Abstract
The reformulated Zagreb indices of a graph are obtained from the original Zagreb indices by replacing vertex degrees with edge degrees, where the degree of an edge is taken as the sum of degrees of its two end vertices minus 2. In this paper, we obtain two upper bounds of the first reformulated Zagreb index among all graphs with p pendant vertices and all graphs having key vertices for which they will become trees after deleting their one key vertex. Moreover, the corresponding extremal graphs which attained these bounds are characterized.
Highlights
Some constants are used to characterize some properties of the graph of a molecule, which are usually called topological indices
In 1977, Kier and Kall [4] extended the concept of molecular connectivity index and defined the zeroth-order general Randicindex
If G ∈ Gpn, there will be a connected subgraph H0 with order n − p for which G can be reconstructed by linking p vertices to some vertices H0
Summary
Some constants are used to characterize some properties of the graph of a molecule, which are usually called topological indices. Note that the first Zagreb index is the zeroth-order general Randicindex for α 2. For a given G, let L(G) be its line graph. Let Gpn be the set of connected graphs with p( ≥ 2) pendant vertices.
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