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7. From a graph G P(G) = (Vp, Ep), is formed as follows: (V, E), with vertices v; E V and edges e; E E, the prism of the graph, Let V' be a copy of V with vertices v, and E' a copy of E with edges e'. Now V, = VUV', and E, = EU E'U {(vi, v²)\v; E V}. An example of a graph and its prism is shown below. (a) If a graph G has n vertices and k edges, find formulae for the number of vertices and the number of edges in P(G) in terms of n and k. (b) Explain why none of the following can be the prism of any connected graph: i. A graph with 9 vertices. ii. A graph with 10 vertices and 10 edges. iii. A simple graph with 3 edges.

7. From a graph G P(G) = (Vp, Ep), is formed as follows: (V, E), with vertices v; E V and edges e; E E, the prism of the graph, Let V' be a copy of V with vertices v, and E' a copy of E with edges e'. Now V, = VUV', and E, = EU E'U {(vi, v²)\v; E V}. An example of a graph and its prism is shown below. (a) If a graph G has n vertices and k edges, find formulae for the number of vertices and the number of edges in P(G) in terms of n and k. (b) Explain why none of the following can be the prism of any connected graph: i. A graph with 9 vertices. ii. A graph with 10 vertices and 10 edges. iii. A simple graph with 3 edges.

Advanced Engineering Mathematics
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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7. From a graph G = (V, E), with vertices v; E V and edges e; E E, the prism of the graph, P(G) = (Vp, Ep), is formed as follows: Let V' be a copy of V with vertices v, and E' a copy of E with edges e. Now Vp = VUV', and E, = EU E'U {(vi, v?)|v¿ E V}. An example of a graph and its prism is shown below. (a) If a graph G has n vertices and k edges, find formulae for the number of vertices and the number of edges in P(G) in terms of n and k. (b) Explain why none of the following can be the prism of any connected graph: i. A graph with 9 vertices. ii. A graph with 10 vertices and 10 edges. iii. A simple graph with 3 edges.
Transcribed Image Text:7. From a graph G = (V, E), with vertices v; E V and edges e; E E, the prism of the graph, P(G) = (Vp, Ep), is formed as follows: Let V' be a copy of V with vertices v, and E' a copy of E with edges e. Now Vp = VUV', and E, = EU E'U {(vi, v?)|v¿ E V}. An example of a graph and its prism is shown below. (a) If a graph G has n vertices and k edges, find formulae for the number of vertices and the number of edges in P(G) in terms of n and k. (b) Explain why none of the following can be the prism of any connected graph: i. A graph with 9 vertices. ii. A graph with 10 vertices and 10 edges. iii. A simple graph with 3 edges.
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