Problem 1. Prove the following or give a counterexample: If G = (V, E) is a graph, and v, w e V arevertices such that there is a path from v to w, then there is a simple path from v to w.

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Answered: Problem 1. Prove the following or give…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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If u and v are vertices of a connected graph G, then d(u, v) is the length of a shortest path in G joining u and v. We call d(u, v) the distance from u to vShow that for any three vertices d(u, v) + d(v, w ) > = d(u, w)
Problem 2. Given the vertices, connect them with edges in order to get a planar graph without isolated vertices (all vertices must be included in the graph).
O (d) (e) 2. For the following pairs of graphs either A). Prove that they are isomorphic by appropriately labeling the vertices of the two graphs in such a way that they have the same edge sets, or B). Justify why they are not isomorphic. In each case encircle either the letter A if they are isomorphic, or B if they are not. (a) A B (b) A A A A B B B B
can you please answer a and b
9. Let G2 be depicted in the following diagram: b h a d 9(d) Give an example of a walk of length five in G2 that is not a path.
2. Show that each of the following graphs are not planar by finding a subgraph that is isomorphic to a subdivision of either K5 or K3,3. Be sure to label the vertices! (a) (b) (c) E D H 15. A E H 3 E D F G E B D B H
a. Let V = {x, y, z, u, v}. How many distinct graphs are there on the vertex set V with exactly 6 edges? (Note that for graphs G₁ = (V, E₁) and G₂ = (V, E₂) to be distinct, they need only have E₁ # E2, but they may be isomorphic.) b. In the graph below, give an example of a trail that is not a path. I Y U Z

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Problem 1. Prove the following or give a counterexample: If G = (V, E) is a graph, and v, w E V are vertices such that there is a path from v to w, then there is a simple path from v to w.