General Sum-connectivity Index Research Articles

New bounds for variable topological indices and applications

AbstractOne of the most important information related to molecular graphs is given by the determination (when possible) of upper and lower bounds for their corresponding topological indices. Such bounds allow to establish the approximate range of the topological indices in terms of molecular structural parameters. The purpose of this paper is to provide new inequalities relating several classes of variable topological indices including the first and second general Zagreb indices, the general sum-connectivity index, and the variable inverse sum deg index. Also, upper and lower bounds on the inverse degree in terms of the first general Zagreb are found. Moreover, the characterization of extremal graphs with respect to many of these inequalities is obtained. Finally, some applications are given.

On Some Bounds of Degree Based Topological Indices for Total Graphs

In this paper, we discuss the concept of total graph and computed some topological indices. If $\Theta$ is a simple graph, then the elements of $\Theta$ are the vertices $\Theta_V$ and edges $\Theta_E$. For $ e=u\acute{u}\in \Theta_E$, the vertex $u$ and edge $e$, as well as $\acute{u}$ and $e$, are incident. We define the general harmonic $(GH)$ index and general sum connectivity $(GS)$ index for graph $\Theta$ regarding incident vertex-edge degrees as: $H_s^{\alpha}(\Theta)=\sum_{e\acute{u}}\big(\frac{2}{\aleph_{\acute{u}}+\aleph_{e}}\big)^{\alpha}$ and $\hat{\chi}_s^{\alpha}(\Theta)=\sum_{e\acute{u}}(\aleph_{\acute{u}}+\aleph_{e})^{\alpha}$, where $\alpha$ is any real number. In this article, we derive the closed formulas for a few standard graphs for $(GH)$ and $(GS)$ indices and then go on to calculate the lowest and the greatest general harmonic index, as well as the general sum-connectivity index, for various graphs that correspond to their total graphs.

Some sufficient conditions for graphs to be [formula omitted]-leaf-connected

Let G be a graph and S be a subset of V(G). A graph G is called S-leaf-connected if G has a spanning tree T such that S is the set of leaves of T. For k≥2, we say a graph G is k-leaf-connected if |V(G)|>k and given any subset S of V(G) with |S|=k, G is S-leaf-connected. In the present note, we give some sufficient conditions on eccentric connectivity index, eccentric distance sum, connective eccentricity index and difference of Zagreb indices for connected graphs to be k-leaf-connected. These results generalize the sufficient conditions on these indices for connected graphs to be Hamilton-connected. At last, we obtain a sufficient condition for a graph to be k-leaf-connected in terms of the general sum-connectivity index of the complement graph.

Computation of multiplicative topological aspects of hex-derived networks

A chemical network is numerically represented by a topological index in chemical graph theory. As opposed to its chemical representation, a topological descriptor correlates with specific physical properties of the underlying chemical molecules. In this article, third type of hex-derived networks HDN3(r), THDN3(r), are described. The goal of this study is to develop some updated and closed formulas based on multiplicative graph invariants. Such as ordinary geometric-arithmetic (OGA), general version of harmonic index (GHI), sum connectivity index (SI), general sum connectivity index (GSI), first and second Gourava and hyper-Gourava indices, Shegehalli and Kanabur indices, first generalized version of Zagreb index (GZI), and forgotten index (FI) for the hex-derived HDN3(r), THDN3(r), networks. Moreover, various types of edge for computing have been discovered and analyzed along with the order and size. The calculation of multiplicative topological features in networks that are generated from hexagonal structures is the main task of this work. Gaining more insight into the structural characteristics and possible uses of these networks requires examining the interaction between topological aspects and multiplication processes. To interpret the chemical compounds', physical and biological attributes, we can integrate the analysis of the networks stated above with the chemical compounds and their graphical structures. These results can be utilized to evaluate the biological and physio-chemical activities of compounds.

Degree-based indices of hypergraphs: Definitions and first results

For a [Formula: see text]-uniform hypergraph [Formula: see text], we introduce degree-based indices such as the general sum-connectivity index [Formula: see text] and the general Randić index [Formula: see text], where [Formula: see text], [Formula: see text] is the set of hyperedges of [Formula: see text] and [Formula: see text] is the degree of a vertex [Formula: see text] in [Formula: see text]; [Formula: see text] and [Formula: see text]. Other indices such as the first and second Zagreb index, first and second hyper-Zagreb index, classical sum-connectivity index, classical Randić index and harmonic index of a hypergraph [Formula: see text] are special cases of the general indices. For [Formula: see text], we obtain upper bounds on [Formula: see text] and [Formula: see text] for a uniform hypergraph [Formula: see text] with given order, order and number of isolated vertices, order and maximum degree, order and diameter at least [Formula: see text], and lower bounds for uniform hypergraphs with given order and no isolated vertices, order and minimum degree, and order and maximum possible degree. We also present extremal graphs for all the bounds. Bounds on Zagreb indices follow from our results on general indices.

General sum-connectivity index and general Randic index of trees with given maximum degree

General sum-connectivity index and general Randic index of trees with given maximum degree

A note on general sum-connectivity index

For a simple finite graph G, general sum-connectivity index is defined for any real number α as χα(G) = , which generalises both the first Zagreb index and the ordinary sum-connectivity index. In this paper, we present some new bounds for the general sum-connectivity index. We also present relation between general sum-connectivity index and general Randić index.

General sum-connectivity index of trees with given number of branching vertices

General sum-connectivity index of trees with given number of branching vertices

On topological indices of some chemical graphs

This research delves into the application of topological indices in predicting the physico-chemical properties of chemical materials. These indices have been employed in various QSAR and QSPR studies, providing a mathematical formulation to predict properties such as viscosity and radius of gyrations Rucker and Rucker [J. Chem. Inf. Comput. Sci. 39, 788–802, (1999)]. We explore several indices, including the forgotten topological index, inverse sum index, general sum connectivity index and first multiplicative atom bond connectivity index, and investigate their usefulness in predicting bio-activity and anti-inflammatory activities of certain chemical compounds. Specifically, we focus on the allotrope structure, nanotubes and dominating oxide and silicate chemical networks. We present a graphical trend of these indices and perform a comparison to facilitate the observation of physical-chemical properties of these materials for researchers.

On the general [formula omitted]-type index of connected graphs

Let G=(V,E) be a connected graph, and d(u) the degree of vertex u∈V. We define the general Z-type index of G as Zα,β(G)=∑uv∈E[d(u)+d(v)−β]α, where α and β are two real numbers. This generalizes several famous topological indices, such as the first and second Zagreb indices, the general sum-connectivity index, the reformulated first Zagreb index, and the general Platt index, which have successful applications in QSPR/QSAR research. Hence, we are able to study these indices in a unified approach.Let C(π) the set of connected graphs with degree sequence π. In the present paper, under different conditions of α and β, we show that:(1) There exists a so-called BFS-graph having extremal Zα,β index in C(π);(2) If π is the degree sequence of a tree, a unicyclic graph, or a bicyclic graph, with minimum degree 1, then there exists a special BFS-graph with extremal Zα,β index in C(π);(3) The so-called majorization theorem of Zα,β holds for trees, unicyclic graphs, and bicyclic graphs.As applications of the above results, we determine the extremal graphs with maximum Zα,β index for α>1 and β≤2 in the set of trees, unicyclic graphs, and bicyclic graphs with given number of pendent vertices, maximum degree, independence number, matching number, and domination number, respectively. These extend the main results of some published papers.

Topological Evaluation of Four Para-Line Graphs Absolute Pentacene Graphs Using Topological Indices

A real-number to molecular structure mapping is a topological index. It is a graph invariant method for describing physico-chemical properties of molecular structures specific substances. In that article, We examined pentacene’s chemical composition. The research on the subsequent indices is reflected in our paper, we conducted an analysis of several indices including general randic connectivity index, first general zagreb index, general sum-connectivity index, atomic bond connectivity index, geometric-arithmetic index, fifth class of geometric-arithmetic indices, hyper-zagreb index, first and second multiple zagreb indices for a four para-lines graphs of linear [n]-pentacene and multi-pentacene.

General approach for obtaining extremal results on degree-based indices illustrated on the general sum-connectivity index

Among bipartite graphs with given order and matching number/vertex cover number/edge cover number/independence number, among multipartite graphs with given order, and among graphs with given order and chromatic number, we present the graphs having the maximum degree-based index if that index satisfies certain conditions. We show that those conditions are satisfied by the general sum-connectivity index χ a for all or some a ≥ 0 .

General sum-connectivity index of a graph and its line graph

For a real number β, the general sum-connectivity index χβ(G) of a graph G is defined as ∑xy∈E(G)(d(x)+d(y))β, where d(x) denote the degree of a vertex x in G. In this paper, we show that for every graph G≇Pn, if β≥0, thenχβ(L(G))≥{χβ(G)forδ(G)≤2,2χβ(G)forδ(G)≥3.In addition, for β<0, we present the lower bound for χβ(L(G)) in terms of χβ(G). Finally, we establish sharp bounds for χβ(G)+χβ(L(G)), and characterize the extremal graphs attaining the bounds.

RELATION BETWEEN GENERAL RANDIC INDEX AND ´ GENERAL SUM CONNECTIVITY INDEX

The general Randi´c index is the sum of weights of (d(u).d(v))k for every edge uv of a molecular graph G. On the other hand general Sum-Connectivity index is the sum of the weights (d(u) + d(v))k for every edge uv of G, where k is a real number and d(u) is the degree of vertex u. Both families of topological indices are well known and closely related. In fact the correlation coefficient value of these two families of indices for the trees representing the Octane Isomers vary between 0.915 to 0.998. In the recent years these families of indices have been extensively explored and studied. The major research on these indices mostly consists of the application in QSPR/QSAR analysis, computation of these indices for various molecular graphs and bounds of the indices for certain graphs, satisfying certain conditions. The main focus of this paper is a comparative study on these two families of indices for various families of graphs. We find a few algebraic relationships between general Randi´c index and general Sum-connectivity index of certain graphs.

Multiplicative Attributes Derived from Graph Invariants for Saztec4 Diamond

This study consists of developing some closed and updated formulas derived from multiplicative graph invariants such as general Randic index GRI R λ 0 ϰ for λ 0 = ± 1 , ± 1 / 2 , ordinary general geometric-arithmetic (OGA), general version of harmonic index (GHI), sum connectivity index (SI), general sum connectivity index (GSI), 1st and 2nd Gourava and hyper-Gourava indices, (ABC) index, Shegehalli and Kanabur indices, 1st generalised version of Zagreb index (GZI), and forgotten index (FI) for the subdivided Aztec diamond network. Aztec diamond is constructed based on the squares boxes. These square boxes are placed at the centre point and nourish the condition s − 1 / 2 + r − 1 / 2 ≤ n . Furthermore, we put a new vertex of degree-2 at each edge of the small boxes, squares in shapes. A new structure is obtained that has the same properties as its parental graph and is called a subdivided Aztec diamond and symbolised as Saztecn. Subsequently, we compute the multiplicative topological attributes to get some new formulas. For this purpose, a simple, connected, and the finite graph is considered by supposing it Y as the graph of the Saztecn. The order and size have also been discussed in this study and found three different kinds of edges (2, 2), (2, 3), and (2, 4) for computing. The discussion on the networks mentioned above provides us with essential results that can be used in the determination of bio and physio activities and can be interspersed with the molecular compounds and their graphical structures better to understand their physical as well as biological properties.

On general Sombor index of graphs

In this paper, we extend the recently introduced vertex-degree-based topological index, the Sombor index, and we call it general Sombor index. The general Sombor index generalizes both the forgotten index and the Sombor index. We present the bounds in terms of other important graph parameters for general Sombor index. We also explore the Nordhaus–Gaddum-type result for the general Sombor index. We present further the relations between general Sombor index and other generalized indices: general Randić index and general sum-connectivity index.

On Bond Incident Degree Indices of Connected Graphs with Fixed Order and Number of Pendent Vertices

The graphs having the maximum value of certain bond incident degree indices (including the second Zagreb index, general sum-connectivity index, and general zeroth-order Randić index) in the class of all connected graphs with fixed order and number of pendent vertices are characterized in this paper. The problem of finding graphs having the minimum values of the second Zagreb index and general zeroth-order Randić index from the aforementioned class of connected graphs is also addressed. One of the obtained results about the first Zagreb index has already been proved in the papers [I. Gutman, M. Kamran Jamil, N. Akhter, Trans. Combin. 4 (2015) 43-48] and [M. Enteshari, B. Taeri, MATCH Commun. Math. Comput. Chem. 86 (2021) 17-28]; however, it is proved here by another method with a short proof. Moreover, one of the obtained results concerning the second Zagreb index gives a partial solution to a problem attacked in the aforementiond paper of Enteshari and Taeri.

On General Sum-Connectivity Index of Trees of Fixed Maximum Degree and Order

On General Sum-Connectivity Index of Trees of Fixed Maximum Degree and Order

Topological Properties of Concealed Non-Kekulean Benzenoid Hydrocarbon

A topological descriptor is regarded as a numerical parameter derived by using mathematical tools from the molecular graph of different chemical structures. The chemical graph theory is a paramount section of mathematical chemistry. In this section, there are many topological parameters, that have very useful characteristics to study the molecular structure of chemicals. In this article, we investigate topological indices of concealed non-kekulean benzenoid hydrocarbon graph. We will compute general Randic index, general sum connectivity index, sum connectivity index, general version of harmonic index, harmonic index, first, second and third Zagreb indices, SK, SK 1, SK 2 indices, augmented Zagreb index, atomic bond connectivity index, geometric arithmetic index, first and second gourava indices, first and second hyper-gourava indices and 4th version of atomic bond connectivity, 5th version of geometric arithmetic index, and the Sanskurti index.

Topological Invariants for the General Structure of Grape Seed Proanthocyanidins

In this paper, we formulate the degree-based topological indices such as first Zagreb index, second Zagreb index, third Zagreb index, atom bond connectivity index, geometric-arithmetic (GA) index, general sum connectivity index, hyper-Zagreb index, augmented Zagreb index, sum connectivity index and general Randic index Rλ (℘), for λ = © −1, 1, − 1 2 , 1 2 ª for grape seed Proanthocyanidins network. By using above-mentioned topological indices, this study provides us different outcomes.