Journal of Advances in Mathematics and Computer Science
In this paper, the norm attaining operators in Frechet spaces are considered. These operators are... more In this paper, the norm attaining operators in Frechet spaces are considered. These operators are characterized based on their density, normality, linearity and compactness. It is shown that the image is dense for a normal and injective operator in a Frechet space, as well as its inverse given that the operator is self-adjoint. A norm attaining operator in a Frechet space is also shown to be normal if its adjoint also attains its norm in the Frechet space, and the condition under which the norm attainability and the normality of an operator in a Frechet space coincides is given. Furthermore, a norm attaining operator between Frechet spaces is linear and bounded as well as its inverse. If a norm attaining, normal and dense operator is of finite rank, then it is compact. The study of norm attaining operators is applicable in algorithm concentration as seen in describing sphere packing.
Fractional variants of distance-based parameters have application in the fields of sensor network... more Fractional variants of distance-based parameters have application in the fields of sensor networking, robot navigation, and integer programming problems. Complex networks are exceptional networks which exhibit significant topological features and have become quintessential research area in the field of computer science, biology, and mathematics. Owing to the possibility that many real-world systems can be intelligently modeled and represented as complex networks to examine, administer and comprehend the useful information from these real-world networks. In this paper, local fractional strong metric dimension of certain complex networks is computed. Building blocks of complex networks are considered as the symmetric networks such as cyclic networks C n , circulant networks C n 1,2 , mobious ladder networks M 2 n , and generalized prism networks G m n . In this regard, it is shown that LSFMD of C n n ≥ 3 and G m n n ≥ 6 is 1 when n is even and n / n − 1 when n is odd, whereas LSFMD of...
This paper presents the classication of the invariant subgroups of the automorphism groups of the... more This paper presents the classication of the invariant subgroups of the automorphism groups of the regular elements obtained from nite local near-rings, the appropriate algebraic structure to study non-linear functions on finite groups. Just as rings of matrices operate on vector spaces, near-rings operate on groups. In this paper, we construct classes of zero symmetric local near-ring of characteristic pk; k = 1; 2 ; k \(\ge\) 3 admitting frobenius derivations, characterize the structures of the cyclic groups generated by the regular elements R(N) as well as the structures and the orders of the automorphism groups Aut(R(N)) of the regular elements.
Let N be a zero-symmetric local near-ring. An element \({x}\) \(\in\) N is either regular, zero o... more Let N be a zero-symmetric local near-ring. An element \({x}\) \(\in\) N is either regular, zero or a zero divisor. In this paper, we construct a class of zero symmetric local near-ring of characteristic pk; k \(\ge\) 3 admitting an identity frobenius derivation, characterize the structures and orders of the set R(N), the regular compartment with an aim of advancing the classication problem of algebraic structures. The number theoretic notions relating the number of regular elements to Euler's phi-function and the arithmetic functions of Galois near-rings are adopted. Using the Fundamental Theorem of nitely generated Abelian groups, the structures of R(N) are proved to be isomorphic to cyclic groups of various orders. The study also extends to the automorphism groups Aut(R(N)) of the regular elements.
Journal of Advances in Mathematics and Computer Science
The characterization of nite local rings via the well known structures of their zero divisor grap... more The characterization of nite local rings via the well known structures of their zero divisor graphs and cayley graphs remains an open problem. Some classes of completely primary finite rings which are local, have however been characterized by the compartments of their units and zero divisors where the classication of the unit groups have been done using the Fundamental Theorem of finitely generated Abelian Groups while the zero divisors have been characterized via the zero divisor graphs. This paper characterizes the zero divisor graphs \(\Gamma\)(R) and cayley graphs CAY (R) where R is a finite local ring with 2-radical index of Nilpotence. These two classes of graphs have been completely described and compared using their algebraic properties. Some of the graphs have been drawn for purposes of their comparison. The methods of study involved partitioning the ring under consideration into mutually disjoint subsets of invertible elements and zero divisors and determining their graphs...
A topological index is a numerical quantity associated with the molecular structure of a chemical... more A topological index is a numerical quantity associated with the molecular structure of a chemical compound. This number remains fixed with respect to the symmetry of a molecular graph. Diverse research studies have shown that the topological indices of symmetrical graphs are interrelated with several physiochemical properties such as boiling point, density, and heat of formation. Peripherality is also an important tool to study topological aspects of molecular graphs. Recently, a bond-additive topological index called the Mostar index that measures the peripherality of a graph is investigated which attained wide attention of researchers. In this article, we compute the Mostar index of cycle-related structures such as the Jahangir graph and the cycle graph with chord.
Graph invariants provide an amazing tool to analyze the abstract structures of networks. The inte... more Graph invariants provide an amazing tool to analyze the abstract structures of networks. The interaction and interconnection between devices, sensors, and service providers have opened the door for an eruption of mobile over the web applications. Structure of web sites containing number of pages can be represented using graph, where web pages are considered to be the vertices, and an edge is a link between two pages. Figuring resolving partition of the graph is an intriguing inquest in graph theory as it has many applications such as sensor design, compound classification in chemistry, robotic navigation, and Internet network. The partition dimension is a graph parameter akin to the concept of metric dimension, and fault-tolerant partition dimension is an advancement in the line of research of partition dimension of the graph. In this paper, we compute fault-tolerant partition dimension of alternate triangular cycle, mirror graph, and tortoise graphs.
Journal of Advances in Mathematics and Computer Science
In this paper, the norm attaining operators in Frechet spaces are considered. These operators are... more In this paper, the norm attaining operators in Frechet spaces are considered. These operators are characterized based on their density, normality, linearity and compactness. It is shown that the image is dense for a normal and injective operator in a Frechet space, as well as its inverse given that the operator is self-adjoint. A norm attaining operator in a Frechet space is also shown to be normal if its adjoint also attains its norm in the Frechet space, and the condition under which the norm attainability and the normality of an operator in a Frechet space coincides is given. Furthermore, a norm attaining operator between Frechet spaces is linear and bounded as well as its inverse. If a norm attaining, normal and dense operator is of finite rank, then it is compact. The study of norm attaining operators is applicable in algorithm concentration as seen in describing sphere packing.
Fractional variants of distance-based parameters have application in the fields of sensor network... more Fractional variants of distance-based parameters have application in the fields of sensor networking, robot navigation, and integer programming problems. Complex networks are exceptional networks which exhibit significant topological features and have become quintessential research area in the field of computer science, biology, and mathematics. Owing to the possibility that many real-world systems can be intelligently modeled and represented as complex networks to examine, administer and comprehend the useful information from these real-world networks. In this paper, local fractional strong metric dimension of certain complex networks is computed. Building blocks of complex networks are considered as the symmetric networks such as cyclic networks C n , circulant networks C n 1,2 , mobious ladder networks M 2 n , and generalized prism networks G m n . In this regard, it is shown that LSFMD of C n n ≥ 3 and G m n n ≥ 6 is 1 when n is even and n / n − 1 when n is odd, whereas LSFMD of...
This paper presents the classication of the invariant subgroups of the automorphism groups of the... more This paper presents the classication of the invariant subgroups of the automorphism groups of the regular elements obtained from nite local near-rings, the appropriate algebraic structure to study non-linear functions on finite groups. Just as rings of matrices operate on vector spaces, near-rings operate on groups. In this paper, we construct classes of zero symmetric local near-ring of characteristic pk; k = 1; 2 ; k \(\ge\) 3 admitting frobenius derivations, characterize the structures of the cyclic groups generated by the regular elements R(N) as well as the structures and the orders of the automorphism groups Aut(R(N)) of the regular elements.
Let N be a zero-symmetric local near-ring. An element \({x}\) \(\in\) N is either regular, zero o... more Let N be a zero-symmetric local near-ring. An element \({x}\) \(\in\) N is either regular, zero or a zero divisor. In this paper, we construct a class of zero symmetric local near-ring of characteristic pk; k \(\ge\) 3 admitting an identity frobenius derivation, characterize the structures and orders of the set R(N), the regular compartment with an aim of advancing the classication problem of algebraic structures. The number theoretic notions relating the number of regular elements to Euler's phi-function and the arithmetic functions of Galois near-rings are adopted. Using the Fundamental Theorem of nitely generated Abelian groups, the structures of R(N) are proved to be isomorphic to cyclic groups of various orders. The study also extends to the automorphism groups Aut(R(N)) of the regular elements.
Journal of Advances in Mathematics and Computer Science
The characterization of nite local rings via the well known structures of their zero divisor grap... more The characterization of nite local rings via the well known structures of their zero divisor graphs and cayley graphs remains an open problem. Some classes of completely primary finite rings which are local, have however been characterized by the compartments of their units and zero divisors where the classication of the unit groups have been done using the Fundamental Theorem of finitely generated Abelian Groups while the zero divisors have been characterized via the zero divisor graphs. This paper characterizes the zero divisor graphs \(\Gamma\)(R) and cayley graphs CAY (R) where R is a finite local ring with 2-radical index of Nilpotence. These two classes of graphs have been completely described and compared using their algebraic properties. Some of the graphs have been drawn for purposes of their comparison. The methods of study involved partitioning the ring under consideration into mutually disjoint subsets of invertible elements and zero divisors and determining their graphs...
A topological index is a numerical quantity associated with the molecular structure of a chemical... more A topological index is a numerical quantity associated with the molecular structure of a chemical compound. This number remains fixed with respect to the symmetry of a molecular graph. Diverse research studies have shown that the topological indices of symmetrical graphs are interrelated with several physiochemical properties such as boiling point, density, and heat of formation. Peripherality is also an important tool to study topological aspects of molecular graphs. Recently, a bond-additive topological index called the Mostar index that measures the peripherality of a graph is investigated which attained wide attention of researchers. In this article, we compute the Mostar index of cycle-related structures such as the Jahangir graph and the cycle graph with chord.
Graph invariants provide an amazing tool to analyze the abstract structures of networks. The inte... more Graph invariants provide an amazing tool to analyze the abstract structures of networks. The interaction and interconnection between devices, sensors, and service providers have opened the door for an eruption of mobile over the web applications. Structure of web sites containing number of pages can be represented using graph, where web pages are considered to be the vertices, and an edge is a link between two pages. Figuring resolving partition of the graph is an intriguing inquest in graph theory as it has many applications such as sensor design, compound classification in chemistry, robotic navigation, and Internet network. The partition dimension is a graph parameter akin to the concept of metric dimension, and fault-tolerant partition dimension is an advancement in the line of research of partition dimension of the graph. In this paper, we compute fault-tolerant partition dimension of alternate triangular cycle, mirror graph, and tortoise graphs.
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