Abstract
The multiplicative-sum Zagreb index is a graph invariant defined as the product of the sums of the degrees of pairs of adjacent vertices in a graph. In this paper, we compute the multiplicative-sum Zagreb index of some composite graphs such as splice graphs, bridge graphs, and bridge-cycle graphs in terms of the multiplicative-sum Zagreb indices of their components. Then, we apply our results to compute the multiplicative-sum Zagreb index for several classes of chemical graphs and nanostructures.
Highlights
The multiplicative versions of Zagreb indices were introduced by Todeschini and Consonni [20] in 2010
The multiplicative sum Zagreb index of G is denoted by Π∗1(G) and defined as
We present exact formulae for computing the multiplicative sum Zagreb index of some other composite graphs such as splice graphs, bridge graphs, and bridge-cycle graphs
Summary
We recall the definitions of splice, bridge, and bridge-cycle graphs and state some preliminary results about these graphs. The bridge-cycle graph BC = BC(G1, G2, ..., Gd; v1, v2, ..., vd), d ≥ 3, of {Gi}di=1 with respect to the vertices {vi}di=1 is obtained by connecting the vertices v1 and vd by an edge in the bridge graph B = B(G1, G2, ..., Gd; v1, v2, ..., vd). The degree of an arbitrary vertex in splice, bridge, and bridge-cycle graphs are computed. (i) The degree of a vertex u in the splice graph S = (G1.G2)(v1, v2) is given by dS (u) =. (ii) The degree of a vertex u in the bridge graph B = B(G1, G2, ..., Gd; v1, v2, ..., vd) is given by dGi (u) dB(u) = δi + 1. (iii) The degree of a vertex u in the bridge-cycle graph BC = BC(G1, G2, ..., Gd; v1, v2, ..., vd) is given by dBC (u) =. We refer the reader to [1,2,3,4,9] for more information on computing topological indices of splice and link graphs
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