2. (a) Let G be a connected non-complete graph with order n 2 3 and diameter d. Prove that the connectivity K(G) of G satisfies d-1 (b) A connected graph is called unicyclic if it contains exactly one cycle. Prove that the edge-connectivity of any unicyclic graph is at most 2. 2. (a) Let G be a connected non-complete graph with order n 2 3 and diameter d. Prove that the connectivity K(G) of G satisfies d-1 (b) A connected...
Let G be a connected non-complete graph with order n 2 3 and diameter d. Prove that the connectivity κ(G) of G satisfies d-1 Let G be a connected non-complete graph with order n 2 3 and diameter d. Prove that the connectivity κ(G) of G satisfies d-1
2) Let G ME) be an undirected Graph. A node cover of G is a subset U of the vertex set V such that every edge in E is incident to at least one vertex in U. A minimum node cover MNC) is one with the lowest number of vertices. For example {1,3,5,6is a node cover for the following graph, but 2,3,5} is a min node cover Consider the following Greedy algorithm for this problem: Algorithm NodeCover (V,E) Uempty While...
8. This question has two parts. (i) Let G be a graph with minimum vertex degree 8(G) > k. Then prove that G has a path of length at least k. (ii) Let G be a graph of order n. If S(G) > nz?, then prove that G is connected.
6. Prove that the following graphs are connected: (a) The 3 vertex cycle: (b) The following 4 vertex graph: (c) K 7. An edge e of a connected graph G is called a cut edge if the graph G obtained by deleting that edge (V(G) V(G) and E(G) E(G) \<ej) is not connected. Prove that if G1 and G2 are connected simple graphs which are isomorphic and if G1 has a cut edge, then G2 also has a cut edge....
Let G = (V, E) be a weighted undirected connected graph that contains a cycle. Let k ∈ E be the edge with maximum weight among all edges in the cycle. Prove that G has a minimum spanning tree NOT including k.
B-1 Graph coloring Given an undirected graph G (V. E), a k-coloring of G is a function c : V → {0, 1, . . . ,k-1} such that c(u)≠c(v) for every edge (u, v) ∈ E. In other words, the numbers 0.1,... k-1 represent the k colors, and adjacent vertices different colors. must havec. Let d be the maximum degree of any vertex in a graph G. Prove that we can color G with d +1 colors.
Question 13. Prove that if k is odd and G is a k-regular (k - 1)-edge-connected graph, then G has a perfect matching Question 13. Prove that if k is odd and G is a k-regular (k - 1)-edge-connected graph, then G has a perfect matching
2a) Let a, b e R with a < b and let g [a, bR be continuous. Show that g(x) cos(nx) dx→ 0 n →oo. as Hint: Let ε > 0, By uniform continuity of g, there exists δ > 0 such that 2(b - a Choose points a = xo < x1 < . . . < Xm such that Irh-1-2k| < δ. Then we may write rb g (z) cos(nx) dx = An + Bn where 7m (g(x)...
Let n be a positive integer. For each possible pair i, j of integers with 1 sisi<n, find an n x n matrix A with the property that 1 is an eigenvalue of A with g(1) = i and a(1) = j.