Relations : Warshall's Algorithm, Digraphs and Connectivity

Please see the attached file for the fully formatted problems. Let A = {a. b, c, d} and let the relation R be defined on A by the 0 0 1 1 matrix MR = 0 1 0 0 Note, take the nodes in A in the order given. 0 0 1 0 1 0 0 0 (a) Use Warshall’ s Algorithm (Section 7.4 of the text) to determine the transitive closure of R. (b) Draw the digraph of the transitive closure of R and use the digraph to explain the idea of connectivity. Is this graph connected? What does this mean?

Spanning Trees and Graphs

Does every graph have a spanning tree? If not, then can you tell from the number of nodes and the number of edges a graph has whether it has a spanning tree, or do you need more information?

Spanning Tree Graph : Movie Collaboration (Kevin Bacon Game)

If you were required by a professor to find a spanning tree of the movie collaboration graph (where each node corresponds to an actor with finite Kevin Bacon number, and two nodes are connected by an edge if the corresponding actors have been in a movie together), how would you do it? Why would you choose your method over other possible methods?

Tile a plane with n-gons.

Is it possible to tile a plane with (a) regular 5-gons and regular 6-gons? (B) regular 5-gons, regular 6-gons, and triangles? (c) regular 5-gons, regular 6-gons, and regular triangles?

Relations : Properties and Equivalence Classes

Please see the attached file for the fully formatted problem. Exercise 5 (4p) R is the relation defined on Z ts follows: for all m,n E Z, m R n <=>4|(m-n) a. Determine whether the relaition is reflexive. b. Determine whether the relation is symmetric. c. Determine whether the relation is transitive. d. In case the relation is an equivalence relation, describe the distinct equivalence classes.