Math3061 Geometry and topology Semester 2, 2022
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Geometry and topology
Semester 2, 2022
Assignment Topology
Math3061
1. The Dodecahedron ������ and the Dodecahedral graph ������ are
������ = , ������ =
The Euler characteristic ���(��� ) of a polyhedron ��� is defined by
���(��� ) = ��� − ��� + ��� = #vertices − #edges + #faces The Euler characteristic ���(���) of a plane graph ��� is defined by
���(���) = ��� − ��� + ��� = #vertices − #edges + #faces
where “faces” means all faces=connected components in the complement ℝ2 ⧵ ���, including the outside.
a) Compute the Euler characteristic of the polyhedron ������
b) Compute the Euler characteristic of the plane graph ������
c) Compare the results: how are they related?
2. Find a polygonal form of ��� = ���2 # #1���2 # #1��� # #0ℙ2 by drawing it
3. Identify the following surface ���, given in polygonal form, explicitly by finding numbers ���, ���, ��� with
������ = 0 such that ��� = ���2 # #��� ���2 # #������ # #���ℙ2:
��� =
Justify your answer
4. Show that the following assertions are equivalent for a graph ���:
a) ��� is a tree
b) ��� is a minimal connected graph, that is, ��� is connected and removing any edge gives a disconnected graph
2022-10-12