Let C41 be the graph with vertices {0, 1, ..., 40} and edges
(0-1), (1-2),..., (39-40), (40-0),
and let K41 be the complete graph on the same set of 41 vertices.
You may answer the following questions with formulas involving exponents, binomial coefficients, and factorials.
(a) How many edges are there in K41?
(b) How many isomorphisms are there from K41 to K4
(c) How many isomorphisms are there from C41 to C41?
(d) What is the chromatic number x(K41)?
(e) What is the chromatic number x(C41)?
(f) How many edges are there in a spanning tree of K41?
(g) A graph is created by adding a single edge between nonadjacent vertices of a tree with 41 vertices. What is the largest number of cycles the graph might have?
(h) What is the smallest number of leaves possible in a spanning tree of K41?
(i) What is the largest number of leaves possible in a in a spanning tree of K41?
(j) How many spanning trees does C41 have?
k) How many spanning trees does K41 have?
(1) How many length-10 paths are there in K41?
(m) How many length-10 cycles are there in K41?