Chapter 10: Q3RE (page 641)
What is the relationship between the sum of the degreesof the vertices in an undirected graph and the number ofedges in this graph? Explain why this relationship holds.
\[2m = \sum\limits_{v \in V} {\deg (v)} \]
Hence, the sum of the degrees must be even, since m is an integer.
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Find the chromatic number of the given graph.
The distance between two distinct vertices\({{\bf{v}}_{\bf{1}}}\)and\({{\bf{v}}_{\bf{2}}}\)of a connected simple graph is the length (number of edges) of the shortest path between\({{\bf{v}}_{\bf{1}}}\)and\({{\bf{v}}_{\bf{2}}}\). The radius of a graph is the minimum over all vertices\({\bf{v}}\)of the maximum distance from\({\bf{v}}\)to another vertex. The diameter of a graph is the maximum distance between two distinct vertices. Find the radius and diameter of
a)\({{\bf{K}}_{\bf{6}}}\)
b)\({{\bf{K}}_{{\bf{4,5}}}}\)
c)\({{\bf{Q}}_{\bf{3}}}\)
d)\({{\bf{C}}_{\bf{6}}}\)
Determine whether the given simple graph is orientable.
A tournament is a simple directed graph such that if u and v are distinct vertices in the graph, exactly one of (u, v) and (v, u) is an edge of the graph.Show that every tournament has a Hamilton path.
Because traffic is growing heavy in the central part of a city, traffic engineers are planning to change all the streets, which are currently two-way, into one-way streets. Explain how to model this problem
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