Computing Metric Dimension of Certain Families of Toeplitz Graphs

Abstract

Let 𝑄 = {𝑞1, 𝑞2, … 𝑞𝑘 } be an ordered set of vertices of a graph 𝐺 and 𝑎 is any vertex in 𝐺, then 𝑎 has representation w.r.t 𝑄, denoted by 𝑟(𝑎|𝑄) is the 𝑘 −tuple which is (𝑟(𝑎, 𝑞1), 𝑟(𝑎, 𝑞2) … 𝑟(𝑎, 𝑞𝑘)). If the different vertices of 𝐺 have the different representation w.r.t 𝑄, then 𝑄 is known as a resolving set/locating set. A resolving/locating set having the least count of vertices is basis of 𝐺 and count of vertices in this basis is called metric dimension of 𝐺 which is represented as 𝑑𝑖𝑚(𝐺). In our work, we study the metric dimension of certain Toeplitz graphs and find out their metric dimension. Firstly, we find the metric dimension of Toeplitz graph 𝑇𝑛〈1, 𝑡〉 where 𝑡 ≥ 2, which is constant. Secondly, we find the metric dimension of Toeplitz graph 𝑇𝑛〈1, 𝑠, 𝑡〉, where 𝑡 = 3, 4, 5, 6 and 𝑠 = 2, which is also constant.