Arumugam and Mathew [Discret. Math. 2012, 312, 1584–1590] introduced the notion of fractional metric dimension of a connected graph. In this paper, a combinatorial technique is devised to compute it. In addition, using this technique the fractional metric dimension of the generalized Jahangir graph $J_{m,k}$ is computed for $k\ge 0$ and $m=5$ .

In this article, the notion of hesitant fuzzy graph is introduced. It an extended structure of a fuzzy graph that gives more precision, flexibility and compatibility to a system when compared with the system that is designed using fuzzy graphs. Several operations on hesitant fuzzy graphs are defined, namely Cartesian product, composition, tensor product and normal product. Furthermore, the relationship between the degree of vertices of the hesitant fuzzy graphs and the hesitant fuzzy graph obtained by their different products is developed. These operations are highly used in computer science, geometry, algebra, operations research, etc.

Metric-related parameters in graph theory have several applications in robotics, navigation, and chemical strata. An important such parameter is the partition dimension of graphs that plays an important role in engineering, computer science, and chemistry. In the context of chemical and pharmaceutical engineering, these parameters are used for unique representation of chemical compounds and their structural analysis. The structure of benzenoid hydrocarbon molecules is represented in the form of caterpillar trees and studied for various attributes including UV absorption spectrum, molecular susceptibility, anisotropy, and heat of atomization. Several classes of trees have been studied for partition dimension; however, in this regard, the advanced variant, the fault-tolerant partition dimension, remains to be explored. In this paper, we computed fault-tolerant partition dimension for homogeneous caterpillars , , and for , , and , respectively, and it is found to be constant. Further numerical examples and an application are furnished to elaborate the accuracy and significance of the work.

Fault-tolerance of a system measures its working capability in the presence of faulty components in the system. The fault-tolerant partition dimension of a network computes the least number of subcomponents of network required to distinctively identify each node in the presence of faults, having promising applications in telecommunication, robot navigation and geographical routing protocols. In this paper, certain triangular mesh networks including, triangular ladder (Tl_s), triangular mesh (T_s), reflection triangular mesh (rl(T_s)), tower triangular mesh (Tr_s) and reflection tower triangular mesh (rl(Tr_s)) networks are discussed for their partition and fault-tolerant partition resolvability. In this regard, it is shown that the partition dimension of these networks is 3, whereas their fault-tolerant partition dimension is 4.

Apart from many branches in mathematical sciences, the main share of applications of graph theory nowadays is in chemistry. The graph-theoretical ideas have been used by many chemists for more than two centuries but over the last 30 years, the great significance of graph theory in chemistry has been extensively acknowledged. In a chemical graph, the vertices can symbolize atoms, orbitals, bonds, collection of atoms, molecules, or group of molecules and the edges symbolize the connection between chemical substances and are used to describe reactions, chemical bonds, kinetic models, reaction mechanisms, or other interactions and alterations of the chemical substances. Benzene is the archetypal aromatic molecule, showing exceptional chemical stability as compared with other unsaturated hydrocarbons. Fullerenes are a unique family of carbon-based cage molecules having remarkable properties. Fullerene has potential applications in medicinal chemistry due to its size, hydrophobicity, three-dimensionality, and electronic configurations. In this paper, we will calculate the fault-tolerant partition dimension of classes of fullerene graphs, n-linear benzene graph and partition dimension of polyphenyl chains.

Graph invariants provide an amazing tool to analyze the abstract structures of networks. The interaction and interconnection between devices, sensors, and service providers have opened the door for an eruption of mobile over the web applications. Structure of web sites containing number of pages can be represented using graph, where web pages are considered to be the vertices, and an edge is a link between two pages. Figuring resolving partition of the graph is an intriguing inquest in graph theory as it has many applications such as sensor design, compound classification in chemistry, robotic navigation, and Internet network. The partition dimension is a graph parameter akin to the concept of metric dimension, and fault-tolerant partition dimension is an advancement in the line of research of partition dimension of the graph. In this paper, we compute fault-tolerant partition dimension of alternate triangular cycle, mirror graph, and tortoise graphs.