Math test explorations 22

How many edges are there in a graph with vertices of degree 2, 3, 3,3,2,3?

(2+3+3+3+2+3)/2=16/2=8.

A connected graph has 27 vertices and 57 edges. How many edges must be removed to create a spanning tree?

31 edges must be removed.

For a complete graph on 6 vertices, How many degrees does each vertex have? What is the total number of edges for this graph? How many Hamilton circuits does this graph have?

5 days, 15 degrees, 5!=120

The Euler Circuit theorem says that if the degree of all the vertices in a graph are odd, then the graph has an Euler circuit.TRUE OR FALSE?

F

The Euler path theorem says that is there are two vertices of odd degree, then an Euler path exists. It must also be connected. True or False?

F

a tree with n edges have n-1 vertices

F

A graph that represents six cities and the roads among them is not connected. What does this mean in real-world terms?

If the graph that represents six cities and the roads among them is not connected, it means that it is not possible to travel among all cities. More specifically, it means at least one pair of cities has no connecting path of roads.

Why would a city street department want its snow plow operator's path to follow an Euler circuit if possible?

If the snow plow operator followed an Euler circuit, he would end his route at the city garage where he started. Also, no time or gas would be wasted traveling down streets that were already plowed.

On a graph that represents six cities and the roads between them, the valence of vertex A is 5. What does this mean in real-world terms?

If the valence of vertex A is five, it means there are five roads which lead to town A.

Give three real-world applications in which a worker would want to find an Euler circuit on a street network.

Plowing snow, Painting lines, drawing down the center of streetsTrash collection, Delivering mail to all houses on all streets

A tree with n vertices has n-1 edges, true or false?

T

It is possible that the sum of all degrees of all vertices of a graph to be an even number. true or false?

T

The nearest-neighbor algorithm uses the most expensive edge of a graph. true or False?

T

a tree must be a connected graph, true or false?

T

If there is an Euler path or circuit, where should it start and stop?

The Euler Circuit can begin anywhere and must end at the same spot it started.

trees can have at most one vertex with degree greater than 2. True or false

f