On Constructing Graphs with the Same Status Sequence

Abstract

The status of a vertex \(v\) in a graph is the sum of the distances between \(v\) and all 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 non-isomorphic graphs may have the same status sequence. This paper gives a sufficient condition for a graph \(G\) with the property that there exists another graph \(G’\) such that \(G’\) and \(G\) have the same status sequence and \(G’\) is not isomorphic to \(G\).