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A topological index is a real number that relates to a graph that must be a structural invariant. In this paper, we first define a new graph, which is a concept from the coronavirus, called a corona graph. We give some theoretical results for the Wiener and the hyper Wiener index of a graph. Moreover, we calculate some topological indices degree-based on a corona graph. In addition, we are introducing a new topological index SK4, which is inspired by the definition of the SK1 index. [ABSTRACT FROM AUTHOR]

Graph-based molecular structure descriptors (often called "topological indices") are useful for modeling the physical and chemical properties of molecules, designing pharmacologically active compounds, detecting environmentally hazardous substances, etc. The graph invariant G R M α , known under the name general reduced second Zagreb index, is defined as G R M α (Γ) = ∑ u v ∈ E (Γ) (d Γ (u) + α) (d Γ (v) + α) , where d Γ (v) is the degree of the vertex v of the graph Γ and α is any real number. In this paper, among all trees of order n, and all unicyclic graphs of order n with girth g, we characterize the extremal graphs with respect to G R M α (α ≥ − 1 2) . Using the extremal unicyclic graphs, we obtain a lower bound on G R M α (Γ) of graphs in terms of order n with k cut edges, and completely determine the corresponding extremal graphs. Moreover, we obtain several upper bounds on G R M α of different classes of graphs in terms of order n, size m, independence number γ , chromatic number k, etc. In particular, we present an upper bound on G R M α of connected triangle-free graph of order n > 2 , m > 0 edges with α > − 1.5 , and characterize the extremal graphs. Finally, we prove that the Turán graph T n (k) gives the maximum G R M α (α ≥ − 1) among all graphs of order n with chromatic number k. [ABSTRACT FROM AUTHOR]

Let G be a graph with vertex set V (G) and edge set E (G). The general reduced second Zagreb index of G is defined as GRM β (G) = ∑ u v ∈ E (G) (d (u) + β) (d (v) + β) , where β is any real number and d (v) is the degree of the vertex v of G. In this paper, the general reduced second Zagreb index of the Cartesian product, corona product, join of graphs and two new operations of graphs are computed. [ABSTRACT FROM AUTHOR]

Network theory as an important role in the field of electronic and electrical engineering, for example, in signal processing, networking, communication theory, etc. The branch of mathematics known as Graph theory found remarkable applications in this area of study. A topological index (TI) is a real number attached with graph networks and correlates the chemical networks with many physical and chemical properties and chemical reactivity. The Optical Transpose Interconnection System (OTIS) network has received considerable attention in recent years and has a special place among real world architectures for parallel and distributed systems. In this report, we compute redefined first, second and third Zagreb indices of OTIS swapped and OTIS biswapped networks. We also compute some Zagreb polynomials of understudy Networks. [ABSTRACT FROM AUTHOR]

A topological index (TI) is a function from Σ to the set of real numbers, where Σ is the set of finite simple graphs. In fact, it is a final outcome of a logical, systematical and mathematical process that transforms feature encoded in a molecular graph to a fixed real number. Gutman and Trinajstic (1972) first time defined degree based TI named as first Zagreb index to compute the total ?-electron energy of a molecular graph. They also exposed another TI that is renamed as modified first Zagreb connection index in [Ali and Trinajstic; Mol: Inform: 37(2018); 1 - 7]: In this paper, we compute the upper bounds for the Zagreb connection indices i.e. first Zagreb connection index, second Zagreb connection index and modified first Zagreb connection index of the resultant graphs which are obtained by applying the tensor and strong product of two graphs. [ABSTRACT FROM AUTHOR]

Topological indices are real numbers associated with molecular graphs and catch symmetry of molecular structures that give it a scientific dialect to foresee properties. In this work, we compute first and second reverse Zagreb indices, first and second reverse hyper Zagreb indices, reverse Atomic-bond connectivity index and reverse Geometric-arithmetic index for TUC4[m, n]. [ABSTRACT FROM AUTHOR]

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 Z α , β (G) = ∑ u v ∈ 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 Z α , 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 Z α , β index in C (π) ; (3) The so-called majorization theorem of Z α , β holds for trees, unicyclic graphs, and bicyclic graphs. As applications of the above results, we determine the extremal graphs with maximum Z α , 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. [ABSTRACT FROM AUTHOR]

Topological invariants are the graph theoretical tools to the theoretical chemists, that correlates the molecular structure with several chemical reactivity, physical properties or biological activity numerically. A function having a set of networks(graph, molecular structure) as its domain and a set of real numbers as its range is referred as a topological invariant(index). Topological invariants are numerical quantity of a network that are invariant under graph isomorphism. Topological invariants such as Zagreb index, Randić index and multiplicative Zagreb indices are used to predict the bioactiviy of chemical compounds in QSAR/QSPR study. In this paper, we compute the general expression of certain degree based topological invariants such as second Zagreb index, F-index, Hyper-Zagreb index, Symmetric division degree index, irregularity of Splitting graph. And also we obtain upper bound for first and second multiplicative Zagreb indices of Splitting graph of a graph H, (S′(H)). [ABSTRACT FROM AUTHOR]

A topic of current interest in the study of topological indices is to find relations between some index and one or several relevant parameters and/or other indices. In this paper we study two general topological indices A α and B α , defined for each graph H = (V (H) , E (H)) by A α (H) = ∑ i j ∈ E (H) f (d i , d j) α and B α (H) = ∑ i ∈ V (H) h (d i) α , where d i denotes the degree of the vertex i and α is any real number. Many important topological indices can be obtained from A α and B α by choosing appropriate symmetric functions and values of α. This new framework provides new tools that allow to obtain in a unified way inequalities involving many different topological indices. In particular, we obtain new optimal bounds on the variable Zagreb indices, the variable sum-connectivity index, the variable geometric-arithmetic index and the variable inverse sum indeg index. Thus, our approach provides both new tools for the study of topological indices and new bounds for a large class of topological indices. We obtain several optimal bounds of A α (respectively, B α ) involving A β (respectively, B β ). Moreover, we provide several bounds of the variable geometric-arithmetic index in terms of the variable inverse sum indeg index, and two bounds of the variable inverse sum indeg index in terms of the variable second Zagreb and the variable sum-connectivity indices. [ABSTRACT FROM AUTHOR]