A new bound of radius with irregularity index

In this paper, we use a technique introduced in the paper [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 Erdös et al. [P. Erdös, J. Pach, R. Pollack, Z. Tuza, Radius, diameter, and minimum degree, J. Combin. Theory B 47 (1989), 73-79] on radius and minimum degree. To be more detailed, we will prove that if G is a connected graph of order n with the minimum degree δ, then the radius G does not exceed32n-t+1δ+1+1,where t is the irregularity index (that is the number of distinct terms of the degree sequence of G) which has been recently defined in the paper [S. Mukwembi, A note on diameter and the degree sequence of a graph, Appl. Math. Lett. 25 (2012), 175-178]. We claim that our result represent the tightest bound that ever been obtained until now.