数学物理学报(英文版) ›› 2013, Vol. 33 ›› Issue (1): 187-206.doi: 10.1016/S0252-9602(12)60204-5

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ON CLASSES OF REGULAR GRAPHS WITH CONSTANT METRIC DIMENSION

Muhammad IMRAN*, Syed Ahtsham ul Haq BOKHARY, Ali AHMAD, Andrea SEMANICOVÁ-FEÑOVCÍKOVÁ   

  1. Centre for Advanced Mathematics and Physics (CAMP), National University of Sciences and Technology (NUST), Sector H-12, Islamabad, Pakistan;Center for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan;College of Computer and Information System, Jazan University, Jazan, Saudi Arabia;Department of Applied Mathematics and Informatics, Technical University, Letn´a 9, 042 00 Koˇsice, Slovakia
  • 收稿日期:2011-09-05 出版日期:2013-01-20 发布日期:2013-01-20
  • 通讯作者: Muhammad IMRAN,imrandhab@gmail.com E-mail:imrandhab@gmail.com;sihtsham@gmail.com;ahmadsms@gmail.com; andrea.fenovcikova@tuke.sk
  • 基金资助:

    This research is partially supported by National University of Sceinces and Technology (NUST), Islamabad, and grant of Higher Education Commission of Pakistan Ref. No: PM-IPFP/HRD/HEC/2011/3386 and support of Slovak VEGA Grant 1/0130/12.

ON CLASSES OF REGULAR GRAPHS WITH CONSTANT METRIC DIMENSION

Muhammad IMRAN*, Syed Ahtsham ul Haq BOKHARY, Ali AHMAD, Andrea SEMANICOVÁ-FEÑOVCÍKOVÁ   

  1. Centre for Advanced Mathematics and Physics (CAMP), National University of Sciences and Technology (NUST), Sector H-12, Islamabad, Pakistan;Center for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan;College of Computer and Information System, Jazan University, Jazan, Saudi Arabia;Department of Applied Mathematics and Informatics, Technical University, Letn´a 9, 042 00 Koˇsice, Slovakia
  • Received:2011-09-05 Online:2013-01-20 Published:2013-01-20
  • Contact: Muhammad IMRAN,imrandhab@gmail.com E-mail:imrandhab@gmail.com;sihtsham@gmail.com;ahmadsms@gmail.com; andrea.fenovcikova@tuke.sk
  • Supported by:

    This research is partially supported by National University of Sceinces and Technology (NUST), Islamabad, and grant of Higher Education Commission of Pakistan Ref. No: PM-IPFP/HRD/HEC/2011/3386 and support of Slovak VEGA Grant 1/0130/12.

摘要:

In this paper, we are dealing with the study of the metric dimension of some classes of regular graphs by considering a class of bridgeless cubic graphs called the flower snarks Jn, a class of cubic convex polytopes considering the open problem raised in [M. Imran et al., families of plane graphs with constant metric dimension, Utilitas Math., in press] and finally Harary graphs H5,n by partially answering to an open problem proposed in [I. Javaid et al., Families of regular graphs with constant metric dimension, Utilitas Math., 2012, 88: 43–57]. We prove that these classes of regular graphs have constant metric dimension.

关键词: metric dimension, basis, resolving set, cubic graph, flower snark, convex polytope

Abstract:

In this paper, we are dealing with the study of the metric dimension of some classes of regular graphs by considering a class of bridgeless cubic graphs called the flower snarks Jn, a class of cubic convex polytopes considering the open problem raised in [M. Imran et al., families of plane graphs with constant metric dimension, Utilitas Math., in press] and finally Harary graphs H5,n by partially answering to an open problem proposed in [I. Javaid et al., Families of regular graphs with constant metric dimension, Utilitas Math., 2012, 88: 43–57]. We prove that these classes of regular graphs have constant metric dimension.

Key words: metric dimension, basis, resolving set, cubic graph, flower snark, convex polytope

中图分类号: 

  • 05C12