Some Properties of Cubic Fuzzy Graphs with an Application

A fuzzy graph is one of the versatile application tools in the field of mathematics, which allows the user to easily describe the fuzzy relation between any objects. The nature of fuzziness is favorable for any environment, which supports to predict the problem and solving it. Fuzzy graphs are beneficial to give more precision and flexibility to the system as compared to the classical model (i.e.,) crisp theory. A topological index is a numerical quantity for the structural graph of the molecule and it can be represented through Graph theory. Moreover, its application not only in the field of chemistry can also be applied in areas including computer science, networking, etc. A lot of topological indices are available in chemical-graph theory and H. Wiener proposed the first index to estimate the boiling point of alkanes called ‘Wiener index’. Many topological indices exist only in the crisp but it’s new to the fuzzy graph environment. The main aim of this paper is to define the topological indices in fuzzy graphs. Here, indices defined in fuzzy graphs are Modified Wiener index, Hyper Wiener index, Schultz index, Gutman index, Zagreb indices, Harmonic index, and Randić index with illustrations. Bounds for some of the indices are proved. The algorithms for distance matrix and MWI are shown. Finally, the application of these indices is discussed.

Several bioactivities of chemical compounds in a molecular graph can be expected by using many topological descriptors. A topological descriptor is a numeric quantity which quantify the topology of a graph. By defining the metric on a graph related with a vector space, we consider this graph in the context of few topological descriptors, and quantify the Wiener index, hyper Wiener index, Reciprocal complimentary Wiener index, Schultz molecular topological index and Harary index. We also provide the graphical comparison of our results to describe the relationship and dependence of these descriptors on the involved parameters.

A graph is a pair (V, R), where V is a set and R is a relation on V. The elements of V are thought of as vertices of the graph and the elements of R are thought of as the edges Similarly, any fuzzy relation ρ on a fuzzy subset μ of a set V can be regarded as defining a weighted graph, or fuzzy graph, where the edge (x, y) ∈ V × V has weight or strength ρ(x, y) ∈ [0, 1]. In this chapter, we shall use graph terminology and introduce fuzzy analogs of several basic graph-theoretical concepts. For simplicity, we will consider only undirected graphs through out this chapter unless otherwise specified. Therefore, all fuzzy relations are symmetric and all edges are regarded as unordered pairs of vertices. We abuse notation by writing (x, y) for an edge in an undirected graph (V, R), where x, y ∈ V. (We need not consider loops, that is, edges of the form (x, x); we can assume, if we wish, that our fuzzy relation is reflexive.) Formally, a fuzzy graph G = (V, μ, ρ) is a nonempty set V together with a pair of functions μ: V → [0, 1] and μ: V × V → [0, 1] such that for all x, y in V, ρ(x, y) ≤ μ(x) ∧ μ(y). We call μ the fuzzy vertex set of G and ρ the fuzzy edge set of G, respectively. Note that ρ is a fuzzy relation on μ. We will assume that, unless otherwise specified, the underlying set is V and that it is finite. Therefore, for the sake of notational convenience, we omit V in the sequel and use the notation G = (μ,ρ). Thus in the most general case, both vertices and edges have membership value. However, in the special case where μ(x) = 1, for all x ∈ V, edges alone have fuzzy membership. So, in this case, we use the abbreviated notation G = (V, ρ). The fuzzy graph H = (v,τ) is called a partial fuzzy subgraph of G = (μ, ρ) if v ⊆ μ and τ ⊆ ρ. Similarly, the fuzzy graph H = (P, v, τ) is called a fuzzy subgraph of G = (V, μ, ρ) induced by P if P ⊆ V, v(x) = μ(x) for all x ∈ P and τ(x, y) = ρ(x, y) for all x, y ∈ P. For the sake of simplicity, we sometimes call H a fuzzy subgraph of G. It is worth noticing that a fuzzy subgraph (P, v, τ) of a fuzzy graph (V, μ, ρ) is in fact a special case of a partial fuzzy subgraph obtained as follows.KeywordsIncidence MatrixIntersection GraphFuzzy SubsetInterval GraphFuzzy RelationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Wiener index is the oldest and one of the most famous topological indices in mathematical chemistry. Recently, a new extension of the Wiener index, called the Terminal Wiener index TW(T) of a tree T, has been put forward and investigated in several published articles. In the present paper, an eort has been made to nd a generalized formula capable to calculate the Wiener and the Terminal Wiener indices of some rooted trees i.e: the Dendrimer tree and the Complete Rooted tree. Mathematics Subject Classication: 05C10, 05C12

Topological indices are graph invariants computed usually by means of the distances or degrees of vertices of a graph. In chemical graph theory, a molecule can be modeled by a graph by replacing atoms by the vertices and bonds by the edges of this graph. Topological graph indices have been successfully used in determining the structural properties and in predicting certain physicochemical properties of chemical compounds. Wiener index is the oldest topological index which can be used for analyzing intrinsic properties of a molecular structure in chemistry. The Wiener index of a graph $G$ is equal to the sum of distances between all pairs of vertices of $G$. Recently, the entire versions of several indices have been introduced and studied due to their applications. Here we introduce the entire Wiener index of a graph. Exact values of this index for trees and some graph families are obtained, some properties and bounds for the entire Wiener index are established. Exact values of this new index for subdivision and $k$-subdivision graphs and some graph operations are obtained.

In this article, a new idea of fuzzy fractional coloring of fuzzy graph is presented and fuzzy fractional chromatic number is defined. A relationship between fuzzy fractional chromatic number and fuzzy fractional clique number is established. Some properties of fuzzy chromatic number of fuzzy graphs and fuzzy fractional chromatic number of fuzzy graphs are proved and the concept of k-strong adjacent vertices is introduced. Fuzzy chromatic number and fuzzy fractional chromatic number have been calculated on lexicographic product of two fuzzy graphs. Also, fuzzy chromatic number, independence number and fuzzy fractional chromatic number have been investigated on disjoint union of two fuzzy graphs. Lastly, a real life application of fuzzy fractional coloring on fuzzy graph is discussed.

Nowadays the fuzzy graphs gained popularity due to their wide applications in different areas of science, engineering, social sciences, etc.In this paper, we consider covering problem in fuzzy graph. Different types of covering problems have been defined in this paper. Almost all problems are new. For each of these problems separate study is required. Also, the covering set, strong covering set, minimal covering set, etc., are defined and explained with examples. In a graph, each vertex can cover only those vertices that lie within its covering radius along shortest path and no vertex can cover itself. Here, an algorithm is designed to find the different types of covering sets on a fuzzy graph. An imprecise number, instead of a real number, namely interval number and triangular fuzzy number are considered as arc length of a fuzzy graph. To the best of our knowledge no works are available for covering problem on fuzzy graphs/networks. An application of covering set in natural disaster management is discussed to highlight the importance of the covering problem. In this application, an algorithm is designed to find all the facility vertices (or supply vertices) for given vertex and covering radius.

Network similarity measures have proven essential in the field of network analysis. Also, topological indices have been used to quantify the topology of networks and have been well studied. In this paper, we employ a new topological index which we call the Ediz eccentric connectivity index. We use this quantity to define network similarity measures as well. First, we determine the extremal value of the Ediz eccentric connectivity index on some network classes. Second, we compare the network similarity measure based on the Ediz eccentric connectivity index with other well-known topological indices such as Wiener index, graph energy, Randić index, the largest eigenvalue, the largest Laplacian eigenvalue, and connectivity eccentric index. Numerical results underpin the usefulness of the chosen measures. They show that our new measure outperforms all others, except the one based on Wiener index. This means that the measure based on Wiener index is still the best, but the new one has certain advantage to some extent.

For a graph, the forgotten topological index (F-index) is defined as the sum of cubes of degrees of vertices. In 2009, Eliasi and Taeri [M. Eliasi, B. Taeri, Four new sums of graphs and their wiener indices, Discrete Appl. Math. 157 (2009) 794-803] introduced four new sums (F-sums) of graphs. In this paper we study the F-index for the F-sums of some special well-known graphs.

On topological indices for small RNA graphs

A topological index W (Wiener index), which is the sum of all the edges between all pairs of vertices in a chemical graph, is used for characterizing branching in random-flight chains. The chains are composed of statistical bonds (or edges) of a length b jointing N beads (or vertices). The mean square radius of gyration 〈S2〉 of random-flight chains is shown to be given by 〈S2〉=(b/N)2W. On the other hand, the set of partial differential equations describing the motion of the chains, whether linear or with any mode of branching, can be expressed by a connectivity matrix (K). We demonstrate that a relationship between the matrix (K) and the Wiener index is given by W=N Tr (K)−1. It follows that the whole of linear chain theory can be generalized to include any form of branching by replacing the molecular weight or N with the Wiener index W.

Connectivity, average connectivity, and its related concepts are prominent notions in fuzzy graph theory that have applications in various areas, including neural networks, transportation problems, cluster analysis, and telecommunication. This work introduces the concepts of connectivity index, average connectivity index, type 2 connectivity index, etc. to a more generalized variant of a fuzzy graph, namely the multidimensional fuzzy graph. Basic results such as the relation of these parameters to vertices and edges, their bounds in some particular multidimensional fuzzy graphs, etc. are studied with proper illustrations. Theorems on the connectivity index of multidimensional fuzzy graphs are obtained after performing operations like direct product, tensor product, composition, join, and so on. Finally, a decision-making problem that can be effectively handled using the connectivity index is demonstrated and compared with others and similar models.

The distance degree sequence and distance neighborhood degree sequence are two sequences in fuzzy graphs. In graph theory, the distance degree sequence of a vertex involves the concept of the number of vertices at various distances from the given vertex of a graph. The concept of distance degree sequence of a vertex in fuzzy graph is developed by taking into account the membership values of the edges which are incident on the vertices at various distances from the given vertex of fuzzy graphs, and the distance neighborhood degree sequence considers the membership values of the vertices at various distances. The drastic sum of fuzzy graphs is an operation in fuzzy graph theory, which is the extension of the standard operations on graphs using the drastic sum of fuzzy sets. The drastic sum provides a way to combine two fuzzy graphs into a new fuzzy graph by using a specific rule for combining the membership values of edges and vertices between two fuzzy graphs. In this paper, the distance degree sequence and distance neighborhood degree sequence of the vertices in the drastic sum of two fuzzy graphs on star graphs are obtained in terms of the parameters of the given graphs by considering all possible cases of drastic sum in detail.

Chemical compounds are often modeled as polygonal shapes, where a vertex represents an atom and an edge symbolizes a bond. Topological properties of molecular graphs of chemical compounds can be correlated to their chemical properties and biological activities. Topological indices are the oldest and the most widely used to describing these activity relationships. Many topological indices can be expressed in terms of the distance concept in graphs. In this paper we explain a method, using the concept of distance in the graphs of zigzag polyhex nanotubes, which enables us to compute different topological indices simultaneously. Keywords: Topological index, wiener index, hosoya polynomial, distance, nanotube

Relationship of Wiener's index and adjacent eccentric distance sum index with nitroxide free radicals and their precursors as modifiers against oxidative damage