Pairs of a tree and a nontree graph with the same status sequence

Abstract

The status of a vertex x in a graph is the sum of the distances between x and all other vertices. Let G be a connected graph. The status sequence of G is the list of the statuses of all vertices arranged in nondecreasing order. G is called status injective if all the statuses of its vertices are distinct. Let G be a member of a family of graphs ℱ and let the status sequence of G be s.G is said to be status unique in ℱ if G is the unique graph in ℱ whose status sequence is s. In 2011, J.L. Shang and C. Lin posed the following two conjectures. Conjecture 1: A tree and a nontree graph cannot have the same status sequence. Conjecture 2: Any status injective tree is status unique in all connected graphs. We settle these two conjectures negatively. For every integer n≥10, we construct a tree Tn and a unicyclic graph Un, both of order n, with the following two properties:(1) Tn and Un have the same status sequence; (2) for n≥15, if n is congruent to 3 modulo 4 then Tn is status injective and among any four consecutive even orders, there is at least one order n such that Tn is status injective.