Question

Prove that the number χ(G, n) of valid n-colorings of a multigraphs satisfies the formula χ(G, n) = χ(G − e, n) − χ(G/e, n). Explain the meaning of this formula when there are several edges connecting the endpoints of the edge e.

Answer #1

Use induction to prove that every graph G = (V, E) satisfies
χ(G) ≤ ∆(G).

Prove the following bound for the independence number.
If G is a n-vertex graph with e edges and maximum degree ∆ >
0, then
α(G) ≤ n − e/∆.

Use the combinations formula to prove that in general C(n, r) =
C(n, n - r).
Use the permutations formula to evaluate for any whole number n,
P(n, 0). Explain the
meaning of your result.
Use the combinations formula and the definition of 0! to
evaluate, for
any whole number n, C(n, 0). Explain the meaning of your
result.
Suppose you have 35 songs for a playlist consisting of only 5
songs. How many different playlists can you have?

Let G be a connected simple graph with n vertices and m edges.
Prove that G contains at least m−n+ 1 different subgraphs
which are polygons (=circuits). Note: Different polygons
can have edges in common. For instance, a square with a diagonal
edge has three different polygons (the square and two different
triangles) even though every pair of polygons have at least one
edge in common.

Consider a minimum spanning tree for a weighted graph G= (V,
E)and a new edge e, connecting two existing nodes in V. Explain how
to find a minimum spanning tree of the new graph in O(n)time, where
n is the number of nodes in the graph. Prove correctness of the
algorithm and justify the running time

Work out and prove by induction on n a formula for (fg)^(n), the
n-th derivative of the product of two functions f,g
pls
explain each step

Let G be a graph or order n with independence number α(G) =
2.
(a) Prove that if G is disconnected, then G contains K⌈ n/2 ⌉ as
a subgraph.
(b) Prove that if G is connected, then G contains a path (u, v,
w) such that uw /∈ E(G) and every vertex in G − {u, v, w} is
adjacent to either u or w (or both).

Let G = (V,E) be a graph with n vertices and e edges. Show that
the following statements are equivalent:
1. G is a tree
2. G is connected and n = e + 1
3. G has no cycles and n = e + 1
4. If u and v are vertices in G, then there exists a unique path
connecting u and v.

prove that if G is a cyclic group of order n, then for
all a in G, a^n=e.

Find a formula for the number of digits of 2^n.
Now the textbook answer is '1+|_n*lg2_|', the symbol used is
integer floor, and lg2 is log(10)2.
Question: How do I find this formula?? Show me the process of
finding this formula. I know that when n=3, the number of digit is
1; when n goes pass 3, there will be 2 digits; when n goes pass 6,
there will be 3 digits, and so on.

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