# Graph: showing that diameter is greater than average pairwise distance

• MHB
• lemonthree
In summary, the conversation discusses the need to prove a statement and finding a case where the graph diameter is greater than the average pairwise distance. The participants also discuss the formula for average pairwise distance and how it relates to the maximum number of edges in a connected graph. They then attempt to find the maximum value for apd(G) and discuss the distances between vertices in a graph. They also make corrections to their calculations and clarify their approach.
lemonthree

I need to prove the above statement. I have a very strong gut feeling that the above equation is not true, and so I need to find a case where the graph diameter is greater than the average pairwise distance.

First off, I would like to clarify about the average pairwise distance, which is given below

Given that the denominator is C(n,2), I am assuming that the average pairwise distance will be taking the maximum number of edges? So in this case, the connected graph has edges connecting every single vertex to each other, always?
But how could this be? What if there was some vertex, $v_{1}$ and $v_{2}$ that is not connected?

For now I can only say that in a connected graph there is a finite distance between any two vertices, but not every pair of vertices is adjacent.

Ah, that makes sense. Any 2 vertices will definitely have a path connecting them, just that they do not have to be "directly" connected to each other?

Hi, I have attempted and I am stuck at this part.

Fix $k$. Consider G, with a path $v_{1},v_{2},...v_{k-1}$. Attach each node $u_{1},u_{2},...u_{n-(k-1)}$to $v_{1}$. This means we have attached $(n-k+1)$nodes to $v_{1}$.

$diam(G) = d(u_{1},v_{k-1}) = k-1$.

So the graph looks like this:

And now we need to find the maximum value for apd(G). How can we find the sum of the distance of all the vertices? I know for sure that there will definitely be an element from $v_{1},v_{2},...v_{k-1}$ since the distance must be at least 1.

Let me edit that! I feel like I could be using better values instead.

lemonthree said:
Hi, I have attempted and I am stuck at this part.

Fix $k$ Consider G, with a path $v_{1},v_{2},...v_{k}$ Attach each node $u_{1},u_{2},...u_{n-k}$ to $v_{1}$. This means we have attached $(n-k)$ nodes to $v_{1}$.

$diam(G) = d(u_{1},v_{k-1}) = k$.

So the graph looks like this:

And now we need to find the maximum value for apd(G). How can we find the sum of the distance of all the vertices? I know for sure that there will definitely be an element from $v_{1},v_{2},...v_{k}$ since the distance must be at least 1.

I changed the $(k-1)$ to $k$ instead, in turn changing $diam(G)$ to $k$ as I think it would make things easier. Let me attempt to find the maximum value of $apd(G)$, please correct me if I'm wrong.

We want the sum of all the distance between u and v, for all u, v that exists in G. So there are $C(n,2)$ pairs. Each of these pairs has a distance of at most $k$. Therefore, the sum is $C(n,2) * k$

By the $apd(G)$ formula, $\frac{C(n,2) * k}{C(n,2)} = k$

Therefore, we have that $\frac{diam(G)}{apd(G)} > 1$

## What is a graph?

A graph is a visual representation of data that shows the relationship between different variables or points. It typically consists of points or nodes connected by lines or edges.

## How is diameter defined in a graph?

The diameter of a graph is the longest distance between any two nodes in the graph. It is a measure of the maximum distance between any two points in the graph.

## What is average pairwise distance?

The average pairwise distance is the average distance between all pairs of nodes in a graph. It is calculated by summing up all the distances between pairs of nodes and dividing by the total number of pairs.

## Why is it important to show that diameter is greater than average pairwise distance?

Showing that the diameter is greater than the average pairwise distance in a graph can provide important insights into the structure and connectivity of the data. It can also help identify any outliers or nodes that are significantly far apart from the rest of the graph.

## How is this relationship typically depicted in a graph?

This relationship is usually depicted by plotting the diameter and average pairwise distance on a graph, with the diameter on the y-axis and the average pairwise distance on the x-axis. The graph will show a clear distinction between the two measures, with the diameter being greater than the average pairwise distance.

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