Abstract:
Let 𝑄 = {𝑞1, 𝑞2, β¦ 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 βtuple which is
(𝑟(𝑎, 𝑞1), 𝑟(𝑎, 𝑞2) β¦ 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 𝑇𝑛β©1, 𝑡βͺ where 𝑡 β₯ 2, which is constant. Secondly, we find the metric dimension of Toeplitz graph 𝑇𝑛β©1, 𝑠, 𝑡βͺ, where 𝑡 = 3, 4, 5, 6 and 𝑠 = 2, which is also constant.