2. Give an example of a graph with at least four vertices, or prove that none exists, such that:(a) There are no vertices of odd degree.(b) There are no vertices of even degree.(c) There is exactly one vertex of odd degree.(d) There is exactly one vertex of even degree.(e) There are exactly two vertices of odd degree.

The objective is to give an example of a graph with at leas four vertices, or prove that none…

Answered: 2. Give an example of a graph with at…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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(b) A planar graph has 10 edges and its dual graph has j edges, for some integer j. What is this j? Answer: j =
indicate if each of the two graphs are equal. Justify youranswer.
2) Parts (a) and (b) on this page are separate. (a) An undirected simple graph has 1000 vertices of degree 6 and 200 vertices of de- gree 2. This accounts for all vertices. How many edges does it have? Work this out to a final numeric answer. (b) How many directed simple graphs (no loops or multiple edges) are there on vertex set {0, 1,...,9}? Express your answer to (b) using an appropriate formula with specific numbers plugged in, such as V100! + 2, rather than evaluating it to a final numerical result.
For which of the following does there exist a simple graph G = - (V, E) satisfying the specified conditions? Select one: O A. It has 7 vertices, 10 edges, and more than two components. B. It has 8 vertices, 8 edges, and no cycles. O C. It has 6 vertices, 11 edges, and more than one component. O D. It is connected and has 10 edges, 5 vertices and fewer than 6 cycles. O E. It has 3 components, 20 vertices and 16 edges.
The number of non-isomorphic simple graphs on 8 vertices, all of which have degree 2, is Select one: O a. 16 b. 7 O c. 5 O d. 8 O e. 3
A graph is bipartite if its vertex set can be partitioned into two sets V₁ and V2 such all edges are between V₁ and V2 (i.e. there are no edges joining vertices inside V₁, and the same for V2). (a) Draw a bipartite graph with 5 vertices and 5 edges. (b) What is the maximum number of edges for a bipartite graph with 2n vertices (suppose n > 1)?
Use the formula deg(v) = 2|E(G)| to find the number of edges of the following vЄV (G) graphs. Classify (count) the vertices by number of neighbors. (a) V(G) = [100]. Edges: for all n and m in [100], n ‡ m, n is adjacent to m if and only if |nm| ≤ 4. (b) V(G) = [10] × [10]. Edges: for all (a, b) and (c,d) in [10] × [10], (a, b) ‡ (c,d), (a, b) is adjacent to (c,d) if and only if a = c or b = d. (c) V(G) = [10] × [10]. Edges: for all (a, b) and (c,d) in [10] × [10], (a, b) ‡ (c,d), (a, b) is adjacent to (c,d) if and only if |ac| + |bd| = 1. (d) V (G) = [10] × [10]. Edges: for all (a, b) and (c,d) in [10] × [10], (a, b) is adjacent to (c,d) if and only if |a - c + b-d ≤ 2. (a, b) ‡ (c,d),
True or False: There exists a simple graph with 9 vertices each of degree 5.
31. In any undirected graph the sum of degrees of all the nodes A. must be even B. are twice the number of edges C. must be odd D. need not be even
3. Prove the following statement: If G is a graph with at least 2 vertices, then a(G) > 2 or w(G) 2 2.
Q₁. Construct a connected graph with at least 12 vertices such that the minimal dominating set has 6 vertices and the minimum dominating set has 4 vertices.

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