On the Number of Resolving Pairs in Graphs

Abstract

Let $G$ be a connected graph. An unordered pair $\{x,y\}$ of vertices of $G$ is a resolving pair if no vertex of $G$ has the same distance to $x$ and $y$. It was conjectured in [8] that the number of resolving pairs of a connected graph of order $n$ is bounded above by $\lfloor \frac{n^2}{4}\rfloor$. In this note we show that there exists a constant $c>0$ such that for sufficiently large $n$ every graph of order $n$ contains at most ${n \atopwithdelims ()2} - cn^{3/2}$ resolving pairs. We further show that the exponent $3/2$ is best possible by exhibiting an infinite sequence of graphs with at least ${n \atopwithdelims ()2}-c n^{3/2} \sqrt{\log n}$ resolving pairs, where $c$ is a positive constant, thus disproving the above conjecture.