A note on diameter and the degree sequence of a graph

In this note, we use a technique introduced by Dankelmann and Entringer [P. Dankelmann, R.C. Entringer, Average distance, minimum degree and spanning trees, J. Graph Theory 33 (2000) 1–13] to obtain a strengthening of an old classical theorem by Erdős, Pach, Pollack and Tuza [P. Erdős, J. Pach, R. Pollack, Z. Tuza, Radius, diameter, and minimum degree, J. Combin. Theory B 47 (1989) 73–79] on diameter and minimum degree. To be precise, we will prove that if G is a connected graph of order n and minimum degree δ , then its diameter does not exceed 3 ( n − t ) δ + 1 + O ( 1 ) , where t is the number of distinct terms of the degree sequence of G . The featured parameter, t , is attractive in nature and promising; more discoveries on it in relation to other graph parameters are envisaged.