Science and Education Publishing Turkish Journal of Analysis and Number Theory 2333-1232 4 1 2016 1 15 On the Bounds of the First Reformulated Zagreb Index 8 15 EN T. Mansour M. A. Rostami E. Suresh Department of Mathematics, Velammal Engineering College, Surapet, Chennai-66, Tamil Nadu, India G. B. A. Xavier TJANT2016412 10.12691/tjant-4-1-2 2015 9 2 2015 11 7 2016 1 13 The edge version of traditional first Zagreb index is known as first reformulated Zagreb index. In this paper, we analyze and compare various lower and upper bounds for the first reformulated Zagreb index and we propose new lower and upper bounds which are stronger than the existing and recent results [Appl. Math. Comp. 273 (2016) 16-20]. In addition, we prove that our bounds are superior in comparison with the other existing bounds.