Some Resolving Parameters in a Class of Cayley Graphs

Resolving parameters are a fundamental area of combinatorics with applications not only to many branches of combinatorics but also to other sciences. In this study, we construct a class of Toeplitz graphs and will be denoted by T 2 n W so that they are Cayley graphs. First, we review some of the features of this class of graphs. In fact, this class of graphs is vertex transitive, and by calculating the spectrum of the adjacency matrix related with them, we show that this class of graphs cannot be edge transitive. Moreover, we show that this class of graphs cannot be distance regular, and because of the difficulty of the computing resolving parameters of a class of graphs which are not distance regular, we regard this as justification for our focus on some resolving parameters. In particular, we determine the minimal resolving set, doubly resolving set, and strong metric dimension for this class of graphs.

[1]  Azriel Rosenfeld,et al.  Localization in graphs , 1994 .

[2]  A. Zafari,et al.  An Interesting Property of a Class of Circulant Graphs , 2017 .

[3]  A. Hora,et al.  Distance-Regular Graphs , 2007 .

[4]  Tomás Vetrík,et al.  Computing the metric dimension of the categorial product of some graphs , 2017, Int. J. Comput. Math..

[5]  S. Morteza Mirafzal,et al.  On the spectrum of a class of distance-transitive graphs , 2017, Electron. J. Graph Theory Appl..

[6]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[7]  Ali Zafari,et al.  Metric Dimension, Minimal Doubly Resolving Sets, and the Strong Metric Dimension for Jellyfish Graph and Cocktail Party Graph , 2020, Complex..

[8]  Ali Zafari,et al.  Computing Minimal Doubly Resolving Sets and the Strong Metric Dimension of the Layer Sun Graph and the Line Graph of the Layer Sun Graph , 2020, Complex..

[9]  Frank Harary,et al.  Which graphs have integral spectra , 1974 .

[10]  Zsolt Tuza,et al.  Hamiltonian properties of Toeplitz graphs , 1996, Discret. Math..

[11]  P. Cameron,et al.  Base size, metric dimension and other invariants of groups and graphs , 2011 .

[12]  W. T. Tutte.  Connectivity in graphs , 1966 .

[13]  L. Beineke,et al.  Topics in algebraic graph theory , 2004 .

[14]  A. Neumaier,et al.  Distance Regular Graphs , 1989 .

[15]  Muhammad Kamran Siddiqui,et al.  On the Metric Dimension of Generalized Petersen Multigraphs , 2018, IEEE Access.

[16]  後藤 達生.  On metric dimension , 2003 .

[17]  F. Smithies.  A HILBERT SPACE PROBLEM BOOK , 1968 .

[18]  David R. Wood,et al.  On the Metric Dimension of Cartesian Products of Graphs , 2005, SIAM J. Discret. Math..

[19]  N. Duncan.  Leaves on trees , 2014 .

[20]  Robert F. Bailey,et al.  The metric dimension of small distance-regular and strongly regular graphs , 2013, Australas. J Comb..

[21]  András Sebö,et al.  On Metric Generators of Graphs , 2004, Math. Oper. Res..

[22]  Thomas Erlebach,et al.  Network Discovery and Verification , 2005, IEEE Journal on Selected Areas in Communications.

[23]  R. Frucht,et al.  On the groups of repeated graphs , 1949 .

[24]  Muhammad Faisal Nadeem,et al.  Computing Metric Dimension of Certain Families of Toeplitz Graphs , 2019, IEEE Access.