Second Zagreb and Sigma Indices of Semi and Total Transformations of Graphs

Abstract

The study of structure-property relations including the transformations of molecules is of utmost importance in correlations with corresponding physicochemical properties. The graph topological indices have been used effectively for such study and, in particular, bond-based indices play a vital role. The bond-additive topological indices of a molecular graph are defined as a sum of edge measures over all edges in which edge measures can be computed based on degrees, closeness, peripherality, and irregularity. In this study, we provide the mathematical characterization of the transformation of a structure that can be accomplished by the novel edge adjacency and incidence relations. We derive the exact expressions of bond type indices such as second Zagreb, sigma indices, and their coindices of total transformation and two types of semitransformations of the molecules which in turn can be used to characterize the topochemical and topostructural properties.

On Transformed Graphs

“…We begin with ordinary graphs and we end up by obtaining a complex network structure from the base graphs. The base graph can be transformed by implementing any of the following combinations [26]: We have, x, y, z ∈ {+, −} and v, w ∈ V (T xyz (H)); Lemma 1 (from [1]) Let H be a graph with size q and order p. Then |V (Txyz(H))| = q + p and the edge cardinality is given as:…”

“…Proof The proof follows from the notion of edge cardinality cases as mentioned in lemma (1). The first Zagreb index of the transformation graphs have been determined in [3].…”

“…, and T +++ (H) ∼ = T −−− (H) and edge cardinality cases as mentioned in lemma (1). The first Zagreb index, the second Zagreb index and the forgotten index of the transformation graphs have been determined in [1,3]. We have…”

“…A graph H can be transformed into another network T (H), such that the transformed graph T and the original graph H are in a one-to-one correspondence. More detailed information on this matter is found in [1][2][3][4][5]. Topological indices are numerical values calculated from the structure of the underlying graph, and reflect certain structural features of this graph.…”

Topological Aspects of Dendrimers via Connection-Based Descriptors

“…Recently, Sattar et al [26][27][28] worked on computing the general expression of ZIs for two types of dendrimers. For the other terminologies not discussed in this paper, the readers are referred to [29][30][31]. The motivation for this article is as follows:…”