Graphs status unique in connected graphs

Abstract

The status of a vertex in a graph is the sum of the distances between that vertex and all other vertices. The status sequence of a graph is the list of the statuses of all vertices arranged in nondecreasing order. It is well-known that nonisomorphic graphs may have the same status sequence. Let F be a family of graphs. A graph G is said to be status unique in F if G is a graph of F and G is uniquely determined in F by its status sequence. A spider is a tree in which exactly one vertex has degree exceeding two. A weakly status injective tree is a tree in which any two vertices with the same status have degree one. Previous studies have shown that both spiders and weakly status injective trees are status unique in the family of all trees. This study presents several types of graphs that are status unique in the family of all connected graphs.