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Papers by gohar ali

Journal of Mathematics

Metric dimension is the extraction of the affine dimension (obtained from Euclidean space E d ) t... more Metric dimension is the extraction of the affine dimension (obtained from Euclidean space E d ) to the arbitrary metric space. A family ℱ = G n of connected graphs with n ≥ 3 is a family of constant metric dimension if dim G = k (some constant) for all graphs in the family. Family ℱ has bounded metric dimension if dim G n ≤ M , for all graphs in ℱ . Metric dimension is used to locate the position in the Global Positioning System (GPS), optimization, network theory, and image processing. It is also used for the location of hospitals and other places in big cities to trace these places. In this paper, we analyzed the features and metric dimension of generalized convex polytopes and showed that this family belongs to the family of bounded metric dimension.

Journal of Mathematics

Topological indices are numeric values associated with a graph and characterize its structure. Th... more Topological indices are numeric values associated with a graph and characterize its structure. There are various topological indices in graph theory such as degree-based, distance-based, and counting-related topological indices. Among these indices, degree-based indices are very interesting and studied well in literature. In this work, we studied the generalized form of harmonic, geometric-arithmetic, Kulli–Basava indices, and generalized power-sum-connectivity index for special graph that are bridge graph over path, bridge graph over cycle, bridge graph over complete graph, wheel graph, gear graph, helm graph, and square lattice graph. We found exact values for the stated indices and for the stated special graphs. We also investigated the generalized form of the indices for various properties of alkane isomers, from which we obtained interesting results which are closed to that of experimental obtained results.

Journal of Mathematics, 2021

A � [a ij ] � 0, if v i � v j or d(v i , v j) ≥ 2 1, if d(v i , v j) � 1.. Here in this paper, we... more A � [a ij ] � 0, if v i � v j or d(v i , v j) ≥ 2 1, if d(v i , v j) � 1.. Here in this paper, we discussed new type of adjacency matrix known by 1-2 adjacency matrix defined as A 1,2 (G) � [a ij ] � 0, if v i � v j or d(v i , v j) ≥ 3 1, if d(v i , v j) � 2 , from eigenvalues of the graph, we mean eigenvalues of the 1-2 adjacency matrix. Let T c n be the set of the complement of trees of order n. In this paper, we characterized a unique graph whose least eigenvalue is minimal among all the graphs in T c n .

In this work, generalized geometry of second-order tangent groups and affine configuration comple... more In this work, generalized geometry of second-order tangent groups and affine configuration complexes is proposed. Initially, geometry for higher weights n=4 and weights n=5 is presented through some interesting and suitable homomorphisms, finally, this geometry is extended and generalized for any weight n.

Journal of Mathematics, 2021

LetGbe a graph andH⊆Gbe subgraph ofG. The graphGis said to bea,d-Hantimagic total graph if there ... more LetGbe a graph andH⊆Gbe subgraph ofG. The graphGis said to bea,d-Hantimagic total graph if there exists a bijective functionf:VH∪EH⟶1,2,3,…,VH+EHsuch that, for all subgraphs isomorphic toH, the totalHweightsWH=WH=∑x∈VHfx+∑y∈EHfyforms an arithmetic sequencea,a+d,a+2d,…,a+n−1d, whereaanddare positive integers andnis the number of subgraphs isomorphic toH. Ana,d-Hantimagic total labelingfis said to be super if the vertex labels are from the set1,2,…,|VG. In this paper, we discuss supera,d-C3-antimagic total labeling for generalized antiprism and a supera,d-C8-antimagic total labeling for toroidal octagonal map.

International Journal of Advanced Trends in Computer Science and Engineering, 2021

The structure of any finite molecular graph which represent numerical quantities are known as top... more The structure of any finite molecular graph which represent numerical quantities are known as topological indices. The importance of topological indices is generally linked with QSAR/QSPR. In this paper, we compute general Zagreb (M஑_) index, general Randic connectivity (R஑_) index, general sum-connectivity (χ஑) index, atom-bond connectivity (ABC) index, and geometric-arithmetic (GA) index,ABCସ, GAହ, multiple Zagreb indices and Zagreb polynomial indices of the of dual graph of benzene ring embedded in P-type-surface in 2D network.

Mathematical Problems in Engineering, 2021

Hypergraph is a generalization of graph in which an edge can join any number of vertices. Hypergr... more Hypergraph is a generalization of graph in which an edge can join any number of vertices. Hypergraph is used for combinatorial structures which generalize graphs. In this research work, the notion of hypergraphical metric spaces is introduced, which generalizes many existing spaces. Some fixed point theorems are studied in the corresponding spaces. To show the authenticity of the established work, nontrivial examples and applications are also provided.

International Journal of Mathematics and Soft Computing, 2011

In this paper we find the 2−rainbow domination number of the 4-regular Harary graphs H 4,n , n ≥ ... more In this paper we find the 2−rainbow domination number of the 4-regular Harary graphs H 4,n , n ≥ 5. Upper bound for the 2-rainbow domination number of P 1 × P m is found. At the end we find the lower bound for the 2-rainbow domination number of k-regular graphs.

Ars Combinatoria -Waterloo then Winnipeg-

If G is a connected graph, the distance d(u,v) between two vertices u,v∈V(G) is the length of the... more If G is a connected graph, the distance d(u,v) between two vertices u,v∈V(G) is the length of the shortest path between them. Let W={w 1 ,w 2 ,⋯,w k } be an ordered set of vertices of G and let v be a vertex of G. The representation r(v∣W) of v with respect to W is the k-tuple (d(v,w 1 ),d(v,w 2 ),⋯,d(v,w k )). If distinct vertices of G have distinct representations with respect to W, then W is called a resolving set or locating set for G. A resolving set of minimum cardinality is called a basis for G and this cardinality is the metric dimension of G, denoted by dim(G). A family G of connected graphs is a family with constant metric dimension if dim(G) does not depend upon the choice of G in G. In this paper we are dealing with the study of metric dimension of Möbius ladders. We prove that Möbius ladders M n constitute a family of cubic graphs with constant metric dimension and only three vertices suffice to resolve all the vertices of Möbius ladder M n except when n≡2(mod8). It is ...

ISRN Combinatorics, 2013

A family 𝒢 of connected graphs is a family with constant metric dimension if dim(G) is finite and... more A family 𝒢 of connected graphs is a family with constant metric dimension if dim(G) is finite and does not depend upon the choice of G in 𝒢. In this paper, we show that the graph An∗ and the graph Anp obtained from the antiprism graph have constant metric dimension.

Mathematical Sciences Letters, 2013

In this paper we compute the metric dimension of two families of graphs constructed from antipris... more In this paper we compute the metric dimension of two families of graphs constructed from antiprism graph.

Open Journal of Discrete Mathematics, 2012

If is a connected graph, the distance between two vertices G  ,

Discrete Mathematics, 2009

Let G = (V , E) be a finite, simple and non-empty (p, q)-graph of order p and size q. An (a, d)-v... more Let G = (V , E) be a finite, simple and non-empty (p, q)-graph of order p and size q. An (a, d)-vertex-antimagic total labeling is a bijection f from V (G) ∪ E(G) onto the set of consecutive integers 1, 2,. .. , p + q, such that the vertex-weights form an arithmetic progression with the initial term a and the common difference d, where the vertex-weight of x is the sum of values f (xy) assigned to all edges xy incident to vertex x together with the value assigned to x itself, i.e. f (x). Such a labeling is called super if the smallest possible labels appear on the vertices. In this paper, we will study the properties of such labelings and examine their existence for disconnected graphs.

Complexity, 2021

In this work, we introduce a new topological index called a general power sum-connectivity index ... more In this work, we introduce a new topological index called a general power sum-connectivity index and we discuss this graph invariant for some classes of extremal graphs. This index is defined by Y α G = ∑ u v ∈ E G d u d u + d v d v α , where d u and d v represent the degree of vertices u and v , respectively, and α ≥ 1 . A connected graph G is called a k -generalized quasi-tree if there exists a subset V k ⊂ V G of cardinality k such that the graph G − V k is a tree but for any subset V k − 1 ⊂ V G of cardinality k − 1 , the graph G − V k − 1 is not a tree. In this work, we find a sharp lower and some sharp upper bounds for this new sum-connectivity index.

Results in Physics, 2021

In the work, author's presents a very significant and important issues related to the health of m... more In the work, author's presents a very significant and important issues related to the health of mankind's. Which is extremely important to realize the complex dynamic of inflected disease. With the help of Caputo fractional derivative, We capture the epidemiological system for the transmission of Novel Coronavirus-19 Infectious Disease (nCOVID-19). We constructed the model in four compartments susceptible, exposed, infected and recovered. We obtained the conditions for existence and Ulam's type stability for proposed system by using the tools of non-linear analysis. The author's thoroughly discussed the local and global asymptotical stabilities of underling model upon the disease free, endemic equilibrium and reproductive number. We used the techniques of Laplace Adomian decomposition method for the approximate solution of consider system. Furthermore, author's interpret the dynamics of proposed system graphically via Mathematica, from which we observed that disease can be either controlled to a large extent or eliminate, if transmission rate is reduced and increase the rate of treatment.

IEEE Access, 2020

In this paper, we obtain the results of controllability for first order impulsive non–linear time... more In this paper, we obtain the results of controllability for first order impulsive non–linear time varying delay dynamic systems and Hammerstein impulsive system on time scale, using the theory of fixed points such as Banach fixed point theorem combined with Lipchitz conditions and non linear functional analysis. We also provide examples to support our theoretical results.

Complexity

Let G n , m represent the family of square power graphs of order n and size m , obtained from the... more Let G n , m represent the family of square power graphs of order n and size m , obtained from the family of graphs F n , k of order n and size k , with m ≥ k . In this paper, we discussed the least eigenvalue of graph G in the family G n , m c . All graphs considered here are undirected, simple, connected, and not a complete K n for positive integer n .

Journal of Mathematics

A subset S of V G is called a total dominating set of a graph G if every vertex in V G is adjacen... more A subset S of V G is called a total dominating set of a graph G if every vertex in V G is adjacent to a vertex in S . The total domination number of a graph G denoted by γ t G is the minimum cardinality of a total dominating set in G . The maximum order of a partition of V G into total dominating sets of G is called the total domatic number of G and is denoted by d t G . Domination in graphs has applications to several fields. Domination arises in facility location problems, where the number of facilities (e.g., hospitals and fire stations) is fixed, and one attempts to minimize the distance that a person needs to travel to get to the closest facility. In this paper, the numerical invariants concerning the total domination are studied for generalized Petersen graphs.

Journal of Advances in Mathematics and Computer Science

The aim of present paper is to investigate the bioconvection squeezing nanofluid flow between two... more The aim of present paper is to investigate the bioconvection squeezing nanofluid flow between two parallel plates' channels. One of the plates is stretched and the other is fixed. In this study water is considered as a base fluid because microorganisms can survive only in water. The significant influences of thermophoresis and Brownian motion have also been taken in nanofluid model. A highly nonlinear and coupled system of partial differential equations presenting the model of bioconvection flow between parallel plates is reduced to a nonlinear and coupled system (non-dimensional bioconvection flow model) of ordinary differential equations with the help of feasible non-dimensional variables. The acquired nonlinear system has been solved via homotopy analysis method (HAM). The convergence of the method has been shown numerically. Also, influence of various parameters has been discussed for the nondimensional velocity, temperature, concentration and density of the motile microorganisms both for suction and injection cases. The variation of the Skin friction, Nusselt number, Sherwood number and

IEEE Access

A family G of connected graphs is a family with unbounded metric dimension if dim(G) is not const... more A family G of connected graphs is a family with unbounded metric dimension if dim(G) is not constant and depends on the order of graph. In this paper, we compute the metric dimension of the splitting graphs S(P n) and S(C n) of a path and cycle. We prove that the metric dimension of these graphs varies and depends on the number of vertices of the graph.

Journal of Mathematics

Metric dimension is the extraction of the affine dimension (obtained from Euclidean space E d ) t... more Metric dimension is the extraction of the affine dimension (obtained from Euclidean space E d ) to the arbitrary metric space. A family ℱ = G n of connected graphs with n ≥ 3 is a family of constant metric dimension if dim G = k (some constant) for all graphs in the family. Family ℱ has bounded metric dimension if dim G n ≤ M , for all graphs in ℱ . Metric dimension is used to locate the position in the Global Positioning System (GPS), optimization, network theory, and image processing. It is also used for the location of hospitals and other places in big cities to trace these places. In this paper, we analyzed the features and metric dimension of generalized convex polytopes and showed that this family belongs to the family of bounded metric dimension.

Journal of Mathematics

Topological indices are numeric values associated with a graph and characterize its structure. Th... more Topological indices are numeric values associated with a graph and characterize its structure. There are various topological indices in graph theory such as degree-based, distance-based, and counting-related topological indices. Among these indices, degree-based indices are very interesting and studied well in literature. In this work, we studied the generalized form of harmonic, geometric-arithmetic, Kulli–Basava indices, and generalized power-sum-connectivity index for special graph that are bridge graph over path, bridge graph over cycle, bridge graph over complete graph, wheel graph, gear graph, helm graph, and square lattice graph. We found exact values for the stated indices and for the stated special graphs. We also investigated the generalized form of the indices for various properties of alkane isomers, from which we obtained interesting results which are closed to that of experimental obtained results.

Journal of Mathematics, 2021

A � [a ij ] � 0, if v i � v j or d(v i , v j) ≥ 2 1, if d(v i , v j) � 1.. Here in this paper, we... more A � [a ij ] � 0, if v i � v j or d(v i , v j) ≥ 2 1, if d(v i , v j) � 1.. Here in this paper, we discussed new type of adjacency matrix known by 1-2 adjacency matrix defined as A 1,2 (G) � [a ij ] � 0, if v i � v j or d(v i , v j) ≥ 3 1, if d(v i , v j) � 2 , from eigenvalues of the graph, we mean eigenvalues of the 1-2 adjacency matrix. Let T c n be the set of the complement of trees of order n. In this paper, we characterized a unique graph whose least eigenvalue is minimal among all the graphs in T c n .

In this work, generalized geometry of second-order tangent groups and affine configuration comple... more In this work, generalized geometry of second-order tangent groups and affine configuration complexes is proposed. Initially, geometry for higher weights n=4 and weights n=5 is presented through some interesting and suitable homomorphisms, finally, this geometry is extended and generalized for any weight n.

Journal of Mathematics, 2021

LetGbe a graph andH⊆Gbe subgraph ofG. The graphGis said to bea,d-Hantimagic total graph if there ... more LetGbe a graph andH⊆Gbe subgraph ofG. The graphGis said to bea,d-Hantimagic total graph if there exists a bijective functionf:VH∪EH⟶1,2,3,…,VH+EHsuch that, for all subgraphs isomorphic toH, the totalHweightsWH=WH=∑x∈VHfx+∑y∈EHfyforms an arithmetic sequencea,a+d,a+2d,…,a+n−1d, whereaanddare positive integers andnis the number of subgraphs isomorphic toH. Ana,d-Hantimagic total labelingfis said to be super if the vertex labels are from the set1,2,…,|VG. In this paper, we discuss supera,d-C3-antimagic total labeling for generalized antiprism and a supera,d-C8-antimagic total labeling for toroidal octagonal map.

International Journal of Advanced Trends in Computer Science and Engineering, 2021

The structure of any finite molecular graph which represent numerical quantities are known as top... more The structure of any finite molecular graph which represent numerical quantities are known as topological indices. The importance of topological indices is generally linked with QSAR/QSPR. In this paper, we compute general Zagreb (M஑_) index, general Randic connectivity (R஑_) index, general sum-connectivity (χ஑) index, atom-bond connectivity (ABC) index, and geometric-arithmetic (GA) index,ABCସ, GAହ, multiple Zagreb indices and Zagreb polynomial indices of the of dual graph of benzene ring embedded in P-type-surface in 2D network.

Mathematical Problems in Engineering, 2021

Hypergraph is a generalization of graph in which an edge can join any number of vertices. Hypergr... more Hypergraph is a generalization of graph in which an edge can join any number of vertices. Hypergraph is used for combinatorial structures which generalize graphs. In this research work, the notion of hypergraphical metric spaces is introduced, which generalizes many existing spaces. Some fixed point theorems are studied in the corresponding spaces. To show the authenticity of the established work, nontrivial examples and applications are also provided.

International Journal of Mathematics and Soft Computing, 2011

In this paper we find the 2−rainbow domination number of the 4-regular Harary graphs H 4,n , n ≥ ... more In this paper we find the 2−rainbow domination number of the 4-regular Harary graphs H 4,n , n ≥ 5. Upper bound for the 2-rainbow domination number of P 1 × P m is found. At the end we find the lower bound for the 2-rainbow domination number of k-regular graphs.

Ars Combinatoria -Waterloo then Winnipeg-

If G is a connected graph, the distance d(u,v) between two vertices u,v∈V(G) is the length of the... more If G is a connected graph, the distance d(u,v) between two vertices u,v∈V(G) is the length of the shortest path between them. Let W={w 1 ,w 2 ,⋯,w k } be an ordered set of vertices of G and let v be a vertex of G. The representation r(v∣W) of v with respect to W is the k-tuple (d(v,w 1 ),d(v,w 2 ),⋯,d(v,w k )). If distinct vertices of G have distinct representations with respect to W, then W is called a resolving set or locating set for G. A resolving set of minimum cardinality is called a basis for G and this cardinality is the metric dimension of G, denoted by dim(G). A family G of connected graphs is a family with constant metric dimension if dim(G) does not depend upon the choice of G in G. In this paper we are dealing with the study of metric dimension of Möbius ladders. We prove that Möbius ladders M n constitute a family of cubic graphs with constant metric dimension and only three vertices suffice to resolve all the vertices of Möbius ladder M n except when n≡2(mod8). It is ...

ISRN Combinatorics, 2013

A family 𝒢 of connected graphs is a family with constant metric dimension if dim(G) is finite and... more A family 𝒢 of connected graphs is a family with constant metric dimension if dim(G) is finite and does not depend upon the choice of G in 𝒢. In this paper, we show that the graph An∗ and the graph Anp obtained from the antiprism graph have constant metric dimension.

Mathematical Sciences Letters, 2013

In this paper we compute the metric dimension of two families of graphs constructed from antipris... more In this paper we compute the metric dimension of two families of graphs constructed from antiprism graph.

Open Journal of Discrete Mathematics, 2012

If is a connected graph, the distance between two vertices G  ,

Discrete Mathematics, 2009

Let G = (V , E) be a finite, simple and non-empty (p, q)-graph of order p and size q. An (a, d)-v... more Let G = (V , E) be a finite, simple and non-empty (p, q)-graph of order p and size q. An (a, d)-vertex-antimagic total labeling is a bijection f from V (G) ∪ E(G) onto the set of consecutive integers 1, 2,. .. , p + q, such that the vertex-weights form an arithmetic progression with the initial term a and the common difference d, where the vertex-weight of x is the sum of values f (xy) assigned to all edges xy incident to vertex x together with the value assigned to x itself, i.e. f (x). Such a labeling is called super if the smallest possible labels appear on the vertices. In this paper, we will study the properties of such labelings and examine their existence for disconnected graphs.

Complexity, 2021

In this work, we introduce a new topological index called a general power sum-connectivity index ... more In this work, we introduce a new topological index called a general power sum-connectivity index and we discuss this graph invariant for some classes of extremal graphs. This index is defined by Y α G = ∑ u v ∈ E G d u d u + d v d v α , where d u and d v represent the degree of vertices u and v , respectively, and α ≥ 1 . A connected graph G is called a k -generalized quasi-tree if there exists a subset V k ⊂ V G of cardinality k such that the graph G − V k is a tree but for any subset V k − 1 ⊂ V G of cardinality k − 1 , the graph G − V k − 1 is not a tree. In this work, we find a sharp lower and some sharp upper bounds for this new sum-connectivity index.

Results in Physics, 2021

In the work, author's presents a very significant and important issues related to the health of m... more In the work, author's presents a very significant and important issues related to the health of mankind's. Which is extremely important to realize the complex dynamic of inflected disease. With the help of Caputo fractional derivative, We capture the epidemiological system for the transmission of Novel Coronavirus-19 Infectious Disease (nCOVID-19). We constructed the model in four compartments susceptible, exposed, infected and recovered. We obtained the conditions for existence and Ulam's type stability for proposed system by using the tools of non-linear analysis. The author's thoroughly discussed the local and global asymptotical stabilities of underling model upon the disease free, endemic equilibrium and reproductive number. We used the techniques of Laplace Adomian decomposition method for the approximate solution of consider system. Furthermore, author's interpret the dynamics of proposed system graphically via Mathematica, from which we observed that disease can be either controlled to a large extent or eliminate, if transmission rate is reduced and increase the rate of treatment.

IEEE Access, 2020

In this paper, we obtain the results of controllability for first order impulsive non–linear time... more In this paper, we obtain the results of controllability for first order impulsive non–linear time varying delay dynamic systems and Hammerstein impulsive system on time scale, using the theory of fixed points such as Banach fixed point theorem combined with Lipchitz conditions and non linear functional analysis. We also provide examples to support our theoretical results.

Complexity

Let G n , m represent the family of square power graphs of order n and size m , obtained from the... more Let G n , m represent the family of square power graphs of order n and size m , obtained from the family of graphs F n , k of order n and size k , with m ≥ k . In this paper, we discussed the least eigenvalue of graph G in the family G n , m c . All graphs considered here are undirected, simple, connected, and not a complete K n for positive integer n .

Journal of Mathematics

A subset S of V G is called a total dominating set of a graph G if every vertex in V G is adjacen... more A subset S of V G is called a total dominating set of a graph G if every vertex in V G is adjacent to a vertex in S . The total domination number of a graph G denoted by γ t G is the minimum cardinality of a total dominating set in G . The maximum order of a partition of V G into total dominating sets of G is called the total domatic number of G and is denoted by d t G . Domination in graphs has applications to several fields. Domination arises in facility location problems, where the number of facilities (e.g., hospitals and fire stations) is fixed, and one attempts to minimize the distance that a person needs to travel to get to the closest facility. In this paper, the numerical invariants concerning the total domination are studied for generalized Petersen graphs.

Journal of Advances in Mathematics and Computer Science

The aim of present paper is to investigate the bioconvection squeezing nanofluid flow between two... more The aim of present paper is to investigate the bioconvection squeezing nanofluid flow between two parallel plates' channels. One of the plates is stretched and the other is fixed. In this study water is considered as a base fluid because microorganisms can survive only in water. The significant influences of thermophoresis and Brownian motion have also been taken in nanofluid model. A highly nonlinear and coupled system of partial differential equations presenting the model of bioconvection flow between parallel plates is reduced to a nonlinear and coupled system (non-dimensional bioconvection flow model) of ordinary differential equations with the help of feasible non-dimensional variables. The acquired nonlinear system has been solved via homotopy analysis method (HAM). The convergence of the method has been shown numerically. Also, influence of various parameters has been discussed for the nondimensional velocity, temperature, concentration and density of the motile microorganisms both for suction and injection cases. The variation of the Skin friction, Nusselt number, Sherwood number and

IEEE Access

A family G of connected graphs is a family with unbounded metric dimension if dim(G) is not const... more A family G of connected graphs is a family with unbounded metric dimension if dim(G) is not constant and depends on the order of graph. In this paper, we compute the metric dimension of the splitting graphs S(P n) and S(C n) of a path and cycle. We prove that the metric dimension of these graphs varies and depends on the number of vertices of the graph.