph is π π π π π [π π π π π π π π π π π π π π π π π π π π π π π π π π π π π π π] Incidence Matrices: We order the vertices as A, B, C, D, E, F. and the edges as E1, E2, E3, E4, E5, E6, E7, E8. The Incidence matrix representing this graph is π π π π π [π π π π π π π π π π π π π π π π π π π π π π π π π π π π π π π π π π π π π π π π π π π] 2- Is it possible to construct a graph with degree sequence degn ={ 3,3,3,2,2,1,1}? No, οdeg(vi )=2m which is even (Note that this applies even if multiple edges and loops are present) 3+3+3+2+2+1+1=15 odd (not even) 3- List all (not isomorphic) connected graphs with 4 vertices. 4- Let T be a tree with 5 vertices of degree 3, 5 vertices of degree 4, and 5 vertices of degree 5. All other vertices are of degree 1. Find the number of the leaves (vertices of degree1) in the tree T using hand-shaking theorem. - Let L be the number of leaves in the tree T. The number of edges (E) in T is n-1 (Theorem2 CH11.1) where n is the number of vertices: E = n-1 = (5 + 5 + 5 + L) -1 - οπππ(π£π ) = 2πΈ From the hand-shaking theorem it follows οπππ(π£π )=(5*3 + 5*4 + + 5*5 + L*1) = 2*(5+5+5+L-1). After Solving this equation: L=32 T contains 32 leaves. 5- Find in which order are the vertices of the ordered rooted tree visited using a preorder, inorder and postorder traversal. Preorder: a, b, e, j, k, n, o, p, f, c, d, g, l, m, h, i. Inorder: j, e, n, k, o, p, b, f, a, c, l, g, m, d, h, i. Postorder: j , n, o, p, k, e, f , b, c, l, m, g, h, i, d, a. Note: See example 2, 3 and 4 in text book (11.3).
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