Sunny Kumar Sharma

In this article, we consider two interesting classes of an infinite planar graphs (viz., the flower graph Sn = ℱn×3 and the graph Dn* , constructed using prism...

In recent years, nanomaterials such as carbon nanotubes, nanocones, boron nanowires and nanoclusters have been widely studied because of their unique structures and useful electronic properties. Among them carbon based nanostructures are especially important for applications in nanotechnology. To study these materials graph theory provides different numerical measures that describe their structure. One such measure is the face neighbourhood index which not only counts the faces in a planar graph but also considers how they are connected through the vertices on their boundaries. This makes it a more detailed tool than earlier indices for understanding molecular geometry. In this work, we calculate a generalized form of the face neighbourhood index for carbon nanocones CNC _m [n]. These results can help researchers better explain the physical and chemical behavior of nanocones and guide future applications in nanoscience.

Graph with no edge intersection other than endpoints and that can be embedded on a plane is termed as a planar graph. A subset of vertices of a graph G is said to be an edge resolving set, if each edge in the graph is uniquely identified by its distances to the vertices in the edge resolving set. The smallest cardinality of an edge resolving set for G is called the edge metric dimension and is denoted by edim(G). In this article, we determine the edge metric dimension and independent edge metric dimensions for 3-cycle and 4-cycle extended kayak paddle graphs.

A metric basis is a subset of vertex set in a graph [ ], such that every vertex in the graph [ ] has a unique distance vector to the vertices in the subset. The cardinality of the metric basis is called the metric dimension of [ ], and is presented by [ ]. In this paper, we define the hellebores flower graph, and it is denoted by [ ]. For this graph, we investigate its metric basis as well as the metric dimension, and we also show that it is unbounded and is equal to [ ] if [ ] and [ ] otherwise.

Let [ ] denote an undirected, simple, nontrivial, and connected graph with an edge set [ ] and the vertex set [ ]. A subset of distinct ordered vertices [ ] is termed as resolving set for [ ] if for every two distinct vertices say [ ] and [ ], there exist a vertex [ ] such that [ ]. The minimum resolving set of [ ] among all the resolving sets, is referred to as the metric basis. The cardinality of metric basis set is known as the metric dimension (MD) of [ ]. In this study, we calculated the MD for a family of planar graphs exhibiting rotational symmetry, demonstrating that each graph in this family possesses an independent minimum resolving set with a cardinality of three.