MHB What Are Key Strategies for Solving Discrete Math Graph Theory Problems?

AI Thread Summary
Key strategies for solving discrete math graph theory problems include understanding the concepts of graph complements, isomorphic graphs, and self-complementary graphs. It is essential to calculate the number of unordered pairs of vertices in a graph and apply this to determine the edges in the complement of a graph. Drawing graphs to visualize relationships and ensuring that transformations maintain isomorphism are crucial steps in problem-solving. Additionally, constructing explicit bijections can help prove properties like self-complementarity in specific graph configurations. Engaging with each problem individually and demonstrating effort in understanding the underlying principles will enhance problem-solving skills.
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First, I'd like to point out http://www.mathhelpboards.com/misc.php?do=vsarules #8: "Do not ask more than two questions in a post" and rule #11: "Show some effort". I recommend starting separate threads for problems 2–5 and describing what you have done and/or what difficulty you are having.

I'll give some hints about problem 1. For A, answer the following questions.

  1. How many (unordered) pairs of vertices does a graph with $n$ vertices have? Let's denote this number by $C(n)$.
  2. If a graph $G$ has $e$ edges, how many edges does the complement of $G$ have? Make sure you know what a graph complement is. The answer should use $C(n)$.
  3. What can be said about the number of edges in isomorphic graphs?
  4. Suppose a self-complementary graph with $n$ vertices has $e$ edges. Write an equation on $e$ using the definition of a self-complementary graph and points 2 and 3 above.

For B, draw a graph connecting each pair of vertices using either solid or dashed line. The number of edges of each type should be what you found in point A. The solid graph should be isomorphic to the dashed one. Put 4 vertices in the corners of a square. Make it so that rotating the picture by 90 degrees maps solid edges into dashed ones and vice versa. For 5 vertices, prove that a regular pentagon is self-complementary (construct an explicit bijection between vertices and check that it is an isomorphism).

For C, use the fact that $C(n)$ is an integer.
 
I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...

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