The position of a moving point in a connected graph can be identified by computing the distance from the point to a set of sonar stations which have been appropriately situated in the graph. Let Q = {q ₁ , q ₂ , ... , q _k } be an ordered set of vertices of a graph G and a is any vertex in G, then the code/representation of a w.r.t Q is the k-tuple (r(a, q ₁ ), r(a, q ₂ ), ... , r(a, q _k )), denoted by r(a|Q). If the different vertices of G have the different representations w.r.t Q, then Q is known as a resolving set/locating set. A resolving/locating set having the least number of vertices is the basis for G and the number of vertices in the basis is called metric dimension of G and it is represented as dim(G). In this paper, the metric dimension of Toeplitz graphs generated by two and three parameters denoted by T _n 〈1, t〉 and T _n 〈1, 2, t〉, respectively is discussed and proved that it is constant.

A carbon nanotube is made up of a molecular structure of a regular hexagonal lattice. This tube can be extended by closing one- or both end of the tube, denoted by semi-capped and capped nanotube. Due to its different structures, it is categorized into three forms, despite every form or shape, it has a wide range of usage. As a memory device, current-carrying device, regenerative study of medicines, and highly recommended in tissue engineering. Due to its numerous applications, there is immense interest to study its physical and chemical properties. As nanotube and its shape is motivated by a molecular structure and topological descriptors are also tools to study different molecular structures, we use this tool to analyze different nanotube structures like armchair carbon, semi-capped, and capped nanotube by Zagreb indices and its co-indices. Moreover, we also compared capped (semi or full) and uncapped armchair carbon nanotube and provided sufficient results of the comparison.