I am trying to find the radius and diameter of a regular graph $G$ with $d(v_i) < (n-1)/2$. I know for $d(v) \geq (n-1)/2$, $\rm{diam}(G) \leq 2$ and $\rm{radius}(G)=\rm{diam}(G).$ If we are not taking the complete graphs.
I am unable to do the same for the regular graphs of valence smaller than $(n-1)/2$.
Are there any regular graphs $G$ for which the diameter and radius are equal and have valence smaller than $(n-1)/2$. ? Can anybody help me. Any hint or suggestion to proceed? Thanks for your help.
Best Answer
There are such graphs.
I'll give you a hint.
Hint 1. Which graph is the first one you should think of when trying to find a counterexample or disprove something?
Hint 2. 10 vertices, 3-regular.