Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let $$G$$G be a connected graph. An unordered pair $$\{x,y\}$${x,y} of vertices of $$G$$G is a resolving pair if no vertex of $$G$$G has the same distance to $$x$$x and $$y$$y. It was conjectured in [8] that the number of resolving pairs of a connected graph of order $$n$$n is bounded above by $$\lfloor \frac{n^2}{4}\rfloor $$?n24?. In this note  show that there exists a constant $$c>0$$c>0 such that for sufficiently large $$n$$n every graph of order $$n$$n contains at most $${n \atopwithdelims ()2} - cn^{3/2}$$n2-cn3/2 resolving pairs.  further show that the exponent $$3/2$$3/2 is best possible by exhibiting an infinite sequence of graphs with at least $${n \atopwithdelims ()2}-c n^{3/2} \sqrt{\log n}$$n2-cn3/2logn resolving pairs, where $$c$$c is a positive constant, thus disproving the above conjecture.
['Peter Dankelmann']
I am writing a research paper on a idea On the Number of Resolving Pairs in Graphs
[{"from": "human", "value": "I am writing a research paper on a idea On the Number of Resolving Pairs in Graphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let $$G$$G be a connected graph. An unordered pair $$\\{x,y\\}$${x,y} of vertices of $$G$$G is a resolving pair if no vertex of $$G$$G has the same distance to $$x$$x and $$y$$y. It was conjectured in [8] that the number of resolving pairs of a connected graph of order $$n$$n is bounded above by $$\\lfloor \\frac{n^2}{4}\\rfloor $$?n24?. In this note  show that there exists a constant $$c>0$$c>0 such that for sufficiently large $$n$$n every graph of order $$n$$n contains at most $${n \\atopwithdelims ()2} - cn^{3/2}$$n2-cn3/2 resolving pairs.  further show that the exponent $$3/2$$3/2 is best possible by exhibiting an infinite sequence of graphs with at least $${n \\atopwithdelims ()2}-c n^{3/2} \\sqrt{\\log n}$$n2-cn3/2logn resolving pairs, where $$c$$c is a positive constant, thus disproving the above conjecture."}]