MHB Determine the diameter of the graph

[mathmari]

Hey!

We have the matrix $$A=\begin{pmatrix}0&1&0&0&0&0&0&1\\1&0&1&0&0&0&0&0\\0&1&0&1&0&0&0&0\\0&0&1&0&1&0&0&0\\0&0&0&1&0&1&0&0\\0&0&0&0&1&0&1&0\\0&0&0&0&0&1&0&1\\1&0&0&0&0&0&1&0\end{pmatrix}$$
and we want to determine the diameter of the graph.

For that do we have to calculate the eigenvalues? Or isn't the diameter related to the number of eigenvalues? (Wondering)

 

[I like Serena]

Hey!

We have the matrix $$A=\begin{pmatrix}0&1&0&0&0&0&0&1\\1&0&1&0&0&0&0&0\\0&1&0&1&0&0&0&0\\0&0&1&0&1&0&0&0\\0&0&0&1&0&1&0&0\\0&0&0&0&1&0&1&0\\0&0&0&0&0&1&0&1\\1&0&0&0&0&0&1&0\end{pmatrix}$$
and we want to determine the diameter of the graph.

For that do we have to calculate the eigenvalues? Or isn't the diameter related to the number of eigenvalues?

Hey mathmari!

I'm not aware (yet) of a relationship between eigenvalues and diameter.
But to find the diameter, we can draw the graph, and figure out its diameter from the graph can't we? (Wondering)

 

[mathmari]

I'm not aware (yet) of a relationship between eigenvalues and diameter.
But to find the diameter, we can draw the graph, and figure out its diameter from the graph can't we? (Wondering)

Do we have the following graph?

View attachment 8334

(Wondering)

 

[I like Serena]

Do we have the following graph?

Yep. (Nod)

 

[mathmari]

Yep. (Nod)

Graph's diameter is the largest number of vertices which must be traversed in order to travel from one vertex to another, right? Is the largest number of vertices equal to 8? (Wondering)

 

[I like Serena]

Graph's diameter is the largest number of vertices which must be traversed in order to travel from one vertex to another, right?

Not vertices but edges.
It's the largest possible 'distance' between 2 vertices. (Nerd)

Is the largest number of vertices equal to 8?

I don't think so.
We have 8 vertices in total that are effectively arranged in a circle.
If we pick 2 of them, what is the distance (the minimum number of edges) between them? (Wondering)

 

[mathmari]

We have 8 vertices in total that are effectively arranged in a circle.
If we pick 2 of them, what is the distance (the minimum number of edges) between them? (Wondering)

The minimum numbers of edges between two vertices is 2, right? Is this the diameter? (Wondering)

 

[I like Serena]

The minimum numbers of edges between two vertices is 2, right? Is this the diameter?

Nope.
The distance between 2 vertices is the minimum number of edges between those vertices.
And the diameter is the maximum distance that we can find.

If we pick 2 neighboring edges (one edge between them), the distance is 1.
But we can also pick vertices that are further apart can't we? (Thinking)

 

[mathmari]

Nope.
The distance between 2 vertices is the minimum number of edges between those vertices.
And the diameter is the maximum distance that we can find.

If we pick 2 neighboring edges (one edge between them), the distance is 1.
But we can also pick vertices that are further apart can't we? (Thinking)

The maximum distance between two vertices is 4, or not? (Wondering)

 

[I like Serena]

The maximum distance between two vertices is 4, or not?

Yes it is. (Nod)

 

[mathmari]

Yes it is. (Nod)

Ok! Thank you! (Smile)