Abstract
Multiplicative Zagreb indices have been studied due to their extensive applications. They play a substantial role in chemistry, pharmaceutical sciences, materials science and engineering, because we can correlate them with numerous physico-chemical properties of molecules. We use graph theory to characterize these chemical structures. The vertices of graphs represent the atoms of a compound and edges of graphs represent the chemical bonds. We present upper and lower bounds on the general multiplicative Zagreb indices for graphs with given number of vertices and cut-edges called bridges. We give all the extremal graphs, which implies that our bounds are best possible.
Highlights
We consider connected graphs without loops and multiple edges
PRELIMINARY RESULTS First, we show that by adding an edge to a graph G, we get a graph with larger general multiplicative Zagreb indices
Note that there is no graph with n − 2 bridges, since every tree has n − 1 bridges and every graph with a cycle has at most n−3 bridges
Summary
We consider connected graphs without loops and multiple edges. Let G be a graph with vertex set V (G) and edge set E(G). We obtain upper and lower bounds on the general multiplicative Zagreb indices for graphs with given number of vertices and bridges. Proof: We show that if G has the smallest Pa1 index (the largest Pa2 index), where a > 0, G contains at most one vertex adjacent to pendant vertices.
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