Optimizing Wiener and Randić Indices of Graphs

Wiener and Randić indices have long been studied in chemical graph theory as connection strength measures of graphs. Later, these indices were used in different fields such as network analysis. We consider two optimization problems related to these indices, with potential applications to network theory, in particular to epidemiological networks. Given a connected graph and a fixed total edge weight, we investigate how individual weights must be assigned to edges, minimizing the connection strength of the graph. In order to measure the connection strength, we use the weighted Wiener index and a modified version of the ordinary Randić index. Wiener index optimization is linear, while Randić index optimization turns out to be both nonlinear and nonconvex. Hence, we adopt the technique of separable programming to generate solutions. We present our experimental results by applying relevant algorithms to several graphs.

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Comparing the molecular graph degeneracy of Wiener, Harary, Balaban, Randi´c and ZEP topological indices

Creative Mathematics and Informatics ◽

10.37193/cmi.2014.02.02 ◽

2014 ◽

Vol 23 (2) ◽

pp. 165-174

Author(s):

ZOITA-MARIOARA BERINDE ◽

Keyword(s):

Topological Index ◽

Molecular Graph ◽

Wiener Index ◽

Topological Indices ◽

Randić Index ◽

Discrimination Power ◽

Harary Index ◽

Molecular Chemistry ◽

Randic Index

The aim of this paper is to show that the ZEP topological index has better discrimination power than four well known topological indices in molecular chemistry: Balaban index, Harary index, Randic index, and Wiener index.

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Topology-Based Analysis of OTIS (Swapped) Networks OKn and OPn

Journal of Chemistry ◽

10.1155/2019/4291943 ◽

2019 ◽

Vol 2019 ◽

pp. 1-11

Author(s):

Hai-Xia Li ◽

Sarfaraz Ahmad ◽

Iftikhar Ahmad

Keyword(s):

Topological Index ◽

Molecular Descriptor ◽

Randić Index ◽

Zagreb Index ◽

Randic Index ◽

Chemical Graph ◽

Zagreb Indices ◽

Second Zagreb Index ◽

Chemical Graph Theory ◽

Augmented Zagreb Index

In the fields of chemical graph theory, topological index is a type of a molecular descriptor that is calculated based on the graph of a chemical compound. In this paper, M-polynomial OKn and OPn networks are computed. The M-polynomial is rich in information about degree-based topological indices. By applying the basic rules of calculus on M-polynomials, the first and second Zagreb indices, modified second Zagreb index, general Randić index, inverse Randić index, symmetric division index, harmonic index, inverse sum index, and augmented Zagreb index are recovered.

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Extremal Trees with Respect to the Difference between Atom-Bond Connectivity Index and Randić Index

Symmetry ◽

10.3390/sym12101591 ◽

2020 ◽

Vol 12 (10) ◽

pp. 1591

Author(s):

Wan Nor Nabila Nadia Wan Zuki ◽

Zhibin Du ◽

Muhammad Kamran Jamil ◽

Roslan Hasni

Keyword(s):

Graph Theory ◽

Connectivity Index ◽

Randić Index ◽

Extremal Graphs ◽

Randic Index ◽

Chemical Graph Theory ◽

The Difference ◽

Atom Bond ◽

Chemical Trees

Let G be a simple, connected and undirected graph. The atom-bond connectivity index (ABC(G)) and Randić index (R(G)) are the two most well known topological indices. Recently, Ali and Du (2017) introduced the difference between atom-bond connectivity and Randić indices, denoted as ABC−R index. In this paper, we determine the fourth, the fifth and the sixth maximum chemical trees values of ABC−R for chemical trees, and characterize the corresponding extremal graphs. We also obtain an upper bound for ABC−R index of such trees with given number of pendant vertices. The role of symmetry has great importance in different areas of graph theory especially in chemical graph theory.

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Some Topological Invariants of Graphs Associated with the Group of Symmetries

Journal of Chemistry ◽

10.1155/2020/6289518 ◽

2020 ◽

Vol 2020 ◽

pp. 1-13

Author(s):

Chang-Cheng Wei ◽

Muhammad Salman ◽

Usman Ali ◽

Masood Ur Rehman ◽

Muhammad Aqeel Ahmad Khan ◽

...

Keyword(s):

Biological Activity ◽

Topological Index ◽

Chemical Reactivity ◽

Wiener Index ◽

Structural Features ◽

Randić Index ◽

Harary Index ◽

Randic Index ◽

Chemical Structures ◽

Connectivity Indices

A topological index is a quantity that is somehow calculated from a graph (molecular structure), which reflects relevant structural features of the underlying molecule. It is, in fact, a numerical value associated with the chemical constitution for the correlation of chemical structures with various physical properties, chemical reactivity, or biological activity. A large number of properties like physicochemical properties, thermodynamic properties, chemical activity, and biological activity can be determined with the help of various topological indices such as atom-bond connectivity indices, Randić index, and geometric arithmetic indices. In this paper, we investigate topological properties of two graphs (commuting and noncommuting) associated with an algebraic structure by determining their Randić index, geometric arithmetic indices, atomic bond connectivity indices, harmonic index, Wiener index, reciprocal complementary Wiener index, Schultz molecular topological index, and Harary index.

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Eccentricity based topological indices of face centered cubic lattice FCC(n)

Main Group Metal Chemistry ◽

10.1515/mgmc-2021-0005 ◽

2020 ◽

Vol 44 (1) ◽

pp. 32-38

Author(s):

Hani Shaker ◽

Muhammad Imran ◽

Wasim Sajjad

Keyword(s):

Wiener Index ◽

Face Centered Cubic ◽

Eccentric Connectivity Index ◽

Zagreb Index ◽

Cubic Lattice ◽

Chemical Graph ◽

Chemical Graph Theory ◽

Wide Range ◽

Unit Cells ◽

Face Centered

Abstract Chemical graph theory has become a prime gadget for mathematical chemistry due to its wide range of graph theoretical applications for solving molecular problems. A numerical quantity is named as topological index which explains the topological characteristics of a chemical graph. Recently face centered cubic lattice FCC(n) attracted large attention due to its prominent and distinguished properties. Mujahed and Nagy (2016, 2018) calculated the precise expression for Wiener index and hyper-Wiener index on rows of unit cells of FCC(n). In this paper, we present the ECI (eccentric-connectivity index), TCI (total-eccentricity index), CEI (connective eccentric index), and first eccentric Zagreb index of face centered cubic lattice.

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Maximally edge-connected graphs and Zeroth-order general Randić index for $$\alpha \le -1$$ α ≤ - 1

Journal of Combinatorial Optimization ◽

10.1007/s10878-014-9728-y ◽

2014 ◽

Vol 31 (1) ◽

pp. 182-195 ◽

Cited By ~ 7

Author(s):

Guifu Su ◽

Liming Xiong ◽

Xiaofeng Su ◽

Guojun Li

Keyword(s):

Randić Index ◽

Zeroth Order ◽

Connected Graphs ◽

Randic Index ◽

General Randić Index ◽

General Randic Index

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Second order Randić index of phenylenes and their corresponding hexagonal squeezes

Journal of Mathematical Chemistry ◽

10.1007/s10910-006-9150-5 ◽

2006 ◽

Vol 42 (4) ◽

pp. 941-947 ◽

Cited By ~ 12

Author(s):

Jie Zhang ◽

Hanyuan Deng ◽

Shubo Chen

Keyword(s):

Second Order ◽

Randić Index ◽

Randic Index

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Ordering of alkane isomers by means of connectivity indices

Journal of the Serbian Chemical Society ◽

10.2298/jsc0202087g ◽

2002 ◽

Vol 67 (2) ◽

pp. 87-97 ◽

Cited By ~ 7

Author(s):

Ivan Gutman ◽

Dusica Vidovic ◽

Anka Nedic

Keyword(s):

Carbon Atom ◽

Organic Molecule ◽

Molecular Graph ◽

Connectivity Index ◽

Randić Index ◽

Randic Index ◽

Connectivity Indices ◽

The Difference

The connectivity index of an organic molecule whose molecular graph is Gis defined as C(?)=?(?u?v)??where ?u is the degree of the vertex u in G, where the summation goes over all pairs of adjacent vertices of G and where ? is a pertinently chosen exponent. The usual value of ? is ?1/2, in which case ?=C(?1/2) is referred to as the Randic index. The ordering of isomeric alkanes according to ??follows the extent of branching of the carbon-atom skeleton. We now study the ordering of the constitutional isomers of alkanes with 6 through 10 carbon atoms with respect to C(?) for various values of the parameter ?. This ordering significantly depends on ?. The difference between the orderings with respect to ??and with respect to C(?) is measured by a function ??and the ?-dependence of ??was established.

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