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Discrete MathematicsAn undirected graph has 8 vertices labelled 1, 2, \dots, 8 and 31 edges. Vertices 1, 3, 5, 7 have degree 8 and vertices 2, 4, 6, 8 have degree 7. What is the degree of vertex 8? (i) 15 (ii) 8 (iii) 5 (iv) 2320222mDiscrete MathematicsA graph which has the same number of edges as its complement must have number of vertices congruent to \_\_\_\_\_\_ or \_\_\_\_\_\_ modulo 4 (for integral values of number of edges). (i) 6k, 6k-1 (ii) 4k, 4k+1 (iii) k, k+2 (iv) 2k+1, k20222mDiscrete Mathematics(i) The number of edges in a bipartite graph with n vertices is at most (i) n^2/2 (ii) n^2/4 (iii) n^2 (iv) 2n20192mDesign and Analysis of Algorithms If all edge weights in a graph are positive, what is the minimum number of edges in a spanning tree for a graph with vertices? (i) V (ii) V-1 (iii) V+1 (iv) 2V20252m
PreviousIf C_n is the nth cyclic graph, where n>3 and n is odd, determine the value of \chi(C_n). (i) 32572 (ii) 16631 (iii) 3 (iv) 310Next(a) Let D = \{-48, -14, -8, 0, 1, 3, 16, 23, 26, 32, 36\} Determine which of the following statements are true and which are false. Provide counterexamples for those statements that are false. (i) \forall x \in D, if x is odd, then x > 0 (ii) \forall x \in D, if x is less than 0, then x is even (iii) \forall x \in D, if x is even, then x \le 0