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Nadeem A, Kashif A, Zafar S, Zahid Z. On 2-partition dimension of the circulant graphs. JOURNAL OF INTELLIGENT & FUZZY SYSTEMS 2021. [DOI: 10.3233/jifs-201982] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
Abstract
The partition dimension is a variant of metric dimension in graphs. It has arising applications in the fields of network designing, robot navigation, pattern recognition and image processing. Let G (V (G) , E (G)) be a connected graph and Γ = {P1, P2, …, Pm} be an ordered m-partition of V (G). The partition representation of vertex v with respect to Γ is an m-vector r (v|Γ) = (d (v, P1) , d (v, P2) , …, d (v, Pm)), where d (v, P) = min {d (v, x) |x ∈ P} is the distance between v and P. If the m-vectors r (v|Γ) differ in at least 2 positions for all v ∈ V (G), then the m-partition is called a 2-partition generator of G. A 2-partition generator of G with minimum cardinality is called a 2-partition basis of G and its cardinality is known as the 2-partition dimension of G. Circulant graphs outperform other network topologies due to their low message delay, high connectivity and survivability, therefore are widely used in telecommunication networks, computer networks, parallel processing systems and social networks. In this paper, we computed partition dimension of circulant graphs Cn (1, 2) for n ≡ 2 (mod 4), n ≥ 18 and hence corrected the result given by Salman et al. [Acta Math. Sin. Engl. Ser. 2012, 28, 1851-1864]. We further computed the 2-partition dimension of Cn (1, 2) for n ≥ 6.
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Affiliation(s)
- Asim Nadeem
- Department of Mathematics, University of Management and Technology, Lahore, Pakistan
| | - Agha Kashif
- Department of Mathematics, University of Management and Technology, Lahore, Pakistan
| | - Sohail Zafar
- Department of Mathematics, University of Management and Technology, Lahore, Pakistan
| | - Zohaib Zahid
- Department of Mathematics, University of Management and Technology, Lahore, Pakistan
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Ali F, Salman M, Hafeez A, Huang S. On computation of some distance-based topological indices on circulant networks-II. JOURNAL OF INFORMATION & OPTIMIZATION SCIENCES 2018. [DOI: 10.1080/02522667.2016.1223588] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/17/2022]
Affiliation(s)
- Faisal Ali
- Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Punjab, Pakistan
| | - Muhammad Salman
- Department of Mathematics, Government Post Graduate College, Bahawalpur, Islamabad, Pakistan,
| | - Abdul Hafeez
- Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, 60800 Punjab, Pakistan,
| | - Shuliang Huang
- School of Mathematics and Finance, Chuzhou University, Chuzhou, China,
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Lu R. Fast methods for designing circulant network topology with high connectivity and survivability. JOURNAL OF CLOUD COMPUTING: ADVANCES, SYSTEMS AND APPLICATIONS 2016. [DOI: 10.1186/s13677-016-0056-x] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022] Open
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