(a) Draw a simple graph (no loops or multiple edges) with vertex degrees 3,3,3,3,4,4,4,6. (b) How many edges does its complement have? Explain.

Chess is a board game, where the board is made up of 64 squares arranged in an 8-by-8 grid. One of the pieces is a rook, which can move from its current square any number of spaces either vertically or horizontally (but not diagonally) in a single turn. Discuss how you could use graphs to show that a rook can get from its current square to any other square on the board in at most two turns. You’re encouraged to utilize relevant graph definitions, problems, and algorithms where appropriate.

Explain thoroughly!!

Construct a graph with vertices J, K, L, M, N whose degrees What is the edge set? "

In a high school class, there are students of different talents (ex. music, art, debate, sports, etc.). A committee needs to be formed consisting of the maximum number of students such that the committee is diverse – no two students with the same talent are in the committee. Transform this problem to a graph theoretic problem. What are the vertices of the graph? What are the edges of the graph? What do you have to find in the graph?

What is the minimum number of colors you need to color the vertices of this graph such that if two vertices are adjacent they receive different colors?