Patched Network and Its Vertex-Edge Metric-Based Dimension

Let M = {v1, v2,…, vn} be an ordered set of vertices in a graph G(V,E). Then (d(u, v1), d(u, v2),...d(u, vn)) is called the M-coordinates of a vertex u of G. The set M is called a resolving set if the vertices of G have distinct M-coordinates. A metric basis is a resolving set M with minimum cardinality. If M is a metric basis then it is clear that for each pair of vertices u and v in the set of vertices V of G not in M, there is a vertex m in M such that the distance between u and m is not equal to the distance between v and m. The cardinality of a metric basis of G is called metric dimension. The members of a metric basis are called landmarks. A metric dimension problem is to find a metric basis. The problem of finding metric dimension is an NP-Complete for general graphs. In this paper we have studied the metric dimension of a new graph called Octo–Nano windows , HDN like networks namely Equilateral Triangular Tetra sheets and Rectangular Tetra Sheet networks.

International Journal of Multidisciplinary Research and Analysis

Networks play an important role in electrical and electronic engineering. It depends on what area of electrical and electronic engineering, for example, there is a lot more abstract mathematics in communication theory and signal processing and networking, etc. Networks involve nodes communicating with each other. Graph theory has found considerable use in this area. In this paper, we introduce some new Networks such as Graph-PW, Network Symmetric Digraph-PW, Change Network Graph-PW, and Change Network Symmetric Digraph- PW. Moreover, several theorems and results of these networks have been studied.

Journal of Chemistry, 2022

e field of graph theory is extensively used to investigate structure models in biology, computer programming, chemistry, and combinatorial optimization. In order to work with the chemical structure, chemists require a mathematical form of the compound. e chemical structure can be depicted using nodes (which represent the atom) and links (which represent the many types of bonds). As a result, a graph theoretic explanation of this problem is to give representations for the nodes of a graph such that different nodes have unique representations. is graph theoretic study is referred to as the metric dimension. In this article, we have computed the edge version of the metric dimension and doubly resolving sets for the family of cycle with chord C t n for n ≥ 6 and 2 ≤ t ≤ ⌊n/2⌋.

Engineering and Applied Science Letters, 2018

The application of graph theory in chemical and molecular structure research far exceeds people's expectations, and it has recently grown exponentially. In the molecular graph, atoms are represented by vertices and bonded by edges. In this report, we study the several Zagreb polynomials and Redefined Zagreb indices of Oxide Network.

Scientific Reports

Resolving set and metric basis has become an integral part in combinatorial chemistry and molecular topology. It has a lot of applications in computer, chemistry, pharmacy and mathematical disciplines. A subset S of the vertex set V of a connected graph G resolves G if all vertices of G have different representations with respect to S. A metric basis for G is a resolving set having minimum cardinal number and this cardinal number is called the metric dimension of G. In present work, we find a metric basis and also metric dimension of 1-pentagonal carbon nanocones. We conclude that only three vertices are minimal requirement for the unique identification of all vertices in this network.

Journal of Chemistry, 2021

Recently, there has been increasing attention on the system network due to its promising applications in parallel hanging architectures such as distributed computing (Day (2004), Day and Al-Ayyoub (2002)). Related networks differ in the circumstances of topology, and the descriptors were freshly examined by Hayat and Imran (2014) and Hayat et al. (2014). Distance-based descriptors, counting-related descriptors, and degree-based descriptors are all examples of topological descriptors. These topological characteristics are linked to chemical features of a substance, such as stability, strain energy, and boiling point. The specifications for the 1st Zagreb alpha, 1st Zagreb beta, 2nd Zagreb, sum-connectivity, geometric-arithmetic, Randic, harmonic, and atom-bond connectivity indices for mesh networks M N m × n based on VE and EV degree are discussed in this paper.

The field of mathematics plays vital role in various fields. One of the important areas in mathematics is graph theory which is used in structural models. This structural arrangements of various objects or technologies lead to new inventions and modifications in the existing environment for enhancement in those fields. This Paper describes the description of graph theory.

Advances and Applications in Discrete Mathematics

Let G be a simple graph. G L is the Laplacian matrix of G and G a 1 p p PG Z Z ℾ 2 p p Z Z and , p p PG Z Z where p is a prime. We also find their girth, vertex connectivity and discuss planarity and Eulerian properties.

IEEE Access, 2022

A molecular (chemical) graph is a simple connected graph, where the vertices represent the compound's atoms and the edges represent bonds between the atoms, and the degree (valence) of every vertex (atom) is not more than four. In this paper, we determine the edge metric basis and edge metric dimension (EMD) of the complex molecular graph of a one-heptagonal carbon nanocone (HCN 7 (q)). We prove that only three non-adjacent vertices are the minimum requirement for the identification of all the edges in HCN 7 (q), uniquely. INDEX TERMS Connected graph, edge metric basis, edge metric dimension, independent set, oneheptagonal carbon nanocone, resolving set.

Symmetry, 2021

Graph theory can be used to optimize interconnection network systems. The compatibility of such networks mainly depends on their topology. Topological indices may characterize the topology of such networks. In this work, we studied a symmetric network θϕ formed by ϕ time repetition of the process of joining θ copies of a selected graph Ω in such a way that corresponding vertices of Ω in all the copies are joined with each other by a new edge. The symmetry of θϕ is ensured by the involvement of complete graph Kθ in the construction process. The free hand to choose an initial graph Ω and formation of chemical graphs using θϕΩ enhance its importance as a family of graphs which covers all the pre-defined graphs, along with space for new graphs, possibly formed in this way. We used Zagreb connection indices for the characterization of θϕΩ. These indices have gained worth in the field of chemical graph theory in very small duration due to their predictive power for enthalpy, entropy, and ...