A trivalent graph with $2k$ nodes has maximum diameter $2k-1$, where the diameter is defined by the maximum distance between two nodes. So, my question is the following: 


What is the lower bound on the diameter of a trivalent graph with $2k$ nodes? 



Lower bound on diameter of trivalent graphs
If diameter equals $d$, the total number $n$ of vertices does not exceed $1+3+3\cdot 2+\dots+3\cdot 2^{d-1}=3\cdot 2^d-2$, this gives you a lower estimate on $d$ which is less or more sharp.


