**Abstract :**Let [latex]G[/latex] be a simple connected molecular graph with vertex set [latex]V(G)[/latex] and edge set [latex]E(G)[/latex]. One important modification of classical Zagreb index, called hyper Zagreb index [latex]HM(G)[/latex] is defined as the sum of squares of the degree sum of the adjacent vertices, that is, sum of the terms [latex]{[{{d}_{G}}(u)+{{d}_{G}}(v)]^2}[/latex] over all the edges of [latex]G[/latex], where [latex]d_G(v)[/latex] denote the degree of the vertex [latex]u[/latex] of [latex]G[/latex]. In this paper, the hyper Zagreb index of certain bridge and chain graphs are computed and hence using the derived results we compute the hyper Zagreb index of several classes of chemical graphs and nanostructures.