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Let G be a connected graph with vertex set V(G) and edge set E(G). Recently, the Revan vertex degree concept is defined in Chemical Graph Theory. The first and second Revan indices of G are defined as R1(G) = ∑uv∈E$\begin{array}{} \displaystyle \sum\limits_{uv\in E} \end{array}$[rG(u) + rG(v)] and R2(G) = ∑uv∈E$\begin{array}{} \displaystyle \sum\limits_{uv\in E} \end{array}$[rG(u)rG(v)], where uv means that the vertex u and edge v are adjacent in G. The first and second hyper-Revan indices of G are defined as HR1(G) = ∑uv∈E$\begin{array}{} \displaystyle \sum\limits_{uv\in E} \end{array}$[rG(u) + rG(v)]2 and HR2(G) = ∑uv∈E$\begin{array}{} \displaystyle \sum\limits_{uv\in E} \end{array}$[rG(u)rG(v)]2. In this paper, we compute the first and second kind of Revan and hyper-Revan indices for the octahedral and icosahedral networks.

Abstract: The Randic (product) connectivity index and its derivative called the sum-connectivity index are well-known topological indices and both of these descriptors correlate well among themselves and with the [pi]-electronic [...]

Abstract: Graph theory plays a vital role in modeling and designing any chemical structure or chemical network. Chemical graph theory helps in understanding the molecular structural properties of a molecular [...]

Topological indices are numerical parameters of a graph that characterize its molecular topology and are usually graph invariant. In a QSAR/QSPR study, the physico-chemical properties and topological indices such as the Randic, atom-bond connectivity (ABC), and geometric-arithmetic (GA) indices are used to predict the bioactivity of chemical compounds. Graph theory has found a considerable use in this important area of research. All of the studied interconnection networks in this paper are constructed by the Star of David network. In this paper, we study the general Randic:, first Zagreb, ABC, GA, [ABC.sub.4] and [GA.sub.5], indices for the first, second, and third types of dominating David derived networks and give closed formulas of these indices for these networks. These results are useful in network science to understand the underlying topologies of these networks. Key words: general Randic: index, atom-bond connectivity (ABC) index, geometric-arithmetic (GA) index, Star of David, dominating David derived network. Les indices topologiques sont les parametres numeriques d'un graphe qui caracterisent sa topologie moleculaire et sont habituellement des invariants de graphe. Dans une etude de relations quantitatives de QSAR/QSPR, nous avons utilise les proprietes physico-chimiques et les indices topologiques tels que l'indice de Randic:, l'indice de connectivite atome-liaison (ABC) et l'indice arithmetico-geometrique (GA) pour predire l'activite biologique de composes chimiques. De fait, l'utilisation de la theorie des graphes s'est accrue considerablement dans cet important domaine de recherche. Dans le present article, tous les reseaux d'interconnexions etudies sont construits a partir du reseau en etoile de David. Dans le cadre des presents travaux, nous avons calcule l'indice de Randic: generalise, le premier indice de Zagreb et les indices ABC, GA, [ABC.sub.4] et [GA.sub.5] pour les premier, deuxieme et troisieme types de reseaux a dominante derivee de David et nous avons presente les formules fermees de ces indices pour les reseaux precites. Ces resultats sont utiles pour la recherche portant sur les reseaux et permettent de comprendre les topologies qui sous-tendent ces reseaux. [Traduit par la Redaction] Mots-cles: indice de Randic: generalise, indice de connectivite atome-liaison (ABC), indice arithmetico-geometrique (GA), etoile de David, reseau a dominante derivee de David., 1. Introduction and preliminary results Graph theory has provided chemists with a variety of useful tools such as topological indices. Molecules and molecular compounds are often modeled by a molecular [...]

Topological indices are numerical parameters of a graph that characterize its topology and are usually graph invariant. In a QSAR/QSPR study, physicochemical properties and topological indices such as Randic, atom-bond connectivity (ABC), and geometric-arithmetic (GA) indices are used to predict the bioactivity of chemical compounds. Graph theory has found a considerable use in this area of research. In this paper, we study different interconnection networks and derive analytical closed results of the general Randic: index ([R.sub.α](G)) for α = 1, 1/2, -1, - [1/2] only, for dominating oxide network (DOX), dominating silicate network (DSL), and regular triangulene oxide network (RTOX). All of the studied interconnection networks in this paper are motivated by the molecular structure of a chemical compound, Si[O.sub.4] . We also compute the general first Zagreb, ABC, GA, [ABC.sub.4], and GA, indices and give closed formulae of these indices for these interconnection networks. Key words: general Randic: index, atom-bond connectivity (ABC) index, geometric-arithmetic (GA) index, hierarchical networks. Les indices topologiques sont les parametres numeriques d'un graphe qui caracterisent sa topologie et sont habituellement des invariants de graphe. Dans une etude de QSAR/QSPR, nous avons utilise les proprietes physico-chimiques et les indices topologiques tels que l'indice de Randic:, l'indice de connectivite atome-liaison (ABC) et l'indice arithmeticogeometrique (GA) pour predire l'activite biologique de composes chimiques. L'utilisation de la theorie des graphes s'est accrue considerablement dans ce domaine de recherche. Dans le present article, nous etudions differents reseaux d'interconnexions et derivons les resultats analytiques fermes de l'indice de Randic: generalise ([R.sub.α](G)) pourα = 1, 1/2, -1, -1/2 seulement, pour un reseau a dominante oxyde (DOX), un reseau a dominante silicate et un reseau d'oxyde de triangulene regulier. Tous les reseaux d'interconnexions etudies dans ces travaux sont motives par la structure moleculaire d'un compose chimique, le Si[O.sub.4]. Nous avons egalement calcule le premier indice de Zagreb generalise et les indices ABC, GA, [ABC.sub.4] et GA et avons presente les formules fermees de ces indices pour ces reseaux. [Traduit par la Redaction] Mots-cles : indice de Randic: generalise, indice de connectivite atome-liaison (ABC), indice arithmetico-geometrique, reseaux hierarchiques., 1. Introduction and preliminary results There is much mathematics involved in electrical and electronic engineering. It depends on what area of electrical and electronic engineering, for example, there is much [...]

Let m, n be positive integers and m ≤ n. Suppose we select two elements at random (with replacement) from the ring Zn. Then a question arise that "what is the probability that the product of these two elements is m?" We derive explicit formulas to compute the probability that the product of two elements chosen at random (with replacement) from Zn is m. Also we obtain some bounds for this probability. [ABSTRACT FROM AUTHOR]

Let G = (V,E) be a simple connected graph with vertex set V and edge set E. For two vertices u and v in a graph G, the distance d(u, v) is the shortest path between u and v in G. Graph theory has much advancements in the field of theoretical chemistry. Recently, chemical graph theory is becoming very popular among researchers because of its wide applications of mathematics in chemistry. One of the important distance based topological index is the Wiener index, defined as the sum of distances between all pairs of vertices of G, defined as... [ABSTRACT FROM AUTHOR]

For a connected graph G, a subset W = {w1, w2, w3, ... , wξ} of the vertices of G is the resolving set for G if for a, b ∈ V(G), we have d(a, wξ) ≠ d(b, wξ) for all wξ ∈ W. Metric basis for G is the minimum number of vertices in W and metric dimension is the cardinality of such a set denoted by β (G). In this paper we compute the metric dimension of P(n, 2)ʘK1. [ABSTRACT FROM AUTHOR]