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Papers by Muhammad Javaid

Research paper thumbnail of ON SUPER (a, d)-EAT VALUATION OF SUBDIVIDED CATERPILLAR

Let G = (V (G), E(G)) be a graph with v = |V (G)| vertices and e = |E(G)| edges. A bijective func... more Let G = (V (G), E(G)) be a graph with v = |V (G)| vertices and e = |E(G)| edges. A bijective function λ : V (G) ∪ E(G) ↔ {1, 2, . . . , v + e} is called an (a, d)edge antimagic total (EAT) labeling(valuation) if the weight of all the edges {w(xy) : xy ∈ E(G)} form an arithmetic sequence starting with first term a and having common difference d, where w(xy) = λ(x) + λ(y) + λ(xy). And, if λ(V ) = {1, 2, . . . , v} then G is super (a, d)-edge antimagic total(EAT) graph. In this paper, we determine the super (a,d)-edge antimagic total (EAT) labeling of the subdivided caterpillar for different values of the parameter d.

Research paper thumbnail of Topological indices of rhombus type silicate and oxide networks

Canadian Journal of Chemistry, 2017

For a molecular graph, a numeric quantity that characterizes the whole structure of a graph is ca... more For a molecular graph, a numeric quantity that characterizes the whole structure of a graph is called a topological index. In the studies of quantitative structure – activity relationship (QSAR) and quantitative structure – property relationship (QSPR), topological indices are utilized to guess the bioactivity of chemical compounds. In this paper, we compute general Randić, first general Zagreb, generalized Zagreb, multiplicative Zagreb, atom-bond connectivity (ABC), and geometric arithmetic (GA) indices for the rhombus silicate and rhombus oxide networks. In addition, we also compute the latest developed topological indices such as the fourth version of ABC (ABC4), the fifth version of GA (GA5), augmented Zagreb, and Sanskruti indices for the foresaid networks. At the end, a comparison between all the indices is included, and the result is shown with the help of a Cartesian coordinate system.

Research paper thumbnail of Topological Properties of Four-Layered Neural Networks

Journal of Artificial Intelligence and Soft Computing Research, 2019

A topological property or index of a network is a numeric number which characterises the whole st... more A topological property or index of a network is a numeric number which characterises the whole structure of the underlying network. It is used to predict the certain changes in the bio, chemical and physical activities of the networks. The 4-layered probabilistic neural networks are more general than the 3-layered probabilistic neural networks. Javaid and Cao [Neural Comput. and Applic., DOI 10.1007/s00521-017-2972-1] and Liu et al. [Journal of Artificial Intelligence and Soft Computing Research, 8(2018), 225-266] studied the certain degree and distance based topological indices (TI’s) of the 3-layered probabilistic neural networks. In this paper, we extend this study to the 4-layered probabilistic neural networks and compute the certain degree-based TI’s. In the end, a comparison between all the computed indices is included and it is also proved that the TI’s of the 4-layered probabilistic neural networks are better being strictly greater than the 3-layered probabilistic neural net...

Research paper thumbnail of Multiplicative Zagreb Indices of Molecular Graphs

Journal of Chemistry, 2019

Mathematical modeling with the help of numerical coding of graphs has been used in the different ... more Mathematical modeling with the help of numerical coding of graphs has been used in the different fields of science, especially in chemistry for the studies of the molecular structures. It also plays a vital role in the study of the quantitative structure activities relationship (QSAR) and quantitative structure properties relationship (QSPR) models. Todeshine et al. (2010) and Eliasi et al. (2012) defined two different versions of the 1st multiplicative Zagreb index as ∏Γ=∏p∈VΓdΓp2 and ∏1Γ=∏pq∈EΓdΓp+dΓq, respectively. In the same paper of Todeshine, they also defined the 2nd multiplicative Zagreb index as ∏2Γ=∏pq∈EΓdΓp×dΓq. Recently, Liu et al. [IEEE Access; 7(2019); 105479–-105488] defined the generalized subdivision-related operations of graphs and obtained the generalized F-sum graphs using these operations. They also computed the first and second Zagreb indices of the newly defined generalized F-sum graphs. In this paper, we extend this study and compute the upper bonds of the f...

Research paper thumbnail of M-polynomial method for topological indices of 2D-lattice of three-layered single-walled titania nanotubes

Journal of Information and Optimization Sciences, 2019

A molecular network can be uniquely identified by a number, polynomial or matrix. A topological i... more A molecular network can be uniquely identified by a number, polynomial or matrix. A topological index (TI) is a number that characterizes a molecular network completely which is used to predict the physical features of the certain changes such as bioactivities and chemical reactivities in the chemical compound. Javaid and Cao [Neural Comput. and Applic., 30(2018), 3869-3876] studied the first Zagreb index, second Zagreb index, general Randic index, and augmented Zagreb index for the 3-layered probabilistic neural networks (PNN). In this paper, we prove the M-polynomial of the 3-layered PNN and use it as a latest developed tool to compute the certain degree based TI's. At the end, a comparison is also shown to find the better one among all the obtained results.

Research paper thumbnail of Super Eat Labeling of Subdivided Stars

Journal of the Indonesian Mathematical Society, 2014

Kotzig and Rosa (1970) conjectured that every tree admits edge-magic total labeling. Enomoto et a... more Kotzig and Rosa (1970) conjectured that every tree admits edge-magic total labeling. Enomoto et al. (1998) proposed the conjecture that every tree is super edge-magic total. In this paper, we describe super (a, d)-edge-antimagic total labelings on a subclass of the subdivided stars denoted by T (n, n, n, n, n 5 , n 6 ..., nr) for d ∈ {0, 1, 2}, where n ≥ 3 odd, r ≥ 5 and nm = 2 m−4 (n − 1) + 1 for 5 ≤ m ≤ r.

Research paper thumbnail of On super edge-antimagic total labeling of subdivided stars

Discussiones Mathematicae Graph Theory, 2014

In 1980, Enomoto et al. proposed the conjecture that every tree is a super (a, 0)-edge-antimagic ... more In 1980, Enomoto et al. proposed the conjecture that every tree is a super (a, 0)-edge-antimagic total graph. In this paper, we give a partial support for the correctness of this conjecture by formulating some super (a, d)edge-antimagic total labelings on a subclass of subdivided stars denoted by T (n, n + 1, 2n + 1, 4n + 2, n 5 , n 6 ,. .. , n r) for different values of the edgeantimagic labeling parameter d, where n ≥ 3 is odd, n m = 2 m−4 (4n + 1) + 1, r ≥ 5 and 5 ≤ m ≤ r.

Research paper thumbnail of Modified Zagreb connection indices of the T-sum graphs

DE GRUYTER, 2020

The quantitative structures activity relationships (QSAR) and quantitative structures property re... more The quantitative structures activity relationships (QSAR) and quantitative structures property relationships (QSPR) between the chemical compounds are studied with the help of topological indices (TI's) which are the fixed real numbers directly linked with the molecular graphs. Gutman and Trinajstic (1972) defined the first degree based TI to measure the total π-electrone energy of a molecular graph. Recently, Ali and Trinajstic (2018) restudied the connection based TI's such as first Zagreb connection index, second Zagreb connection index and modified first Zagreb connection index to find entropy and accentric factor of the octane isomers. In this paper, we study the modified second Zagreb connection index and modified third Zagreb connection index on the T-sum (molecular) graphs obtained by the operations of subdivision and product on two graphs. At the end, as the applications of the obtained results for the modified Zagreb connection indices of the T-sum graphs of the particular classes of alkanes are also included. Mainly, a comparision among the Zagreb indices, Zagreb connection indices and modified Zagreb connection indices of the T-sum graphs of the particular classes of alkanes is performed with the help of numerical tables, 3D plots and line graphs using the statistical tools.

Research paper thumbnail of ON SUPER (a, d)-EAT VALUATION OF SUBDIVIDED CATERPILLAR

Let G = (V (G), E(G)) be a graph with v = |V (G)| vertices and e = |E(G)| edges. A bijective func... more Let G = (V (G), E(G)) be a graph with v = |V (G)| vertices and e = |E(G)| edges. A bijective function λ : V (G) ∪ E(G) ↔ {1, 2, . . . , v + e} is called an (a, d)edge antimagic total (EAT) labeling(valuation) if the weight of all the edges {w(xy) : xy ∈ E(G)} form an arithmetic sequence starting with first term a and having common difference d, where w(xy) = λ(x) + λ(y) + λ(xy). And, if λ(V ) = {1, 2, . . . , v} then G is super (a, d)-edge antimagic total(EAT) graph. In this paper, we determine the super (a,d)-edge antimagic total (EAT) labeling of the subdivided caterpillar for different values of the parameter d.

Research paper thumbnail of Topological indices of rhombus type silicate and oxide networks

Canadian Journal of Chemistry, 2017

For a molecular graph, a numeric quantity that characterizes the whole structure of a graph is ca... more For a molecular graph, a numeric quantity that characterizes the whole structure of a graph is called a topological index. In the studies of quantitative structure – activity relationship (QSAR) and quantitative structure – property relationship (QSPR), topological indices are utilized to guess the bioactivity of chemical compounds. In this paper, we compute general Randić, first general Zagreb, generalized Zagreb, multiplicative Zagreb, atom-bond connectivity (ABC), and geometric arithmetic (GA) indices for the rhombus silicate and rhombus oxide networks. In addition, we also compute the latest developed topological indices such as the fourth version of ABC (ABC4), the fifth version of GA (GA5), augmented Zagreb, and Sanskruti indices for the foresaid networks. At the end, a comparison between all the indices is included, and the result is shown with the help of a Cartesian coordinate system.

Research paper thumbnail of Topological Properties of Four-Layered Neural Networks

Journal of Artificial Intelligence and Soft Computing Research, 2019

A topological property or index of a network is a numeric number which characterises the whole st... more A topological property or index of a network is a numeric number which characterises the whole structure of the underlying network. It is used to predict the certain changes in the bio, chemical and physical activities of the networks. The 4-layered probabilistic neural networks are more general than the 3-layered probabilistic neural networks. Javaid and Cao [Neural Comput. and Applic., DOI 10.1007/s00521-017-2972-1] and Liu et al. [Journal of Artificial Intelligence and Soft Computing Research, 8(2018), 225-266] studied the certain degree and distance based topological indices (TI’s) of the 3-layered probabilistic neural networks. In this paper, we extend this study to the 4-layered probabilistic neural networks and compute the certain degree-based TI’s. In the end, a comparison between all the computed indices is included and it is also proved that the TI’s of the 4-layered probabilistic neural networks are better being strictly greater than the 3-layered probabilistic neural net...

Research paper thumbnail of Multiplicative Zagreb Indices of Molecular Graphs

Journal of Chemistry, 2019

Mathematical modeling with the help of numerical coding of graphs has been used in the different ... more Mathematical modeling with the help of numerical coding of graphs has been used in the different fields of science, especially in chemistry for the studies of the molecular structures. It also plays a vital role in the study of the quantitative structure activities relationship (QSAR) and quantitative structure properties relationship (QSPR) models. Todeshine et al. (2010) and Eliasi et al. (2012) defined two different versions of the 1st multiplicative Zagreb index as ∏Γ=∏p∈VΓdΓp2 and ∏1Γ=∏pq∈EΓdΓp+dΓq, respectively. In the same paper of Todeshine, they also defined the 2nd multiplicative Zagreb index as ∏2Γ=∏pq∈EΓdΓp×dΓq. Recently, Liu et al. [IEEE Access; 7(2019); 105479–-105488] defined the generalized subdivision-related operations of graphs and obtained the generalized F-sum graphs using these operations. They also computed the first and second Zagreb indices of the newly defined generalized F-sum graphs. In this paper, we extend this study and compute the upper bonds of the f...

Research paper thumbnail of M-polynomial method for topological indices of 2D-lattice of three-layered single-walled titania nanotubes

Journal of Information and Optimization Sciences, 2019

A molecular network can be uniquely identified by a number, polynomial or matrix. A topological i... more A molecular network can be uniquely identified by a number, polynomial or matrix. A topological index (TI) is a number that characterizes a molecular network completely which is used to predict the physical features of the certain changes such as bioactivities and chemical reactivities in the chemical compound. Javaid and Cao [Neural Comput. and Applic., 30(2018), 3869-3876] studied the first Zagreb index, second Zagreb index, general Randic index, and augmented Zagreb index for the 3-layered probabilistic neural networks (PNN). In this paper, we prove the M-polynomial of the 3-layered PNN and use it as a latest developed tool to compute the certain degree based TI's. At the end, a comparison is also shown to find the better one among all the obtained results.

Research paper thumbnail of Super Eat Labeling of Subdivided Stars

Journal of the Indonesian Mathematical Society, 2014

Kotzig and Rosa (1970) conjectured that every tree admits edge-magic total labeling. Enomoto et a... more Kotzig and Rosa (1970) conjectured that every tree admits edge-magic total labeling. Enomoto et al. (1998) proposed the conjecture that every tree is super edge-magic total. In this paper, we describe super (a, d)-edge-antimagic total labelings on a subclass of the subdivided stars denoted by T (n, n, n, n, n 5 , n 6 ..., nr) for d ∈ {0, 1, 2}, where n ≥ 3 odd, r ≥ 5 and nm = 2 m−4 (n − 1) + 1 for 5 ≤ m ≤ r.

Research paper thumbnail of On super edge-antimagic total labeling of subdivided stars

Discussiones Mathematicae Graph Theory, 2014

In 1980, Enomoto et al. proposed the conjecture that every tree is a super (a, 0)-edge-antimagic ... more In 1980, Enomoto et al. proposed the conjecture that every tree is a super (a, 0)-edge-antimagic total graph. In this paper, we give a partial support for the correctness of this conjecture by formulating some super (a, d)edge-antimagic total labelings on a subclass of subdivided stars denoted by T (n, n + 1, 2n + 1, 4n + 2, n 5 , n 6 ,. .. , n r) for different values of the edgeantimagic labeling parameter d, where n ≥ 3 is odd, n m = 2 m−4 (4n + 1) + 1, r ≥ 5 and 5 ≤ m ≤ r.

Research paper thumbnail of Modified Zagreb connection indices of the T-sum graphs

DE GRUYTER, 2020

The quantitative structures activity relationships (QSAR) and quantitative structures property re... more The quantitative structures activity relationships (QSAR) and quantitative structures property relationships (QSPR) between the chemical compounds are studied with the help of topological indices (TI's) which are the fixed real numbers directly linked with the molecular graphs. Gutman and Trinajstic (1972) defined the first degree based TI to measure the total π-electrone energy of a molecular graph. Recently, Ali and Trinajstic (2018) restudied the connection based TI's such as first Zagreb connection index, second Zagreb connection index and modified first Zagreb connection index to find entropy and accentric factor of the octane isomers. In this paper, we study the modified second Zagreb connection index and modified third Zagreb connection index on the T-sum (molecular) graphs obtained by the operations of subdivision and product on two graphs. At the end, as the applications of the obtained results for the modified Zagreb connection indices of the T-sum graphs of the particular classes of alkanes are also included. Mainly, a comparision among the Zagreb indices, Zagreb connection indices and modified Zagreb connection indices of the T-sum graphs of the particular classes of alkanes is performed with the help of numerical tables, 3D plots and line graphs using the statistical tools.