1. ## Graph Theory Problem

Is it possible to have a geodesic whose length is diam (G) for a graph G or is this a contradiction? A u-v geodesic is a u-v path of length d(u,v), but diam(G) is the greatest distance between any vertices of a connected graph G....So is it possible for me to have a geodesic with length diam(G)?

More specifically the problem gives me a picture of a graph and asks me to find an example of a geodesic with length diam (G) or explain why none exist. The graph is like

r-s-t
| | |
u-v-w
| | |
x-y-z

Those are the 9 vertices and the lines are edges. Also, there is an edge between r and v and between t and v.

Thank you.

2. Originally Posted by zhupolongjoe
Is it possible to have a geodesic whose length is diam (G) for a graph G or is this a contradiction? A u-v geodesic is a u-v path of length d(u,v), but diam(G) is the greatest distance between any vertices of a connected graph G....So is it possible for me to have a geodesic with length diam(G)?

More specifically the problem gives me a picture of a graph and asks me to find an example of a geodesic with length diam (G) or explain why none exist. The graph is like

r-s-t
| | |
u-v-w
| | |
x-y-z

Those are the 9 vertices and the lines are edges. Also, there is an edge between r and v and between t and v.

Thank you.
What is diam(G) for this graph? Can you find two points whose shortest path between them is equal to that?

3. Well diam(G) here would be 4. Because of the diagonals on the top, I can not find any two vertices like that. Each pair of vertices can be attained in a path of 3 or less. So does that mean that such does not exist?