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Chapter 11: Problem 6
Find all (loop-free) nonisomorphic undirected graphs with four vertices. How many of these graphs are connected?
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a) How many paths of length 5 are there in the complete bipartite graph
\(K_{3,7}\) ? (Remember that a path such as \(v_{1} \rightarrow v_{2} \rightarrow
v_{3} \rightarrow v_{4} \rightarrow v_{5} \rightarrow v_{6}\) is considered to
be the same as the path \(\left.v_{6} \rightarrow v_{5} \rightarrow v_{4}
\rightarrow v_{3} \rightarrow v_{2} \rightarrow v_{1} .\right)\)
b) How many paths of length 4 are there in \(K_{3,7}\) ?
c) Let \(m, n, p \in \mathbf{Z}^{+}\)with \(2 m
a) Find the number of edges in \(Q_{8}\). b) Find the maximum distance between pairs of vertices in \(Q_{8}\). Give an example of one such pair that achieves this distance. c) Find the length of a longest path in \(Q_{8}\).
Let \(G=(V, E)\) be a loop-free connected undirected graph with \(|V| \geq 2\). Prove that \(G\) contains two vertices \(v, w\), where \(\operatorname{deg}(v)=\operatorname{deg}(w)\)
a) If the edges of \(K_{6}\) are painted either red or blue, prove that there is a red triangle or a blue triangle that is a subgraph. b) Prove that in any group of six people there must be three who are total strangers to one another or three who are mutual friends.
a) What is the dimension of the hypercube with 524,288 edges? b) How many vertices are there for a hypercube with \(4,980,736\) edges?
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