On the Metric Index of Circulant Networks–An Algorithmic Approach

Abstract

A vertex v of a graph G uniquely determines (resolves) a pair (v 1 , v 2) of vertices of G if the distance between v and v 1 is different from the distance between v and v 2. The metric index is a distancebased topological index of a graph G, which is the least number of vertices in G chosen in such a way that each vertex of G can be determined uniquely by its distances to the chosen vertices. The metric index of a family of graphs is said to be constant if it does not increase with an increase in the number of vertices of graphs in the family. Otherwise, it is said to be unbounded. In this paper, we develop an algorithm to construct a larger order circulant network from the smallest order circulant network. Then, we consider two families of circulant networks: one in the context of constant metric index; and other in the perspective of the unbounded metric index to counter the popular belief that the metric index of circulant networks will never depend upon the number of vertices. INDEX TERMS Topological index, metric index, circulant networks.

An Algorithmic Approach to Compute the Metric Index of Chordal Ring Networks

“…In fact, this constructive algorithm is supportive in proving our results by using mathematical induction to develop the base for induction process and to develop the step next to the supposition step of the induction. Moreover, there is no need to provide the comparison of this algorithm as well as its time complexity which can lead/support/effect the NP-completeness of the problem, because this algorithm is not a computer programming based, no such algorithm exists previously for the underline networks, and its construction procedure for networks of various families will vary according to the structure of networks (as two such algorithms are established for circulant networks in [6], [7], and each of them has different construction proce-dures because of the structural difference in the networks). Secondly, we provide distinct metric coding of the vertices of chordal ring networks with respect to a chosen ordered set, due to this we would claim for the upper bound of the metric index of chordal ring networks in our results.…”

“…In 2008, Javaid et al initiated the study of constant metric index by investigating it for three families of regular networks, and asked for the investigation of more such families possessing the constant metric index [9]. In this regard, a number of research articles have been published on various families of regular networks such as families of Harary networks, Caylay networks, circulant networks, generalized Petersen networks fullerence networks, to name a few [6], [7], [9], [17], [26]- [31]. With this article, we extend the study of constant metric index by exploring one more family of regular networks, namely a family of choral ring networks CR η (1, 3, 5).…”

“…We prove that a set of least number of nodes which defines the metric index in the basic chordal ring network of a family is adequate for the whole family of chordal ring networks. To the best of our knowledge, we were the pioneers of this technique [6], [7].…”

Desenvolvimento De Transdutores Magnéticos Em Malha Fechada Baseados No Efeito Da Magnetoimpedância Gigante

“…Tal efeito consiste na repulsão do fluxo magnético do interior do supercondutor e na quantização do fluxo magnético (conservação do fluxo magnético em uma malha fechada de material supercondutor). O campo magnético externo é repelido de forma dinâmica, por meio de correntes criadas na superfície do supercondutor, a fim de cancelá-lo [22,143]. A Figura 100 ilustra este comportamento.…”

“…A Figura 1 apresenta e compara os valores típicos de resolução, fundo de escala e largura de banda de alguns tipos de magnetômetros [3,[17][18][19][20]. Conforme evidenciado na Figura 1, na atualidade, o SQUID (Superconducting Quantum Interference Device) é o transdutor de campo magnético mais sensível, com a capacidade de atingir resoluções da ordem de femtoteslas (10 -15 T) e operar em faixas de frequências de 0 a 20 kHz [3,21,22]. Os princípios de operação dos magnetômetros SQUID são baseados na teoria da supercondutividade, sendo necessário o uso de sistemas de resfriamento criogênicos, o que eleva consideravelmente o custo de operação de tais transdutores magnéticos.…”