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Fuzzy graphs (FGs) are efficient mathematical models for analyzing many problems of daily life. Various problems in the fields of physics, telecommunication science, psychology and genetics can be modeled and solved with the help of fuzzy graph theory. One of the most widely used types of FG is vague graph (VG). VGs play an important role in various fields such as computer science, psychology, medicine, and political sciences and are used to find effective people in an organization or social institution. Today, connectivity indices are used in computer networks and economics to find the most suitable place for construction. These indices also have many applications in medical sciences, including ultrasound and radiology. Hence, in this research, we introduce connectivity index (CI), average connectivity index (ACI), and randic index (RI) in VGs with some examples. Likewise, some new vertices such as enhancing vertex (EV), reducing vertex (RV), and neutral vertex (NV) are described. Finally, an application of CI in the selecting town for building a school is given. [ABSTRACT FROM AUTHOR]

Today, fuzzy graphs (FGs) have made significant progress in other sciences and help researchers to always make the best choice in complex problems. A type of FG that is widely used in medical and psychological sciences is vague graph (VG). VGs play an important role in various fields such as computer science, psychology, medicine, and political sciences and are used to find effective people in an organization or social institution. Regularity is very important in the VG theory because it can help a lot in identifying people with high ability in an institute or social group. In addition to regularity, the degree of nodes and edges also play an effective role in this relationship and can evaluate the ability and knowledge of people. Therefore, in this paper, first perfectly regular vague graph (PRVG) is introduced and investigated the regularity of vertices. The totally accurate communication between all connected vertices is described by defining completely open neighborhood degree (COND) and completely closed neighborhood degree (CCND) of vertices and edges in a VG. In the following, ((s1, s2),m, (t1, t2))- regular vague graph and totally ((s1, s2),m, (t1, t2))-regular vague graph are introduced and a necessary and sufficient condition under which they are equivalent is presented. Psychologists play a significant role in creating a healthy society in terms of physical and mental health. They help people in the society to better understand their emotions and behaviors and manage them to live a better life. So in the final part, an application of COND and CCND in psychological sciences has been introduced. [ABSTRACT FROM AUTHOR]

Graphs are used to solve many problems in mathematics and computer sciences. Many structures can be displayed with the help of graphs. For example, a directed graph can be used to show how websites are related to each other. Vague graph (VG) is one of the most important graphs in the fuzzy graph (FG)-theory, which can play a significant role in finding the most suitable places in construction and also finding the shortest path in computer networks. Connectivity indices are one of the most widely used topics in graph theory, which are used in other sciences, including computer science and chemistry. One of the most famous indices in the graph is the Wiener index, which belongs to the description of the molecular structure, which is used to design molecules with desirable properties. Therefore, in this paper, we introduce important topological indices such as Wiener index, Wiener absolute index, Randic index, Zegreb index, Harmonic index, and Average Wiener index on VGs and investigate their properties with several examples. Finally, an application of the Wiener index is given to find the most suitable place to build an university. [ABSTRACT FROM AUTHOR]

Graphs are used to solve many problems in mathematics and computer sciences. Many structures can be displayed with the help of graphs. For example, a directed graph can be used to show how websites are related to cach other. Vague graph (VG) is one of the most important graphs in the fuzzy graph (FG)-theory, which can play a significant role in finding the most suitable places in construction and also finding the shortest path in computer networks. Connectivity indices are one of the most widely used topics in graph theory, which are used in other sciences, including computer science and chemistry. One of the most famous indices in the graph is the Wiener index, which belongs to the description of the molecular structure, which is used to design molecules with desirable properties. Therefore, in this paper, we introduce important topological indices such as Wiener index, Wiener absolute index, Randic index, Zegreb index, Harmonic index, and Average Wiener index on VGs and investigate their properties with several examples. Finally, an application of the Wiener index is given to find the most suitable place to build an university. [ABSTRACT FROM AUTHOR]

In 2021, a novel degree-based topological index was introduced by Gutman, called the Sombor index. Recently Kulli and Gutman introduced a vertex-edge variant of the Sombor index, is caled KG-Sombor index. In this paper, we establish lower bound on the KG-Sombor index and determine the extremal trees achieve this bound. [ABSTRACT FROM AUTHOR]

An independent Italian dominating function (IID-function) on a graph G is a function f: V (G) → {0, 1, 2} satisfying the conditions that (i) u∈N (v) f (u) ≥ 2 when f (v) = 0, and (ii) the set of all vertices assigned non-zero values under f is independent. The weight of an IID-function is the sum of its function values over all vertices, and the independent Italian domination number i I (G) of G is the minimum weight of an IID-function on G. In this paper, we initiate the study of the independent Italian bondage number b iI (G) of a graph G having at least one component of order at least three, defined as the smallest size of a set of edges of G whose removal from G increases i I (G). We show that the decision problem associated with the independent Italian bondage problem is NP-hard for arbitrary graphs. Moreover, various upper bounds on b iI (G) are established as well as exact values on it for some special graphs. In particular, for trees T of order at least three, it is shown that b iI (T) ≤ 2. [ABSTRACT FROM AUTHOR]

The concepts of fuzzy graph theory are applied in many areas of computer science including data mining, image segmentation, clustering, and image capturing. A vague graph, a generalization of a fuzzy graph, is more useful for decision making problems, social network models and computer network models. In this paper, we introduce the concept of vague neutrosophic graphs. We present the dominating and independent sets of vague neutrosophic graphs. Some properties of regular, totally regular, irregular, and totally irregular vague neutrosophic graphs are discussed. We develop the vague neutrosophic weighted average operator (Aω) and vague neutrosophic weighted geometric operator (Gω) to aggregate the vague neutrosophic information. Furthermore, we develop a vague neutrosophic multiple criteria decision making approach, in which the evaluation values of alternatives on the attributes take the form of vague neutrosophic numbers to select the most desirable one(s). Finally, a numerical example of the method was given to demonstrate the application and effectiveness of the developed method. [ABSTRACT FROM AUTHOR]

Fuzzy graphs and algorithms based on them can be very useful for the solution of many problems of practical interest. Since the uncertain and imprecise information is an essential characteristic feature virtually all real life problems, mostly uncertain, the modeling of such such problems using fuzzy graphs is difficult, even for an expert. A vague graph, an extension of the basic concept of a fuzzy graph, can be employed to deal with deeper aspects of uncertainty and imprecision for which the use of fuzzy graphs would not fully succeed. Domination is one of the most important issue in graph theory and has found many applications for the formulation and solution of many problems in various areas of science and technology exemplified by computer networks, artificial intelligence, combinatorial analyses, coding theory, etc. The concept of domination has been extended to fuzzy graphs, and vague graphs, to just name a few. We discuss here different concepts and properties related to domination in vague graphs such as an edge dominating set, an edge independent set, regular dominating set, regular independent set, and global dominating set, with some examples. Finally, we show an application of domination in vague graphs in the field of medicine that is related to the Influenza vaccine. [ABSTRACT FROM AUTHOR]

Based on the uncertainty theory, Liu [B. Liu, Some research problems in uncertainty theory, J. Uncertain Syst. 3 2009, 1, 3–10] introduced an uncertain integral for applying uncertain differential equation, finance, control, filtering and dynamical systems. Since uncertain integrals are the important content of uncertainty theory, this paper explores an approach of the relationship between uncertain integrals by the well-known Chebyshev-type inequality. Also, we propose the concept of an uncertain fractional integral which is generalized version of an uncertain integral. The definition of a strong comonotonic uncertain process and some new properties of the uncertain integral were presented in [C. You and N. Xiang, Some properties of uncertain integral, Iran. J. Fuzzy Syst. 15 2018, 2, 133–142]. Based on the strong comonotonic uncertain process, as an application, we provide Chebyshev's inequality for a fractional uncertain integral and an uncertain integral. [ABSTRACT FROM AUTHOR]

Graph theory serves as an engaging arena for the investigation of proof methods within the field of discrete mathematics, and its findings find practical utility in numerous scientific domains. Chemical graph theory is a specialized branch of mathematics that uses graphs to represent and analyze the structure and properties of chemical compounds. Topological indices are mathematical properties of graphs that play a crucial role in chemistry. They provide a unique way to connect the structural characteristics of chemical compounds to their corresponding molecular graphs. The flabellum graph F n (k , j) is obtained with the help of k ≥ 2 duplicates of the cycle graph C n with a common vertex (known as, central vertex). Then, in j of these duplicates, additional edges are added, joining the central vertex to all non-adjacent vertices. In this article, we compute different degree-based topological indices for flabellum graphs, including some well known indices, such as the Randić index, the atom bond connectivity index, the geometric–arithmetic index, and the Zagreb indices. This research provides an in-depth examination of these specific indices within the context of flabellum graphs. Moreover, the behavior of these indices is shown graphically, in terms of the parameters j , k , and n. Additionally, we have extended the concept of the first Zagreb index, to address the issue of cybercrime. This application enables us to identify criminals who exhibit higher levels of activity and engagement in multiple criminal activities when compared to their counterparts. Furthermore, we conducted a comprehensive comparative analysis of the first Zagreb index against the closeness centrality measure. This analysis sheds light on the effectiveness and relevance of the topological index in the context of cybercrime detection and network analysis. [ABSTRACT FROM AUTHOR]

Vague set as a generalization of fuzzy set increase its elasticity drastically. Vague graph is an extended structure of a fuzzy graph which gives more precision, flexibility, and compatibility to the system when compared with the classical, fuzzy and vague models. In this paper we first introduce the notion of perfectly regular vague graph that is both regular and totally regular. Then we present the concepts of strongly edge irregular vague graphs and strongly edge totally irregular vague graphs. A comparative study between strongly edge irregular vague graphs and strongly edge totally irregular vague graphs is done. Finally we give an application of vague digraph in travel time. [ABSTRACT FROM AUTHOR]

Let l , k be two positive integers. An l -distance k -rainbow dominating function on G is a function f that assigns to each vertex a set of colors chosen from the set { 1 , 2 , ... , k } such that for any v ∈ V (G) , f (v) = ∅ implies ⋃ u ∈ N l , G (v) f (u) = { 1 , 2 , ... , k }. The weight of an l -distance k -rainbow dominating function f is the value ω (f) = ∑ v ∈ V | f (v) | over all such functions f. The l -distance k -rainbow domination number of a graph G , denoted by γ r k l (G) , equals the minimum weight of a l -distance k -rainbow dominating function on G. Note that the 1 -distance k -rainbow domination number γ r k 1 (G) is the usual k -rainbow domination number γ r k (G). In this study, the l -distance k -rainbow domination number is defined and its properties are characterized. [ABSTRACT FROM AUTHOR]

An independent Roman dominating function (IRD-function) on a graph G is a function f : V(G) → {0, 1, 2} satisfying the conditions that (i) every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2, and (ii) the set of all vertices assigned non-zero values under f is independent. The weight of an IRD-function is the sum of its function values over all vertices, and the independent Roman domination number iR(G) of G is the minimum weight of an IRD-function on G. In this paper, we initiate the study of the independent Roman bondage number biR(G) of a graph G having at least one component of order at least three, defined as the smallest size of set of edges F ⊆ E(G) for which iR(G − F) > iR(G). We begin by showing that the decision problem associated with the independent Roman bondage problem is NP-hard for bipartite graphs. Then various upper bounds on biR(G) are established as well as exact values on it for some special graphs. In particular, for trees T of order at least three, it is shown that biR(T) ≤ 3, while for connected planar graphs the upper bounds are in terms of the maximum degree with refinements depending on the girth of the graph. [ABSTRACT FROM AUTHOR]