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In this study, we first introduce polygonal cylinder and torus using Cartesian products and topologically identifications and then find their Wiener and hyper-Wiener indices using a quick, interesting technique of counting. Our suggested mathematical structures could be of potential interests in representation of computer networks and enhancing lattice hardware security.
Zhi-Ba Peng; Abdul Rauf Nizami; Zaffar Iqbal; Muhammad Mobeen Munir; Hafiz Muhammad Waqar Ahmed; Jia-Bao Liu. Wiener and Hyper-Wiener Indices of Polygonal Cylinder and Torus. Complexity 2021, 2021, 1 -15.
AMA StyleZhi-Ba Peng, Abdul Rauf Nizami, Zaffar Iqbal, Muhammad Mobeen Munir, Hafiz Muhammad Waqar Ahmed, Jia-Bao Liu. Wiener and Hyper-Wiener Indices of Polygonal Cylinder and Torus. Complexity. 2021; 2021 ():1-15.
Chicago/Turabian StyleZhi-Ba Peng; Abdul Rauf Nizami; Zaffar Iqbal; Muhammad Mobeen Munir; Hafiz Muhammad Waqar Ahmed; Jia-Bao Liu. 2021. "Wiener and Hyper-Wiener Indices of Polygonal Cylinder and Torus." Complexity 2021, no. : 1-15.
Sierpinski networks are networks of fractal nature having several applications in computer science, music, chemistry, and mathematics. These networks are commonly used in chaos, fractals, recursive sequences, and complex systems. In this article, we compute various connectivity polynomials such as M -polynomial, Zagreb polynomials, and forgotten polynomial of generalized Sierpinski networks S k n and recover some well-known degree-based topological indices from these. We also compute the most general Zagreb index known as α , β -Zagreb index and several other general indices of similar nature for this network. Our results are the natural generalizations of already available results for particular classes of such type of networks.
Chengmei Fan; M. Mobeen Munir; Zafar Hussain; Muhammad Athar; Jia-Bao Liu. Polynomials and General Degree-Based Topological Indices of Generalized Sierpinski Networks. Complexity 2021, 2021, 1 -10.
AMA StyleChengmei Fan, M. Mobeen Munir, Zafar Hussain, Muhammad Athar, Jia-Bao Liu. Polynomials and General Degree-Based Topological Indices of Generalized Sierpinski Networks. Complexity. 2021; 2021 ():1-10.
Chicago/Turabian StyleChengmei Fan; M. Mobeen Munir; Zafar Hussain; Muhammad Athar; Jia-Bao Liu. 2021. "Polynomials and General Degree-Based Topological Indices of Generalized Sierpinski Networks." Complexity 2021, no. : 1-10.
Molecular topology is a key to determine the physiochemical aspects of the molecular structure. To determine the degree of irregularity of a certain molecular structure or a network has been a key source of interest for the molecular topologists as it provides an insight of the key features that are used to guess the properties of the structures. In this article, we are interested in formulating closed forms of irregularity measures of some popular benzenoid systems, namely, zigzag, rhombic, and honeycomb benzenoid systems. We also compared our results graphically and concluded that benzenoid system among above is more asymmetric than the others.
Zafar Hussain; Yujun Yang; Mobeen Munir; Zahid Hussain; Muhammad Athar; Ali Ahmed; Haseeb Ahmad. Analysis of irregularity measures of zigzag, rhombic, and honeycomb benzenoid systems. Open Physics 2020, 18, 1146 -1153.
AMA StyleZafar Hussain, Yujun Yang, Mobeen Munir, Zahid Hussain, Muhammad Athar, Ali Ahmed, Haseeb Ahmad. Analysis of irregularity measures of zigzag, rhombic, and honeycomb benzenoid systems. Open Physics. 2020; 18 (1):1146-1153.
Chicago/Turabian StyleZafar Hussain; Yujun Yang; Mobeen Munir; Zahid Hussain; Muhammad Athar; Ali Ahmed; Haseeb Ahmad. 2020. "Analysis of irregularity measures of zigzag, rhombic, and honeycomb benzenoid systems." Open Physics 18, no. 1: 1146-1153.
Resolving set and metric basis has become an integral part in combinatorial chemistry and molecular topology. It has a lot of applications in computer, chemistry, pharmacy and mathematical disciplines. A subset S of the vertex set V of a connected graph G resolves G if all vertices of G have different representations with respect to S. A metric basis for G is a resolving set having minimum cardinal number and this cardinal number is called the metric dimension of G. In present work, we find a metric basis and also metric dimension of 1-pentagonal carbon nanocones. We conclude that only three vertices are minimal requirement for the unique identification of all vertices in this network.
Zafar Hussain; Mobeen Munir; Ashfaq Ahmad; Maqbool Chaudhary; Junaid Alam Khan; Imtiaz Ahmed. Metric basis and metric dimension of 1-pentagonal carbon nanocone networks. Scientific Reports 2020, 10, 1 -7.
AMA StyleZafar Hussain, Mobeen Munir, Ashfaq Ahmad, Maqbool Chaudhary, Junaid Alam Khan, Imtiaz Ahmed. Metric basis and metric dimension of 1-pentagonal carbon nanocone networks. Scientific Reports. 2020; 10 (1):1-7.
Chicago/Turabian StyleZafar Hussain; Mobeen Munir; Ashfaq Ahmad; Maqbool Chaudhary; Junaid Alam Khan; Imtiaz Ahmed. 2020. "Metric basis and metric dimension of 1-pentagonal carbon nanocone networks." Scientific Reports 10, no. 1: 1-7.
For a graph G , an ordered set S ⊆ V G is called the resolving set of G , if the vector of distances to the vertices in S is distinct for every v ∈ V G . The minimum cardinality of S is termed as the metric dimension of G . S is called a fault-tolerant resolving set (FTRS) for G , if S \ v is still the resolving set ∀ v ∈ V G . The minimum cardinality of such a set is the fault-tolerant metric dimension (FTMD) of G . Due to enormous application in science such as mathematics and computer, the notion of the resolving set is being widely studied. In the present article, we focus on determining the FTMD of a generalized wheel graph. Moreover, a formula is developed for FTMD of a wheel and generalized wheels. Recently, some bounds of the FTMD of some of the convex polytopes have been computed, but here we come up with the exact values of the FTMD of two families of convex polytopes denoted as D k for k ≥ 4 and Q k for k ≥ 6 . We prove that these families of convex polytopes have constant FTMD. This brings us to pose a natural open problem about the existence of a polytope having nonconstant FTMD.
Zhi-Bo Zheng; Ashfaq Ahmad; Zaffar Hussain; Mobeen Munir; Muhammad Imran Qureshi; Imtiaz Ali; Jia-Bao Liu. Fault-Tolerant Metric Dimension of Generalized Wheels and Convex Polytopes. Mathematical Problems in Engineering 2020, 2020, 1 -8.
AMA StyleZhi-Bo Zheng, Ashfaq Ahmad, Zaffar Hussain, Mobeen Munir, Muhammad Imran Qureshi, Imtiaz Ali, Jia-Bao Liu. Fault-Tolerant Metric Dimension of Generalized Wheels and Convex Polytopes. Mathematical Problems in Engineering. 2020; 2020 ():1-8.
Chicago/Turabian StyleZhi-Bo Zheng; Ashfaq Ahmad; Zaffar Hussain; Mobeen Munir; Muhammad Imran Qureshi; Imtiaz Ali; Jia-Bao Liu. 2020. "Fault-Tolerant Metric Dimension of Generalized Wheels and Convex Polytopes." Mathematical Problems in Engineering 2020, no. : 1-8.
The concept of minimum resolving set for a connected graph has played a vital role in Robotic navigation, networking, and in computer sciences. In this article, we investigate the values of m and n for which P 2 ⊗ m P n and P 2 ⊗ m C n are connected and find metric dimension in this case. We also conclude that, for each m, we obtain a new regular family of constant metric dimension. We also give a basis for these graphs and presentation of resolving vector in general closed form with respect to the basis.
Song Li; Jia-Bao Liu; Mobeen Munir. On the Metric Dimension of Generalized Tensor Product of Interval with Paths and Cycles. Journal of Mathematics 2020, 2020, 1 -6.
AMA StyleSong Li, Jia-Bao Liu, Mobeen Munir. On the Metric Dimension of Generalized Tensor Product of Interval with Paths and Cycles. Journal of Mathematics. 2020; 2020 ():1-6.
Chicago/Turabian StyleSong Li; Jia-Bao Liu; Mobeen Munir. 2020. "On the Metric Dimension of Generalized Tensor Product of Interval with Paths and Cycles." Journal of Mathematics 2020, no. : 1-6.
Affine monoids are the considered as natural discrete analogues of the finitely generated cones. The interconnection between these two objects has been an active area of research since last decade. Star network is one of the most common in computer network topologies. In this work, we study star topology S n and associate a Coxeter structure of affine type on it. We find a recurrence relation and the Hilbert series of the associated right-angled monoid M S n ∞ . We observe that the growth rate of the monoid M S n ∞ is unbounded.
Jiang-Hua Tang; Zaffar Iqbal; Abdul Rauf Nizami; Mobeen Munir; Faiza Azam; Jia-Bao Liu. Recurrence Relations and Hilbert Series of the Monoid Associated with Star Topology. Journal of Mathematics 2020, 2020, 1 -6.
AMA StyleJiang-Hua Tang, Zaffar Iqbal, Abdul Rauf Nizami, Mobeen Munir, Faiza Azam, Jia-Bao Liu. Recurrence Relations and Hilbert Series of the Monoid Associated with Star Topology. Journal of Mathematics. 2020; 2020 ():1-6.
Chicago/Turabian StyleJiang-Hua Tang; Zaffar Iqbal; Abdul Rauf Nizami; Mobeen Munir; Faiza Azam; Jia-Bao Liu. 2020. "Recurrence Relations and Hilbert Series of the Monoid Associated with Star Topology." Journal of Mathematics 2020, no. : 1-6.
Dendrimers are highly branched organic macromolecules with successive layers of branch units surrounding a central core. Some properties like toxicity, entropy, and heats of vaporization of these dendrimers can be forecasted using topological indices. The present article is devoted to study of irregularity indices of three well-known classes of dendrimers, namely, nanostar dendrimer D[p], fullerene dendrimer NS4[p], and polymer dendrimerNS5[p], where p is the step size. We also see the relation of irregularity of these dendrimers on the step size graphically.
Xie Qing; Zhen Wang; Mobeen Munir; Haseeb Ahmad. Molecular Irregularity Indices of Nanostar, Fullerene, and Polymer Dendrimers. Journal of Chemistry 2020, 2020, 1 -12.
AMA StyleXie Qing, Zhen Wang, Mobeen Munir, Haseeb Ahmad. Molecular Irregularity Indices of Nanostar, Fullerene, and Polymer Dendrimers. Journal of Chemistry. 2020; 2020 ():1-12.
Chicago/Turabian StyleXie Qing; Zhen Wang; Mobeen Munir; Haseeb Ahmad. 2020. "Molecular Irregularity Indices of Nanostar, Fullerene, and Polymer Dendrimers." Journal of Chemistry 2020, no. : 1-12.
Zheng-Qing Chu; Mobeen Munir; Amina Yousaf; Muhammad Imran Qureshi; Jia-Bao Liu. Laplacian and signless laplacian spectra and energies of multi-step wheels. Mathematical Biosciences and Engineering 2020, 17, 3649 -3659.
AMA StyleZheng-Qing Chu, Mobeen Munir, Amina Yousaf, Muhammad Imran Qureshi, Jia-Bao Liu. Laplacian and signless laplacian spectra and energies of multi-step wheels. Mathematical Biosciences and Engineering. 2020; 17 (4):3649-3659.
Chicago/Turabian StyleZheng-Qing Chu; Mobeen Munir; Amina Yousaf; Muhammad Imran Qureshi; Jia-Bao Liu. 2020. "Laplacian and signless laplacian spectra and energies of multi-step wheels." Mathematical Biosciences and Engineering 17, no. 4: 3649-3659.
Zafar Hussain; Mobeen Munir; Waqas Nazeer; Muhammad Shoaib Saleem; Shin Min Kang; Young Chel Kwun. Computational aspects of line graph of carbon nanocones. Journal of the National Science Foundation of Sri Lanka 2019, 47, 435 .
AMA StyleZafar Hussain, Mobeen Munir, Waqas Nazeer, Muhammad Shoaib Saleem, Shin Min Kang, Young Chel Kwun. Computational aspects of line graph of carbon nanocones. Journal of the National Science Foundation of Sri Lanka. 2019; 47 (4):435.
Chicago/Turabian StyleZafar Hussain; Mobeen Munir; Waqas Nazeer; Muhammad Shoaib Saleem; Shin Min Kang; Young Chel Kwun. 2019. "Computational aspects of line graph of carbon nanocones." Journal of the National Science Foundation of Sri Lanka 47, no. 4: 435.
Classical applications of resolving sets and metric dimension can be observed in robot navigation, networking and pharmacy. In the present article, a formula for computing the metric dimension of a simple graph wihtout singleton twins is given. A sufficient condition for the graph to have the exchange property for resolving sets is found. Consequently, every minimal resolving set in the graph forms a basis for a matriod in the context of independence defined by Boutin [Determining sets, resolving set and the exchange property, Graphs Combin., 2009, 25, 789-806]. Also, a new way to define a matroid on finite ground is deduced. It is proved that the matroid is strongly base orderable and hence satisfies the conjecture of White [An unique exchange property for bases, Linear Algebra Appl., 1980, 31, 81-91]. As an application, it is shown that the power graphs of some finite groups can define a matroid. Moreover, we also compute the metric dimension of the power graphs of dihedral groups.
Ghulam Abbas; Usman Ali; Mobeen Munir; Syed Ahtsham Ul Haq Bokhary; Shin Min Kang. Power graphs and exchange property for resolving sets. Open Mathematics 2019, 17, 1303 -1309.
AMA StyleGhulam Abbas, Usman Ali, Mobeen Munir, Syed Ahtsham Ul Haq Bokhary, Shin Min Kang. Power graphs and exchange property for resolving sets. Open Mathematics. 2019; 17 (1):1303-1309.
Chicago/Turabian StyleGhulam Abbas; Usman Ali; Mobeen Munir; Syed Ahtsham Ul Haq Bokhary; Shin Min Kang. 2019. "Power graphs and exchange property for resolving sets." Open Mathematics 17, no. 1: 1303-1309.
. Molecular topology provides a basis for the correlation of physical as well as chemical properties of a certain molecule. Irregularity indices are used as functions in the statistical analysis of the topological properties of certain molecular graphs and complex networks, and hence help us to correlate properties like enthalpy, heats of vaporization, and boiling points etc. with the molecular structure. In this article we are interested in formulating closed forms of imbalance-based irregularity measures of boron nanotubes. These tubes are known as α-boron nanotube, triangular boron nanotubes, and tri-hexagonal boron nanotubes. We also compare our results graphically and come up with the conclusion that alpha boron tubes are the most irregular with respect to most of the irregularity indices.
Bin Yang; Mobeen Munir; Shazia Rafique; Haseeb Ahmad; Jia-Bao Liu. Computational Analysis of Imbalance-Based Irregularity Indices of Boron Nanotubes. Processes 2019, 7, 678 .
AMA StyleBin Yang, Mobeen Munir, Shazia Rafique, Haseeb Ahmad, Jia-Bao Liu. Computational Analysis of Imbalance-Based Irregularity Indices of Boron Nanotubes. Processes. 2019; 7 (10):678.
Chicago/Turabian StyleBin Yang; Mobeen Munir; Shazia Rafique; Haseeb Ahmad; Jia-Bao Liu. 2019. "Computational Analysis of Imbalance-Based Irregularity Indices of Boron Nanotubes." Processes 7, no. 10: 678.
Dendrimers are branched organic macromolecules with successive layers of branch units surrounding a central core. The molecular topology and the irregularity of their structure plays a central role in determining structural properties like enthalpy and entropy. Irregularity indices which are based on the imbalance of edges are determined for the molecular graphs associated with some general classes of dendrimers. We also provide graphical analysis of these indices for the above said classes of dendrimers.
Zafar Hussain; Mobeen Munir; Shazia Rafique; Tayyab Hussnain; Haseeb Ahmad; Young Chel Kwun; Shin Min Kang. Imbalance-Based Irregularity Molecular Descriptors of Nanostar Dendrimers. Processes 2019, 7, 517 .
AMA StyleZafar Hussain, Mobeen Munir, Shazia Rafique, Tayyab Hussnain, Haseeb Ahmad, Young Chel Kwun, Shin Min Kang. Imbalance-Based Irregularity Molecular Descriptors of Nanostar Dendrimers. Processes. 2019; 7 (8):517.
Chicago/Turabian StyleZafar Hussain; Mobeen Munir; Shazia Rafique; Tayyab Hussnain; Haseeb Ahmad; Young Chel Kwun; Shin Min Kang. 2019. "Imbalance-Based Irregularity Molecular Descriptors of Nanostar Dendrimers." Processes 7, no. 8: 517.
Determining the degree of irregularity of a certain molecular structure or a network has been a key source of interest for molecular topologists, but it is also important as it provides an insight into the key features used to guess properties of the structures. In this article, we are interested in formulating closed forms of irregularity measures of some popular benzenoid systems, such as hourglass H (m, n), jagged-rectangular J (m, n), and triangular benzenoid T (m, n) systems. We also compared our results graphically and concluded which benzenoid system among the above listed is more irregular than the others.
Zafar Hussain; Shazia Rafique; Mobeen Munir; Muhammad Athar; Maqbool Chaudhary; Haseeb Ahmad; Shin Min Kang. Irregularity Molecular Descriptors of Hourglass, Jagged-Rectangle, and Triangular Benzenoid Systems. Processes 2019, 7, 413 .
AMA StyleZafar Hussain, Shazia Rafique, Mobeen Munir, Muhammad Athar, Maqbool Chaudhary, Haseeb Ahmad, Shin Min Kang. Irregularity Molecular Descriptors of Hourglass, Jagged-Rectangle, and Triangular Benzenoid Systems. Processes. 2019; 7 (7):413.
Chicago/Turabian StyleZafar Hussain; Shazia Rafique; Mobeen Munir; Muhammad Athar; Maqbool Chaudhary; Haseeb Ahmad; Shin Min Kang. 2019. "Irregularity Molecular Descriptors of Hourglass, Jagged-Rectangle, and Triangular Benzenoid Systems." Processes 7, no. 7: 413.
This paper is concerned with the combinatorial facts of the lattice graphs of Z p 1 × p 2 × ⋯ × p m , Z p 1 m 1 × p 2 m 2 , and Z p 1 m 1 × p 2 m 2 × p 3 1 . We show that the lattice graph of Z p 1 × p 2 × ⋯ × p m is realizable as a convex polytope. We also show that the diameter of the lattice graph of Z p 1 m 1 × p 2 m 2 × ⋯ × p r m r is ∑ i = 1 r m i and its girth is 4.
Jia-Bao Liu; Mobeen Munir; Qurat-Ul-Ain Munir; Abdul Rauf Nizami. Some Metrical Properties of Lattice Graphs of Finite Groups. Mathematics 2019, 7, 398 .
AMA StyleJia-Bao Liu, Mobeen Munir, Qurat-Ul-Ain Munir, Abdul Rauf Nizami. Some Metrical Properties of Lattice Graphs of Finite Groups. Mathematics. 2019; 7 (5):398.
Chicago/Turabian StyleJia-Bao Liu; Mobeen Munir; Qurat-Ul-Ain Munir; Abdul Rauf Nizami. 2019. "Some Metrical Properties of Lattice Graphs of Finite Groups." Mathematics 7, no. 5: 398.
HCV genes interfere with host cellular genes and play crucial role in pathogenesis. The mechanism under which HCV genes induce insulin resistance is not much clear. This study is aimed to examine the role of HCV NS5A in inducing insulin resistance by examining its affect in the phosphorylation level of AKT/PKB. In the present study, HepG2 cells were transfected with HCV NS5A and after 24 hours of transfection, protein was extracted from cells that were pre induced with insulin at three different time intervals i.e. 1hour, 2 hours and 3hours. Dot Blot analysis was performed to study the phosphorylation level of AKT. Results showed that there is clear upregulation of serine 473 phosphorylation level of AKT in NS5A transfected cells as compared with control (without NS5A). In conclusion, upregulation of serine 473 phosphorylation by NS5A of HCV genotype 3a suggests that this gene impairs the normal insulin AKT/PKB signaling pathway that leads towards insulin resistance and Type 2 diabetes mellitus. Therefore, HCV non-structural protein NS5A should be considered as promising candidate to be studied in detail for HCV induced insulin resistance and should be regarded as a therapeutically important target for the prevention of chronic liver diseases.
Faiza Shams; Shazia Rafique; Sadia Zahid; Mobeen Munir; Muhammad Idrees; Muhammad Ilyas; Tayyab Husnain. Advances in the role of HCV nonstructural protein 5a (NS5A) of 3a genotype in inducing insulin resistance by possible phosphorylation of AKT/PKB. Scientific Reports 2019, 9, 6150 .
AMA StyleFaiza Shams, Shazia Rafique, Sadia Zahid, Mobeen Munir, Muhammad Idrees, Muhammad Ilyas, Tayyab Husnain. Advances in the role of HCV nonstructural protein 5a (NS5A) of 3a genotype in inducing insulin resistance by possible phosphorylation of AKT/PKB. Scientific Reports. 2019; 9 (1):6150.
Chicago/Turabian StyleFaiza Shams; Shazia Rafique; Sadia Zahid; Mobeen Munir; Muhammad Idrees; Muhammad Ilyas; Tayyab Husnain. 2019. "Advances in the role of HCV nonstructural protein 5a (NS5A) of 3a genotype in inducing insulin resistance by possible phosphorylation of AKT/PKB." Scientific Reports 9, no. 1: 6150.
Oxide networks have diverse applications in the polymer and pharmaceutical industries. Polynomials and degree-based topological indices have tendencies to correlate properties of molecular graphs. In this article, we formulate the closed forms of Zagreb and forgotten polynomials and topological indices such as Hyper-Zagreb index, first and second multiple Zagreb indices, forgotten index, Albert index, Bell index, IRM(G) of oxide networks. We also compute the F-index of complement of oxide networks, F-coindex of G and F-coindex of complement of oxide networks. We put graphical analysis of each index with respect to the parameter involved in each case.
Zafar Hussain; Mobeen Munir; Muhammad Bilal; Alam Ameer; Shazia Rafique; Shin Min Kang. Computational Analysis of new Degree-based descriptors of oxide networks. Open Chemistry 2019, 17, 177 -182.
AMA StyleZafar Hussain, Mobeen Munir, Muhammad Bilal, Alam Ameer, Shazia Rafique, Shin Min Kang. Computational Analysis of new Degree-based descriptors of oxide networks. Open Chemistry. 2019; 17 (1):177-182.
Chicago/Turabian StyleZafar Hussain; Mobeen Munir; Muhammad Bilal; Alam Ameer; Shazia Rafique; Shin Min Kang. 2019. "Computational Analysis of new Degree-based descriptors of oxide networks." Open Chemistry 17, no. 1: 177-182.
Concept of resolving set and metric basis has enjoyed a lot of success because of multipurpose applications both in computer and mathematical sciences. A system in which failure of any single unit, another chain of units not containing the faulty unit can replace the originally used chain is called fault-tolerant self-stable system. Recent research reveal that the problem of finding metric dimension is NP-hard and problem of computing the exact values of fault tolerant metric dimension seems to be even harder although some bounds can be computed rather easily. In the present article we compute closed formulas for the fault-tolerant metric dimension of gear, anti-web gear and anti-web graphs. We conclude that out of these only anti-web graph has constant fault-tolerant metric dimension.
Jia-Bao Liu; Mobeen Munir; Imtiaz Ali; Zaffar Hussain; Ashfaq Ahmed. Fault-Tolerant Metric Dimension of Wheel related Graphs. 2019, 1 .
AMA StyleJia-Bao Liu, Mobeen Munir, Imtiaz Ali, Zaffar Hussain, Ashfaq Ahmed. Fault-Tolerant Metric Dimension of Wheel related Graphs. . 2019; ():1.
Chicago/Turabian StyleJia-Bao Liu; Mobeen Munir; Imtiaz Ali; Zaffar Hussain; Ashfaq Ahmed. 2019. "Fault-Tolerant Metric Dimension of Wheel related Graphs." , no. : 1.
Stanley depth is a geometric invariant of the module and is related to an algebraic invariant called depth of the module. We compute Stanley depth of the quotient of edge ideals associated with some familiar families of wheel-related graphs. In particular, we establish general closed formulas for Stanley depth of quotient of edge ideals associated with the m t h -power of a wheel graph, for m ≥ 3 , gear graphs and anti-web gear graphs.
Jia-Bao Liu; Mobeen Munir; Raheel Farooki; Muhammad Imran Qureshi; Quratulien Muneer. Stanley Depth of Edge Ideals of Some Wheel-Related Graphs. Mathematics 2019, 7, 202 .
AMA StyleJia-Bao Liu, Mobeen Munir, Raheel Farooki, Muhammad Imran Qureshi, Quratulien Muneer. Stanley Depth of Edge Ideals of Some Wheel-Related Graphs. Mathematics. 2019; 7 (2):202.
Chicago/Turabian StyleJia-Bao Liu; Mobeen Munir; Raheel Farooki; Muhammad Imran Qureshi; Quratulien Muneer. 2019. "Stanley Depth of Edge Ideals of Some Wheel-Related Graphs." Mathematics 7, no. 2: 202.
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 wheel graphs. We also give pictorial dependencies of energies on the involved parameters m ≥ 3 and n .
Jia-Bao Liu; Mobeen Munir; Amina Yousaf; Asim Naseem; Khudaija Ayub. Distance and Adjacency Energies of Multi-Level Wheel Networks. Mathematics 2019, 7, 43 .
AMA StyleJia-Bao Liu, Mobeen Munir, Amina Yousaf, Asim Naseem, Khudaija Ayub. Distance and Adjacency Energies of Multi-Level Wheel Networks. Mathematics. 2019; 7 (1):43.
Chicago/Turabian StyleJia-Bao Liu; Mobeen Munir; Amina Yousaf; Asim Naseem; Khudaija Ayub. 2019. "Distance and Adjacency Energies of Multi-Level Wheel Networks." Mathematics 7, no. 1: 43.