# Ratio of vertices to edges when airplaines can fly from 1 of 4 cities to any other of the 4 cities

This is a true of false question:

There are direct (nonstop) flights amount four cities that make it possible to get from any city to any other city by air. It follows that the beta index of the graph of cities and direct flights is at least $\frac{3}{4}$.

The book says this about beta indexes

"Geographer define the beta index of a graph as the ratio of the number of edges to the number of vertices and view this number as a measure of connectivity of the region".

The answer key said this is true. I said it's false because essentially you have a complete graph with 4 nodes for each of the cities and it has 6 edges. $\frac{6}{4} < \frac {3}{4}$ so what's the deal with the answer key? I'm getting worried because the question before it was screwy too.

• Nothing wrong with the answer key, because $\dfrac64\gt\dfrac34$. – bof Dec 7 '13 at 10:38

## 1 Answer

Only possible explanation I can think of is that you can get from any city to any other city, but not directly, i.e., you can get from $A$ to $C$ , but you may have to go thru $B$ in order to do that. Strange wording, I agree with you, but this interpretation seems to fit somehow. If you have 3 edges in your graph , where each city is a vertex, you have a tree, and a tree is path-connected, so that you can do a path between any pair of cities.

• Thanks. If you see fitting could you upvote the question so it gets more attention? – Celeritas Dec 7 '13 at 10:35
• Sure, no problem. – user99680 Dec 7 '13 at 10:36
• In your example you say "A to C , but you may have to go thru C in" do you mean to change one of the Cs to Bs? – Celeritas Dec 7 '13 at 10:36
• Yes, sorry, let me correct it. – user99680 Dec 7 '13 at 10:37