Locating and Identifying Codes in Circulant Graphs

Discrete Dynamics in Nature and Society ◽

10.1155/2021/4056910 ◽

2021 ◽

Vol 2021 ◽

pp. 1-9

Author(s):

Shu Jiao Song ◽

Weiqian Zhang ◽

Can Xu

Keyword(s):

Circulant Graphs ◽

Identifying Codes ◽

Bounds On Codes ◽

Optimal Bounds

Identifying and locating-dominating codes have been studied widely in circulant graphs. Recently, Ville Junnila et al. (Optimal bounds on codes for location in circulant graphs, Cryptography and Communications; 2019) studied identifying and locating-dominating codes in circulants C n 1 , d , C n 1 , d − 1 , d , and C n 1 , d − 1 , d , d + 1 . In this paper, identifying, locating, and self-identifying codes in the circulant graphs C n k , d , C n k , d − k , d , and C n k , d − k , d , d + k are studied, and this extends Junnila et al.’s results to general cases.

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Optimal bounds on codes for location in circulant graphs

Cryptography and Communications ◽

10.1007/s12095-018-0316-3 ◽

2018 ◽

Vol 11 (4) ◽

pp. 621-640 ◽

Cited By ~ 1

Author(s):

Ville Junnila ◽

Tero Laihonen ◽

Gabrielle Paris

Keyword(s):

Circulant Graphs ◽

Bounds On Codes ◽

Optimal Bounds

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Mathematical Properties of Variable Topological Indices

Symmetry ◽

10.3390/sym13010043 ◽

2020 ◽

Vol 13 (1) ◽

pp. 43

Author(s):

José M. Sigarreta

Keyword(s):

Real Number ◽

Large Class ◽

Symmetric Functions ◽

Topological Indices ◽

Current Interest ◽

Zagreb Indices ◽

Mathematical Properties ◽

Connectivity Indices ◽

Optimal Bounds ◽

New Framework

A topic of current interest in the study of topological indices is to find relations between some index and one or several relevant parameters and/or other indices. In this paper we study two general topological indices Aα and Bα, defined for each graph H=(V(H),E(H)) by Aα(H)=∑ij∈E(H)f(di,dj)α and Bα(H)=∑i∈V(H)h(di)α, where di denotes the degree of the vertex i and α is any real number. Many important topological indices can be obtained from Aα and Bα by choosing appropriate symmetric functions and values of α. This new framework provides new tools that allow to obtain in a unified way inequalities involving many different topological indices. In particular, we obtain new optimal bounds on the variable Zagreb indices, the variable sum-connectivity index, the variable geometric-arithmetic index and the variable inverse sum indeg index. Thus, our approach provides both new tools for the study of topological indices and new bounds for a large class of topological indices. We obtain several optimal bounds of Aα (respectively, Bα) involving Aβ (respectively, Bβ). Moreover, we provide several bounds of the variable geometric-arithmetic index in terms of the variable inverse sum indeg index, and two bounds of the variable inverse sum indeg index in terms of the variable second Zagreb and the variable sum-connectivity indices.

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Minimax-Optimal Bounds for Detectors Based on Estimated Prior Probabilities

IEEE Transactions on Information Theory ◽

10.1109/tit.2012.2201914 ◽

2012 ◽

Vol 58 (9) ◽

pp. 6101-6109 ◽

Cited By ~ 3

Author(s):

Jiantao Jiao ◽

Lin Zhang ◽

Robert D. Nowak

Keyword(s):

Prior Probabilities ◽

Optimal Bounds

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Optimal bounds on finding fixed points of contraction mappings

Theoretical Computer Science ◽

10.1016/j.tcs.2010.01.016 ◽

2010 ◽

Vol 411 (16-18) ◽

pp. 1742-1749 ◽

Cited By ~ 1

Author(s):

Ching-Lueh Chang ◽

Yuh-Dauh Lyuu

Keyword(s):

Fixed Points ◽

Contraction Mappings ◽

Optimal Bounds

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The energy of integral circulant graphs with prime power order

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm110131003s ◽

2011 ◽

Vol 5 (1) ◽

pp. 22-36 ◽

Cited By ~ 21

Author(s):

J.W. Sander ◽

T. Sander

Keyword(s):

Adjacency Matrix ◽

Prime Power ◽

Prime Power Order ◽

Circulant Graph ◽

Closed Formula ◽

Circulant Graphs ◽

Vertex Set

The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs. Such a graph can be characterized by its vertex count n and a set D of divisors of n such that its vertex set is Zn and its edge set is {{a,b} : a, b ? Zn; gcd(a-b, n)? D}. For an integral circulant graph on ps vertices, where p is a prime, we derive a closed formula for its energy in terms of n and D. Moreover, we study minimal and maximal energies for fixed ps and varying divisor sets D.

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