On a conjecture involving the spectral radius and the second Zagreb index of a graph

For a given graph G with n vertices and m edges, let M2 denote the second Zagreb index of G and 1 its spectral radius.We show that the inequality M 2 m 2 2 1 is not valid in general case, but is satisfied by trees and unicyclic graphs, settling thus an old conjecture by one of the present authors.We also pose some related problems for future research.