Abstract
hefirst Zagreb indexM1and the second Zagreb indexM2belongto the class of degree-based topological indices which are defined fora simple connected graphGwith vertex setV={υ1,υ2,···,υn}asM1(G)=Pn ̇ı=1d2 ̇ıandM2(G)=Pυ ̇ı∼υ ̇jd ̇ıd ̇j,whered ̇ıis the degreeof vertexυiandυ ̇ı∼υ ̇jrepresents the adjacency of verticesυ ̇ıandυ ̇jinG. The eccentric connectivity index (ECI) is a distance basedtopological index, denoted byξc,isdefined asξc(G)=Pni=1ε ̇ıd ̇ı,whereε ̇ıis the eccentricity ofυ ̇ıinG. The aim of this paper is toderive the inequalities between ECI and the Zagreb indices. Moreover,we establish the inequalities between some variants of ECI and theZagreb indices
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