Improving the Bounds On Murty_Simon Conjecture

A graph is said to be diameter-\(k\)-critical if its diameter is \(k\) and removal of any of its edges increases its diameter. A beautiful conjecture by Murty and Simon, says that every diameter-2-critical graph of order \(n\) has at most \(\lfloor n^2/4\rfloor\) edges and equality holds only for \(K_{\lceil n/2 \rceil,\lfloor n/2 \rfloor }\). Haynes et al. proved that the conjecture is true for \(\Delta\geq 0.7n\). They also proved that for \(n>2000\), if \(\Delta \geq 0.6789n\) then the conjecture is true. We will improve this bound by showing that the conjecture is true for every \(n\) if \(\Delta\geq\ 0.676n\).