We consider only graphs without loops or multiple edges. Pertinent definitions are given below. For notation and other definitions we generally follow Ore [1].

A connected graph G = (X, E) is said to have the property P if for every maximal tree T of G there exists a vertex aT of G such that distance between aT and x is same in T as in G for every x in X. The following problem has been posed by Ore (see [1] page 103, problem 4): Determine the graphs with property P. This paper presents a solution to the above problem in the finite case.

A (ν, κ, λ)-graph is defined in [3] as a graph on ν points, each of valency κ, and such that for any two points P and Q there are exactly λ points which are joined to both. In other words, if Si is the set of points joined to Pi, then

Si has k elements for any i

SiSj has λ elements if i≠j

The sets Si are the blocks of a (v, k, λ)-configuration, so a necessary condition on v, k, and λ that a graph should exist is that a (v, k, λ)- configuration should exist. Another necessary condition, reported by Bose (see [1]) and others, is that there should be an integer m satisfying have equal parity. We shall prove that these conditions are not sufficient.