Exam 15: Graph Theory

Solve the problem. -A computer network is to be built by connecting two "server" computers to 6 "client" computers. Each client computer is connected to each server by a single cable. Also, the two servers are connected by a single cable. Draw a graph with vertices representing computers and edges representing cables. How many cables are required? Is the graph of the network a complete graph?

Determine whether the graph is connected or disconnected. -

Determine how many components the graph has. -

Solve the problem. -A psychology class decides to compare the social attitudes of students from urban and rural backgrounds. Class members are divided into two groups: 5 students who grew up in an urban setting and 3 students who grew up in a rural environment. Each member of a group is to meet with each member of the other group one time, at which time the pair will interview each other by asking questions about current social issues. After all the interviews are completed, the students will compile the results and draw conclusions. Draw a graph with vertices representing students and edges representing interviews. How many interviews will there be?

Determine how many vertices and how many edges the graph has. -

Use the theorem that relates the sum of degrees to the number of edges to determine the number of edges in the graph. -A graph with 5 vertices, one of degree 4, three of degree 3, and one of degree 1.

Determine whether the two graphs are isomorphic. If they are, illustrate the isomorphism. -Yes (a) (b)

Determine whether the graph is a complete graph. -

Determine how many components the graph has. -

Decide if an Euler circuit exists for the graph. -

Determine whether the two graphs are isomorphic. If they are, illustrate the isomorphism. -(a) (b)

Determine how many components the graph has. -

Identify the cut edges in the graph or say there are none. -

Determine how many vertices and how many edges the graph has. -

Determine whether the graph is connected or disconnected. -

Represent the following with a graph. -

Solve the problem. -Six freshmen move into a dormitory suite at the beginning of the semester. In order to get to know one another, they decide to have breakfast in pairs for the next few days, each student having a different breakfast partner each day until each student has had breakfast with all other students exactly once. Draw a graph with vertices representing students and edges representing breakfast dates. How many breakfasts will occur? How many days will the process take?

Determine how many components the graph has. -

Determine whether the sequence of vertices is i)a walk, ii)a path, iii)a circuit in the given graph. - $\mathrm { L } \rightarrow \mathrm { Y } \rightarrow \mathrm { K } \rightarrow \mathrm { L } \rightarrow \mathrm { J } \rightarrow \mathrm { I }$

There are 9 members on a baseball team. At the end of the game, each member of the team shakes hands with each member of the opposing team. How many handshakes occur?