Problem Set I (Geometric Structures - Spring 2011)

Prepared by:

Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451

email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk

1. (i) Determine if there is a graph which is connected and has no loops or multiple edges for the valence (degree) sequences below and if such a graph exists, draw a diagram of the graph:

a. 5, 4, 4, 3, 3, 3, 2, 1, 1

b. 6, 4, 4, 3, 3, 3, 3, 2, 1, 1

c. 7, 5, 4, 4, 2

(If no such graph is possible, give a brief explanation of why the graph does not exist.)

ii. For each of the valence sequences for which a graph does exist try to draw a diagram of another graph which is not isomorphic to the one you originally drew.

iii. If the graph you drew in (i) is not a plane graph, draw a plane version of the graph you found, if possible. For the plane graphs you draw write down the number of sides of each face in the graph.

2. A graph G has 10 edges and all of its vertices have the same valence.

a. Determine all of the possible valences that the vertices of G can have if G has no-self loops. (Draw diagrams of some typical graphs that achieve the valences you find.)

b. Determine all of the possible valences that the vertices of G can have if G has no multiple edges. (Draw diagrams of some typical graphs that achieve the valences you find.)

c. Determine all of the possible valences that the vertices of G can have if G has no-self loops or multiple edges. (Draw diagrams of some typical graphs that achieve the valences you find.)

d. Determine all of the possible valences that the vertices of G can have if G is connected and has no-self loops. (Draw diagrams of some typical graphs that achieve the valences you find.)

e. Determine all of the possible valences that the vertices of G can have if G is connected and has no multiple edges. (Draw diagrams of some typical graphs that achieve the valences you find.)

f. Determine all of the possible valences that the vertices of G can have if G is connected and has no-self loops or multiple edges. (Draw diagrams of some typical graphs that achieve the valences you find.)

3. The diagram below shows a drawing of a polyhedron with 6 faces. The faces consist of 6 congruent isosceles triangles: ABD, ADC, BDC, ACE, BCE, and AEB. The edges AB, BC, and AC have length 1, and AD, BD, CD, AE, BE, and CE have length 2.

Figure 1

Note that the edges AC, AB, BC do NOT form a face of this polyhedron.

a. Draw at least 5 different nets of this polyhedron and for each of the nets specify which spanning tree's edges in Figure 1 give rise to this net.

(Two nets are considered different if they are not congruent as polygons.)

b. Can you find a complete collection of nets for this polyhedron?

4. Draw a diagram of those points in the taxicab plane the sum of whose distances from (-1, 0) and (1, 0) is 8.

The diagram you get is a taxicab geometry analogue of a well known family of curves in the Euclidean plane.