14.5: Activities
Suppose \(G = (V,E)\) is a graph. Decide the truth of the following statement.
Every pair of a subset \(V' \subseteq V\) and a subcollection \(E' \subseteq E\) defines a subgraph \(G' = (V',E')\) of \(G\text{.}\)
Draw a graph where the nodes are students present in today's class. Draw edges between pairs of students that are in the same group today. Additionally, draw an edge between a member of your group and another student if that pair was in a group together last class.
For each of the following graphs, write out its formal definition as either a (regular) graph, a weigthed graph, or a directed graph, as appropriate.
Consider the website Facebook as a graph where vertices are profiles and edges represent “friendship”.
- Should this graph be a directed graph? Why or why not?
- Is this graph simple? complete? Justify your answers.
- What does the degree of a vertex represent?
- Could this graph have isolated vertices?
- Suppose the following graph is a subgraph of the Facebook graph.
- What is the largest party one of these people could throw where each party-goer is Facebook friends with every other party-goer? Justify your answer.
- Assume all of the people in this graph live in the same geographic area. Which pair of non-friends are most likely to become friends in the future? Which pair of non-friends are least likely to become friends in the future? Justify your answers.