More On The Zagreb Indices Inequality

The Zagreb indices are very popular topological indices in mathematical chemistry and attracted a lot of attention in recent years. The first and second Zagreb indices of a graph G = (V, E) are defined as M-1(G) = Sigma V-v.is an element of d(i)(2) and M-2(G) = Sigma(vi similar to vj) (d(i)d(j)), where d(i) denotes the degree of a vertex v(i) and v(i) similar to v(j) represents the adjacency of vertices v(i) and v(j) in G. It has been conjectured that M1/n = M2/m holds for a connected graph G with n = |V | and m = |E|. Later, it is proved that this inequality holds for some classes of graphs but does not hold in general. This inequality is proved to be true for graphs with d(i) is an element of [h, h +left perpendicular root hright perpendicular] or d(i) is an element of [h, h + z], where h >= z(z- 1)/2. In this paper, we prove that the graphs satisfy the inequality if the sequences (d(i)) and (S-i) have the similar monotonicity, where S.i = P....N(..i) d.. and N(v(i)) = {Vj |..i}. As a consequence, we present an infinite family of connected graphs with d.i. [1,8), for which the inequality holds. Moreover,