Abstract
The Cartesian product and join are two classical operations in graphs. Let be the degree of a vertex in line graph of a graph . The edge versions of atom-bond connectivity () and geometric arithmetic () indices of G are defined as and , respectively. In this paper, and indices for certain Cartesian product graphs (such as , and ) are obtained. In addition, and indices of certain join graphs (such as , , and ) are deduced. Our results enrich and revise some known results.
Highlights
The invariants based on the distance or degree of vertices in molecules are called topological indices
The first topological index, Wiener index, was published in 1947 [1], and the edge version of the Wiener index was proposed by Iranmanesh et al in 2009 [2]
Because the important effects of the topological indices are proved in chemical research, more and more topological indices are studied, including the classical atom-bond connectivity index and the geometric arithmetic index
Summary
The invariants based on the distance or degree of vertices in molecules are called topological indices. Let L( G ) or G L be the line graph of G such that each vertex of L( G ) represents an edge of G and two vertices of L( G ) are adjacent if and only if their corresponding edges share a common endpoint in G [3]. The edge version of the ABC index is:. The recent research on edge version ABC index can be referred to Gao et al [5]. Das [17] obtained the upper and lower bounds of the ABC index of trees. Chen et al [20] obtained some upper bounds for the ABC index of graphs with given vertex connectivity. Our results extend and enrich some known results [5,23,24]
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