In any interconnection network, fault tolerance is the most desirable property to achieve reliabi... more In any interconnection network, fault tolerance is the most desirable property to achieve reliability. Toeplitz networks are used as interconnection networks due their smaller diameter, symmetry, simpler routing, high connectivity, and reliability. The partition dimension of a network is presented as an extension of metric dimension of networks. Its applications can be seen in several areas including robot navigation, network designing, image processing, and chemistry. In this article, the fault tolerant partition dimension, pd 2 T n 1 , t , of Toeplitz networks, is shown to be bounded below by 4 for t ≥ 2 , n ≥ 4 , whereas it is bounded above by 5 for t = 3 , n ≥ 14 . Further, it is shown that the exact value of pd 2 T n 1 , t equals 4 for t = 2 , n ≥ 4 ; t = 3 , n ∈ 5,6 , … , 13 ; and t ≥ 4 , n ∈ t + 2 , t + 3 , t + 4 .
The 2-metric resolvability is an extension of metric resolvability in graphs having several appli... more The 2-metric resolvability is an extension of metric resolvability in graphs having several applications in intelligent systems for example network optimization, robot navigation and sensor networking. Rotationally symmetric graphs are important in intelligent networks due to uniform rate of data transformation to all nodes. In this article, 2-metric dimension of rotationally symmetric plane graphs Rn, Sn and Tn is computed and found to be independent of the number of vertices.
We determine the number of subgroups of finite non-Abelian group defined by the presentation G=(x... more We determine the number of subgroups of finite non-Abelian group defined by the presentation G=(x,y∣x 2 =y 16 =1,yx=xy 7 ). We identify the form and the order of elements of the group G· There are 26 non-trivial subgroups of G and the subgroup lattice diagram is then presented.
In any interconnection network, fault tolerance is the most desirable property to achieve reliabi... more In any interconnection network, fault tolerance is the most desirable property to achieve reliability. Toeplitz networks are used as interconnection networks due their smaller diameter, symmetry, simpler routing, high connectivity, and reliability. The partition dimension of a network is presented as an extension of metric dimension of networks. Its applications can be seen in several areas including robot navigation, network designing, image processing, and chemistry. In this article, the fault tolerant partition dimension, pd 2 T n 1 , t , of Toeplitz networks, is shown to be bounded below by 4 for t ≥ 2 , n ≥ 4 , whereas it is bounded above by 5 for t = 3 , n ≥ 14 . Further, it is shown that the exact value of pd 2 T n 1 , t equals 4 for t = 2 , n ≥ 4 ; t = 3 , n ∈ 5,6 , … , 13 ; and t ≥ 4 , n ∈ t + 2 , t + 3 , t + 4 .
The 2-metric resolvability is an extension of metric resolvability in graphs having several appli... more The 2-metric resolvability is an extension of metric resolvability in graphs having several applications in intelligent systems for example network optimization, robot navigation and sensor networking. Rotationally symmetric graphs are important in intelligent networks due to uniform rate of data transformation to all nodes. In this article, 2-metric dimension of rotationally symmetric plane graphs Rn, Sn and Tn is computed and found to be independent of the number of vertices.
We determine the number of subgroups of finite non-Abelian group defined by the presentation G=(x... more We determine the number of subgroups of finite non-Abelian group defined by the presentation G=(x,y∣x 2 =y 16 =1,yx=xy 7 ). We identify the form and the order of elements of the group G· There are 26 non-trivial subgroups of G and the subgroup lattice diagram is then presented.
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Papers by humera bashir