Zagreb-Type Indices of R-Vertex Join and R-Edge Join of Graphs

Abstract

There are various methods available which are used to search large chemical databases and to predict the physicochemical properties of molecular structures. Using molecular descriptors for this purpose is the simplest of these methods. The Zagreb indices are amongst the oldest molecular descriptors, and their properties have been extensively studied and applied in QSAR/QSPR studies. The Zagreb coindices were recently introduced, attracting the attention of researchers in mathematical chemistry. In this paper, we study Zagreb indices and several other Zagreb-type indices including the general Randić index, sum-connectivity index, F-index, and Zagreb coindices of R-vertex and edge join of two arbitrary graphs.

Zagreb indices of link graphs related to carbon nanotube and fullerene

“…We use the following definitions: Definition 1 -The 𝑅 -graph [8] of a graph 𝐻 is represented by 𝑅(𝐻) and is obtained from 𝐻 by linking a vertex 𝑤 𝑒 ∉ 𝑉(𝐻) corresponding to each edge 𝑒 = 𝑢𝑣 ∈ (𝐻) andjoining 𝑤 𝑒 to end vertices 𝑢 and 𝑣 for each edge 𝑒. Definition 2 -𝑅 -Vertex join [9] of 𝐻 1 and 𝐻 2 is represented by 𝑅(𝐻) is obtained from 𝑅(𝐻 1 ) and 𝐻 2 by linking every vertex of 𝑉(𝐻1) and 𝑉(𝐻2) by an edge. Definition 3 -𝑅 -Edge join [9] of 𝐻 1 and 𝐻 2 is represented by 𝑅(𝐻) is obtained from 𝑅(𝐻 1 ) and 𝐻 2 by joining every new vertex of 𝑅(𝐻1) and 𝑉(𝐻2) by an edge.…”

“…Definition 2 -𝑅 -Vertex join [9] of 𝐻 1 and 𝐻 2 is represented by 𝑅(𝐻) is obtained from 𝑅(𝐻 1 ) and 𝐻 2 by linking every vertex of 𝑉(𝐻1) and 𝑉(𝐻2) by an edge. Definition 3 -𝑅 -Edge join [9] of 𝐻 1 and 𝐻 2 is represented by 𝑅(𝐻) is obtained from 𝑅(𝐻 1 ) and 𝐻 2 by joining every new vertex of 𝑅(𝐻1) and 𝑉(𝐻2) by an edge. Definition 4 -𝑅 -Edge neighborhood link graph [10] between 𝐻 1 and 𝐻 2 is represented by 𝑅 𝐸 (𝐻) is obtained by joining the neighbourhood separated vertices of newly added vertices of 𝑅(𝐻1) to each copy of 𝐻2 by an edge.…”