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A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (28 February 2019) | Viewed by 67527

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Special Issue Editor

Prof. Dr. Frank Werner

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Guest Editor

Faculty of Mathematics, Otto-von-Guericke-University, P.O. Box 4120, D-39016 Magdeburg, Germany
Interests: scheduling; development of exact and approximate algorithms; stability investigations; discrete optimization; scheduling with interval processing times; complex investigations for scheduling problems; train scheduling; graph theory; logistics; supply chains; packing; simulation; applications
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

We invite you to submit your latest research in the area of discrete optimization to this Special Issue, “Discrete Optimization: Theory, Algorithms, and Applications” in the journal Mathematics. We are looking for new and innovative approaches for solving discrete optimization problems exactly or approximately. High-quality papers are solicited to address both theoretical and practical issues of discrete optimization. Submissions are welcome presenting new theoretical results, structural investigations, new models and algorithmic approaches as well new applications of discrete optimization problems. Potential topics include, but are not limited to, integer programming, combinatorial optimization, graph-theoretic problems, matroids, scheduling, and logistics.

Prof. Dr. Frank Werner
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

Nonlinear and Linear Integer Programming
Optimization on Graphs and Networks
Greedy Algorithms, Matroids and Submodular Functions
Polyhedral Combinatorics
Combinatorial Optimization
Scheduling
Robust Discrete Optimization
Optimization under Uncertainty
Computational Complexity
Branch and Bound, Cutting-Plane Methods, Dynamic Programming
Approximation and Randomized Algorithms
Metaheuristics, Matheuristics
Interior Point Methods
Decomposition Methods

Published Papers (19 papers)

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Editorial

Jump to: Research

4 pages, 170 KiB

Open AccessEditorial

Discrete Optimization: Theory, Algorithms, and Applications

by Frank Werner

Mathematics 2019, 7(5), 397; https://doi.org/10.3390/math7050397 - 01 May 2019

Cited by 1 | Viewed by 3357

Abstract

Discrete optimization is an important area of applied mathematics that is at the intersection of several disciplines and covers both theoretical and practical aspects [...] Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

Research

Jump to: Editorial

21 pages, 350 KiB

Open AccessFeature PaperArticle

The Optimality Region for a Single-Machine Scheduling Problem with Bounded Durations of the Jobs and the Total Completion Time Objective

by Yuri N. Sotskov and Natalja G. Egorova

Mathematics 2019, 7(5), 382; https://doi.org/10.3390/math7050382 - 26 Apr 2019

Cited by 7 | Viewed by 2338

Abstract

We study a single-machine scheduling problem to minimize the total completion time of the given set of jobs, which have to be processed without job preemptions. The lower and upper bounds on the job duration is the only information that is available before scheduling. Exact values of the job durations remain unknown until the completion of the jobs. We use the optimality region for the job permutation as an optimality measure of the optimal schedule. We investigate properties of the optimality region and derive

O (n)

-algorithm for calculating a quasi-perimeter of the optimality set (i.e., the sum of lengths of the optimality segments for n given jobs). We develop a fast algorithm for finding a job permutation having the largest quasi-perimeter of the optimality set. The computational results in constructing such permutations show that they are close to the optimal ones, which can be constructed for the factual durations of all given jobs. Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

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11 pages, 299 KiB

Open AccessArticle

Two-Machine Job-Shop Scheduling with Equal Processing Times on Each Machine

by Evgeny Gafarov and Frank Werner

Mathematics 2019, 7(3), 301; https://doi.org/10.3390/math7030301 - 25 Mar 2019

Cited by 8 | Viewed by 3596

Abstract

In this paper, we consider a two-machine job-shop scheduling problem of minimizing total completion time subject to n jobs with two operations and equal processing times on each machine. This problem occurs e.g., as a single-track railway scheduling problem with three stations and [...] Read more.

O (n^{5})

and a heuristic procedure of the complexity

O (n^{3})

. This settles the complexity status of the problem under consideration which was open before and extends earlier work for the two-station single-track railway scheduling problem. We also present computational results of the comparison of both algorithms. For the 30,000 instances with up to 30 jobs considered, the average relative error of the heuristic is less than

1 %

. In our tests, the practical running time of the dynamic programming algorithm was even bounded by

O (n^{4})

. Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

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13 pages, 332 KiB

Open AccessArticle

Further Results on the Resistance-Harary Index of Unicyclic Graphs

by Jian Lu, Shu-Bo Chen, Jia-Bao Liu, Xiang-Feng Pan and Ying-Jie Ji

Mathematics 2019, 7(2), 201; https://doi.org/10.3390/math7020201 - 20 Feb 2019

Cited by 1 | Viewed by 3418

Abstract

The Resistance-Harary index of a connected graph G is defined as

R H (G) = \sum_{{u, v} \subseteq V (G)} \frac{1}{r (u, v)}

, where [...] Read more.

The Resistance-Harary index of a connected graph G is defined as

R H (G) = \sum_{{u, v} \subseteq V (G)} \frac{1}{r (u, v)}

, where

r (u, v)

is the resistance distance between vertices u and v in G. A graph G is called a unicyclic graph if it contains exactly one cycle and a fully loaded unicyclic graph is a unicyclic graph that no vertex with degree less than three in its unique cycle. Let

U (n)

and

U (n)

be the set of unicyclic graphs and fully loaded unicyclic graphs of order n, respectively. In this paper, we determine the graphs of

U (n)

with second-largest Resistance-Harary index and determine the graphs of

U (n)

with largest Resistance-Harary index. Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

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10 pages, 275 KiB

Open AccessArticle

Fractional Metric Dimension of Generalized Jahangir Graph

by Jia-Bao Liu, Agha Kashif, Tabasam Rashid and Muhammad Javaid

Mathematics 2019, 7(1), 100; https://doi.org/10.3390/math7010100 - 18 Jan 2019

Cited by 57 | Viewed by 3389

Abstract

Arumugam and Mathew [Discret. Math. 2012, 312, 1584–1590] introduced the notion of fractional metric dimension of a connected graph. In this paper, a combinatorial technique is devised to compute it. In addition, using this technique the fractional metric dimension of [...] Read more.

J_{m, k}

is computed for

k \geq 0

and

m = 5

. Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

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19 pages, 376 KiB

Open AccessArticle

Fault-Tolerant Resolvability and Extremal Structures of Graphs

by Hassan Raza, Sakander Hayat, Muhammad Imran and Xiang-Feng Pan

Mathematics 2019, 7(1), 78; https://doi.org/10.3390/math7010078 - 14 Jan 2019

Cited by 51 | Viewed by 4334

Abstract

In this paper, we consider fault-tolerant resolving sets in graphs. We characterize n-vertex graphs with fault-tolerant metric dimension n,

n - 1

, and 2, which are the lower and upper extremal cases. Furthermore, in the first part of the paper, [...] Read more.

In this paper, we consider fault-tolerant resolving sets in graphs. We characterize n-vertex graphs with fault-tolerant metric dimension n,

n - 1

, and 2, which are the lower and upper extremal cases. Furthermore, in the first part of the paper, a method is presented to locate fault-tolerant resolving sets by using classical resolving sets in graphs. The second part of the paper applies the proposed method to three infinite families of regular graphs and locates certain fault-tolerant resolving sets. By accumulating the obtained results with some known results in the literature, we present certain lower and upper bounds on the fault-tolerant metric dimension of these families of graphs. As a byproduct, it is shown that these families of graphs preserve a constant fault-tolerant resolvability structure. Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

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9 pages, 377 KiB

Open AccessArticle

Distance and Adjacency Energies of Multi-Level Wheel Networks

by Jia-Bao Liu, Mobeen Munir, Amina Yousaf, Asim Naseem and Khudaija Ayub

Mathematics 2019, 7(1), 43; https://doi.org/10.3390/math7010043 - 04 Jan 2019

Cited by 7 | Viewed by 3928

Abstract

Energies of molecular graphs have various applications in chemistry, polymerization, computer networking and pharmacy. In this paper, we give general closed forms of distance and adjacency energies of generalized wheel networks

W_{n, m}

. Consequently, we give these results for classical [...] Read more.

W_{n, m}

. Consequently, we give these results for classical wheel graphs. We also give pictorial dependencies of energies on the involved parameters

m \geq 3

and

n

. Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

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9 pages, 249 KiB

Open AccessArticle

Construction Algorithm for Zero Divisor Graphs of Finite Commutative Rings and Their Vertex-Based Eccentric Topological Indices

by Kashif Elahi, Ali Ahmad and Roslan Hasni

Mathematics 2018, 6(12), 301; https://doi.org/10.3390/math6120301 - 04 Dec 2018

Cited by 22 | Viewed by 3161

Abstract

Chemical graph theory is a branch of mathematical chemistry which deals with the non-trivial applications of graph theory to solve molecular problems. Graphs containing finite commutative rings also have wide applications in robotics, information and communication theory, elliptic curve cryptography, physics, and statistics. [...] Read more.

Z_{p_{1} p_{2}} \times Z_{q}

, where

p_{1}, p_{2}

, and q are primes. To enhance the importance of these indices a construction algorithm is also devised for zero divisor graphs of commutative rings

Z_{p_{1} p_{2}} \times Z_{q}

. Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

10 pages, 259 KiB

Open AccessArticle

Resistance Distance in H-Join of Graphs G₁,G₂,…,G_k

by Li Zhang, Jing Zhao, Jia-Bao Liu and Micheal Arockiaraj

Mathematics 2018, 6(12), 283; https://doi.org/10.3390/math6120283 - 26 Nov 2018

Cited by 5 | Viewed by 2766

Abstract

In view of the wide application of resistance distance, the computation of resistance distance in various graphs becomes one of the main topics. In this paper, we aim to compute resistance distance in H-join of graphs [...] Read more.

G_{1}, G_{2}, \dots, G_{k}

. Recall that H is an arbitrary graph with

V (H) = {1, 2, \dots, k}

, and

G_{1}, G_{2}, \dots, G_{k}

are disjoint graphs. Then, the H-join of graphs

G_{1}, G_{2}, \dots, G_{k}

, denoted by

⋁_{H} {G_{1}, G_{2}, \dots, G_{k}}

, is a graph formed by taking

G_{1}, G_{2}, \dots, G_{k}

and joining every vertex of

G_{i}

to every vertex of

G_{j}

whenever i is adjacent to j in H. Here, we first give the Laplacian matrix of

⋁_{H} {G_{1}, G_{2}, \dots, G_{k}}

, and then give a

{1}

-inverse

L {(⋁_{H} {G_{1}, G_{2}, \dots, G_{k}})}^{{1}}

or group inverse

L {(⋁_{H} {G_{1}, G_{2}, \dots, G_{k}})}^{#}

L (⋁_{H} {G_{1}, G_{2}, \dots, G_{k}})

. It is well know that, there exists a relationship between resistance distance and entries of

{1}

-inverse or group inverse. Therefore, we can easily obtain resistance distance in

⋁_{H} {G_{1}, G_{2}, \dots, G_{k}}

. In addition, some applications are presented in this paper. Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

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11 pages, 261 KiB

Open AccessArticle

The Extremal Graphs of Some Topological Indices with Given Vertex k-Partiteness

by Fang Gao, Xiaoxin Li, Kai Zhou and Jia-Bao Liu

Mathematics 2018, 6(11), 271; https://doi.org/10.3390/math6110271 - 21 Nov 2018

Cited by 1 | Viewed by 2226

Abstract

The vertex k-partiteness of graph G is defined as the fewest number of vertices whose deletion from G yields a k-partite graph. In this paper, we characterize the extremal value of the reformulated first Zagreb index, the multiplicative-sum Zagreb index, the general Laplacian-energy-like invariant, the general zeroth-order Randić index, and the modified-Wiener index among graphs of order n with vertex k-partiteness not more than

m

. Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

16 pages, 273 KiB

Open AccessArticle

Scheduling and Planning in Service Systems with Goal Programming: Literature Review

by Şeyda Gür and Tamer Eren

Mathematics 2018, 6(11), 265; https://doi.org/10.3390/math6110265 - 19 Nov 2018

Cited by 36 | Viewed by 5196

Abstract

Background: People want to be able to evaluate different kinds of information in a good way. There are various methods that they develop in such situations. Among the optimization methods, the goal programming method is often used when there are multiple objectives that decision makers want to accomplish. Because scheduling and planning problems have multiple objectives that are desired to be achieved, the goal programming method helps the researcher in contradictory situations between these goals. Methods: This study includes, examines, and analyzes recent research on service scheduling and planning. In the literature, service scheduling and planning studies have been examined using goal programming method from past to today. Results: The studies are detailed according to the type of goal programming, according to scheduling types, the purpose used in the studies, and the methods integrated with the goal programming. There are 142 studies in Emerald, Science Direct, Jstor, Springer, Taylor and Francis, Google Scholar, etc. databases that are examined in detail. For readers, diversification has been made to facilitate the identification of these studies and a detailed overview has been presented. Conclusion: As a result of the study, studies with the goal programming in the literature have been seen. The readers’ perspectives for planning and scheduling are discussed. Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

10 pages, 297 KiB

Open AccessArticle

Edge Version of Metric Dimension and Doubly Resolving Sets of the Necklace Graph

by Jia-Bao Liu, Zohaib Zahid, Ruby Nasir and Waqas Nazeer

Mathematics 2018, 6(11), 243; https://doi.org/10.3390/math6110243 - 07 Nov 2018

Cited by 35 | Viewed by 4014

Abstract

Consider an undirected and connected graph

G = (V_{G}, E_{G})

, where

V_{G}

and

E_{G}

represent the set of vertices and the set of edges respectively. The concept of edge version of metric dimension and doubly [...] Read more.

Consider an undirected and connected graph

G = (V_{G}, E_{G})

, where

V_{G}

and

E_{G}

represent the set of vertices and the set of edges respectively. The concept of edge version of metric dimension and doubly resolving sets is based on the distances of edges in a graph. In this paper, we find the edge version of metric dimension and doubly resolving sets for the necklace graph. Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

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10 pages, 607 KiB

Open AccessArticle

Maximizing and Minimizing Multiplicative Zagreb Indices of Graphs Subject to Given Number of Cut Edges

by Shaohui Wang, Chunxiang Wang, Lin Chen, Jia-Bao Liu and Zehui Shao

Mathematics 2018, 6(11), 227; https://doi.org/10.3390/math6110227 - 29 Oct 2018

Cited by 8 | Viewed by 2410

Abstract

Given a (molecular) graph, the first multiplicative Zagreb index

Π_{1}

is considered to be the product of squares of the degree of its vertices, while the second multiplicative Zagreb index

Π_{2}

is expressed as the product of endvertex degree of each [...] Read more.

Given a (molecular) graph, the first multiplicative Zagreb index

Π_{1}

is considered to be the product of squares of the degree of its vertices, while the second multiplicative Zagreb index

Π_{2}

is expressed as the product of endvertex degree of each edge over all edges. We consider a set of graphs

G_{n, k}

having n vertices and k cut edges, and explore the graphs subject to a number of cut edges. In addition, the maximum and minimum multiplicative Zagreb indices of graphs in

G_{n, k}

are provided. We also provide these graphs with the largest and smallest

Π_{1} (G)

and

Π_{2} (G)

G_{n, k}

. Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

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11 pages, 460 KiB

Open AccessArticle

The Double Roman Domination Numbers of Generalized Petersen Graphs P(n, 2)

by Huiqin Jiang, Pu Wu, Zehui Shao, Yongsheng Rao and Jia-Bao Liu

Mathematics 2018, 6(10), 206; https://doi.org/10.3390/math6100206 - 16 Oct 2018

Cited by 12 | Viewed by 2719

Abstract

A double Roman dominating function (DRDF) f on a given graph G is a mapping from

V (G)

{0, 1, 2, 3}

in such a way that a vertex u for which [...] Read more.

A double Roman dominating function (DRDF) f on a given graph G is a mapping from

V (G)

{0, 1, 2, 3}

in such a way that a vertex u for which

f (u) = 0

has at least a neighbor labeled 3 or two neighbors both labeled 2 and a vertex u for which

f (u) = 1

has at least a neighbor labeled 2 or 3. The weight of a DRDF f is the value

w (f) = \sum_{u \in V (G)} f (u)

. The minimum weight of a DRDF on a graph G is called the double Roman domination number

γ_{d R} (G)

of G. In this paper, we determine the exact value of the double Roman domination number of the generalized Petersen graphs

P (n, 2)

by using a discharging approach. Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

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16 pages, 924 KiB

Open AccessArticle

On Metric Dimensions of Symmetric Graphs Obtained by Rooted Product

by Shahid Imran, Muhammad Kamran Siddiqui, Muhammad Imran and Muhammad Hussain

Mathematics 2018, 6(10), 191; https://doi.org/10.3390/math6100191 - 08 Oct 2018

Cited by 12 | Viewed by 3766

Abstract

Let G = (V, E) be a connected graph and d(x, y) be the distance between the vertices x and y in G. A set of vertices W resolves a graph G if every vertex [...] Read more.

Let G = (V, E) be a connected graph and d(x, y) be the distance between the vertices x and y in G. A set of vertices W resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in W. A metric dimension of G is the minimum cardinality of a resolving set of G and is denoted by dim(G). In this paper, Cycle, Path, Harary graphs and their rooted product as well as their connectivity are studied and their metric dimension is calculated. It is proven that metric dimension of some graphs is unbounded while the other graphs are constant, having three or four dimensions in certain cases. Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

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16 pages, 499 KiB

Open AccessArticle

Optimizing Three-Dimensional Constrained Ordered Weighted Averaging Aggregation Problem with Bounded Variables

by Hui-Chin Tang and Shen-Tai Yang

Mathematics 2018, 6(9), 172; https://doi.org/10.3390/math6090172 - 19 Sep 2018

Cited by 3 | Viewed by 2481

Abstract

A single constrained ordered weighted averaging aggregation (COWA) problem is of considerable importance in many disciplines. Two models are considered: the maximization COWA problem with lower bounded variables and the minimization COWA problem with upper bounded variables. For a three-dimensional case of these models, we present the explicitly optimal solutions theoretically and empirically. The bounds and weights can affect the optimal solution of the three-dimensional COWA problem with bounded variables. Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

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14 pages, 281 KiB

Open AccessArticle

Computing The Irregularity Strength of Planar Graphs

by Hong Yang, Muhammad Kamran Siddiqui, Muhammad Ibrahim, Sarfraz Ahmad and Ali Ahmad

Mathematics 2018, 6(9), 150; https://doi.org/10.3390/math6090150 - 30 Aug 2018

Cited by 8 | Viewed by 3694

Abstract

The field of graph theory plays a vital role in various fields. One of the important areas in graph theory is graph labeling used in many applications such as coding theory, X-ray crystallography, radar, astronomy, circuit design, communication network addressing, and data base management. In this paper, we discuss the totally irregular total k labeling of three planar graphs. If such labeling exists for minimum value of a positive integer k, then this labeling is called totally irregular total k labeling and k is known as the total irregularity strength of a graph G. More preciously, we determine the exact value of the total irregularity strength of three planar graphs. Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

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10 pages, 4227 KiB

Open AccessArticle

Computing Topological Indices and Polynomials for Line Graphs

by Shahid Imran, Muhammad Kamran Siddiqui, Muhammad Imran and Muhammad Faisal Nadeem

Mathematics 2018, 6(8), 137; https://doi.org/10.3390/math6080137 - 10 Aug 2018

Cited by 11 | Viewed by 3292

Abstract

A topological index is a number related to the atomic index that allows quantitative structure–action/property/toxicity connections. All the more vital topological indices correspond to certain physico-concoction properties like breaking point, solidness, strain vitality, and so forth, of synthetic mixes. The idea of the hyper Zagreb index, multiple Zagreb indices and Zagreb polynomials was set up in the substance diagram hypothesis in light of vertex degrees. These indices are valuable in the investigation of calming exercises of certain compound systems. In this paper, we computed the first and second Zagreb index, the hyper Zagreb index, multiple Zagreb indices and Zagreb polynomials of the line graph of wheel and ladder graphs by utilizing the idea of subdivision. Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

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13 pages, 7628 KiB

Open AccessArticle

Eccentricity Based Topological Indices of an Oxide Network

by Muhammad Imran, Muhammad Kamran Siddiqui, Amna A. E. Abunamous, Dana Adi, Saida Hafsa Rafique and Abdul Qudair Baig

Mathematics 2018, 6(7), 126; https://doi.org/10.3390/math6070126 - 18 Jul 2018

Cited by 22 | Viewed by 5464

Abstract

Graph theory has much great advances in the field of mathematical chemistry. Chemical graph theory has become very popular among researchers because of its wide applications in mathematical chemistry. The molecular topological descriptors are the numerical invariants of a molecular graph and are very useful for predicting their bioactivity. A great variety of such indices are studied and used in theoretical chemistry, pharmaceutical researchers, in drugs and in different other fields. In this article, we study the chemical graph of an oxide network and compute the total eccentricity, average eccentricity, eccentricity based Zagreb indices, atom-bond connectivity (

A B C

) index and geometric arithmetic index of an oxide network. Furthermore, we give analytically closed formulas of these indices which are helpful in studying the underlying topologies. Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

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