Consider a tree with k nodes and for each node v the vector lv = (lv0, lv1, ..., lvk-1) with lvd the number of leaves (!) with distance d to v. I wonder whether two nodes v, w with lv = lw are conjugate (I guess they are). Can anyone help me to prove this - or give a counter example?
I have a counterexample. It is not enough just to count leaves, since this doesn't take into account the number of possible ways to arrive at those leaves.
Consider the graph below.
A - B - C - D - E - F | | G H | I
I think the vector for C and D both is 002200000, since they each have two leaves at distance 2 and two leaves at distance 3. But they are not conjugate, since C has degree 3 and D has degree 2.
I think this might be a minimal counterexample.