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From a mathematical perspective, the energy of a graph is described as the total absolute values of the eigenvalues of the adjacency matrix of the graph. It was commonly related to the theory of spectral graphs. The energy of the graph is initially characterized by Ivan Gutman in 1978. However, an exploration of energy was inspired by Hukel's work in the 1930s when accustomed to calculating the total amount of π-electron energy of molecules. Since then, several mathematicians have become concern about the energy of the graph and a few modifications have been proposed. Previously, the line graph of prime graph, L(PG(R)) has been introduced. In a prime graph, PG(R) either two different vertices x, y ∈ R are adjacent in the case that yRx = 0 or xRy = 0. Meanwhile, the line graph of prime graph, L(PG(R)) was defined as a graph whose vertex set is made up of prime graph edges, where the two vertices are adjacent if their respective edges in the prime graph share the same vertex. The goal of this paper is to determine the energy of L(PG(Zn)) for n = 3, 4, 5 by using the eigenvalues of the graph. The eigenvalues are obtained by mapping the elements onto their adjacency matrix. Finally, the energy of L(PG(Zn)) for n = 3, 4, 5 is computed. It is found that the energy of L(PG(Zn)) is equal to 2n − 4. [ABSTRACT FROM AUTHOR]

Graph-theoretic concepts are useful in studying various uses in several fields like mechanical & electrical engineering, specifically multi-physics systems. A core aim of the present study is to run an outline of the uses of graph-theoretical ideas in the modeling & investigation of systems in numerous areas. Graph theory (GT) is used in gear train analysis synthesis for decades, & this was an effective & organic modelling tactic in designing gear transmission. The graph-based technique for kinematic & static power examination, power stream, & mechanical proficiency calculation is additionally introduced. The paper additionally features an overall methodology for taking care of designing issues by changing them to issues in other designing areas, via broad discrete numerical models known as GRs. Design synthesis is the central point of interest in the mechanically applied plan to create the plan up-&-comers that meet the plan prerequisites. This paper reviews a novel & calculable synthesis approach of components dependent on GT. It is likewise settled unambiguously the connection between linear graphs & physical systems. Among the continuum of mechanical framework models we can recognize labelling graphs, which can be utilized for detailing & formalizing mechanical assignments. Mechanically coupled frameworks are successfully displayed utilizing hybrid graphs which consolidate attributes of polar & flow graphs & are unrivalled as far as algorithmic methodology. A hybrid used to portray a model of a mechanical framework relies upon the chosen algebraic method & applying matrices is the most helpful strategy to decide the dynamic qualities of the framework [ABSTRACT FROM AUTHOR]

Research on the algebraic graph theory is still being developed by many researchers. Let ℤ be a commutative ring. A graph I(ℤ) is a graph with a vertex set of units ℤ and x, y∈ℤ, x≠y, are adjacent if and only if x. y=1, and all vertices adjacent to 1. The complement of I(ℤ)= (V(I(ℤ)), E(I(ℤ))), denoted by I (ℤ) ¯ = (V (I (ℤ)) ¯ , E (I (ℤ ¯))) , is a graph with V (I (ℤ)) ¯ = V (I (ℤ)) and E (I (ℤ)) ¯ = { x y ≠ E (I (ℤ)) : x , y ∈ V (I (ℤ)) }. In this paper we determine the Laplacian spectrum of the complement of identity graph I (ℤ 2 p) ¯ , for some prime p, that can be constructed by investigating the eigenvalues of I (ℤ 2 p) ¯. The result shows that all eigenvalues of I (ℤ 2 p) ¯ are integers. [ABSTRACT FROM AUTHOR]

In a communication network, channel assignment is an essential component that can be divided into three categories which are static, dynamic, and hybrid channel allocation. In this paper, we focus on the dynamic channel allocation that is applied in Wireless Mesh Networks (WMN) where communication between two nodes occurs when they are both assigned the same channel. Wireless networking is a way to connect various equipment without having to install expensive cables throughout a place. Channel assignment in the networks is an application of graph theory on the vertex coloring. Three types of interference are considered in our work which are adjacent channel, cochannel and cosite constraints. The channels are allocated in such a way to minimize the interference and maximizing the performance of the networks. The interference is said to be minimized when all the constraints are considered, so that the interference can be avoided. The network is optimized if the interference reduced with a minimum number of the channel used. An algorithm called Improved Greedy Algorithm was proposed to solve the multi-channel assignments in wireless mesh networks by considering the adjacent channel, cochannel and cosite constraints. The Improved Greedy Algorithm has been tested with different values of constraints. From the results, it can be seen that the Improved Greedy Algorithm perform significantly well for the channel assignment problem. [ABSTRACT FROM AUTHOR]

Literature review is one of the essential steps in research process which will conceptualize our research progress both quantitative and qualitative way. This paper illustrates the survey about the concepts of Rough sets in the domain of Graphs. Rough set Theory and Graph theory are two major disciplines in computational mathematics. This paper exhibits the exposition to unification process of these two major theories and existing as well as ongoingworks in these domains are well explained. [ABSTRACT FROM AUTHOR]

The first definition of fuzzy graph by Kanfinam (1973) was based on Zadehs fuzzy realition (1971). But it was Azraiel Rosenfield (1975) who considered fuzzy relation on fuzzy sets and developed the theory of fuzzy graphs. Battacharya (1987) established that a fuzzy group can be associated with a fuzzy graph in a natural way. Bhutani (1989) introduced homomorphisms of fuzzy graphs and showed that the collection of all automorphisims of a fuzzy graph is a group under set theoretic product of functions as a binary operation. In this paper, we introduce the aggregation operation on Hesitancy fuzzy graphs and apply the Ordered weighted averaging aggregation operation on Hesitancy fuzzy graphs. [ABSTRACT FROM AUTHOR]

The paper considers some author's approaches to develop a machine translation system to handle texts specialized in construction industry both in Russian and English. Graph theory is the basis for automatic sentence analysis and representation of its structure. At the first stage, the sentence is represented by a line graph. The article discusses how to assign attributes to each word in accordance with its prospective belonging to different parts of speech for their subsequent processing using graph convolutional neural networks (GCN). The initial features of each element of the sentence are compared with the calculated values obtained, and by means of certain parameters for the compiled vectors, they are classified as parts of speech. The paper contains parts of the software listing for the computer implementation of the proposed technology. [ABSTRACT FROM AUTHOR]

In the case of "modeling a manufacturing program represented through the product state concept, the nodes are state characteristics, and the lines are the (inter-) relations between the state characteristics. These can be within the product state, or between neighboring states, or system-wide. Once the graph of the model of a manufacturing program has been obtained, in this case, the visualization model, the resultant graph can be considered complex due to its size and the multiple relations among its nodes". Considering, that "the more nodes and lines a graph has, the more complex the graph is". "Divisor 3-equitable labeling (D3EL) is a bijection f: V(G) → { 1,2, ..., n; } that induces a function f′: E(G)→{0, 1, 2} defined by for each line e = xy, (i) f'(e) = 1 if f(x)|f(y) or f(y)|f(x), (ii) f'(e) = 2 if f(x)/f(y) = 2 or f(y)/f(e) = 2, and (iii) f'(e) = 0 otherwise such that |ef′(i)−ef′(j)| ≤ 1 for all 0 ≤ i, j ≤ 2, where ef′(i) is the number of lines having labels i under f'. A graph that admits D3EL is called a divisor 3-equitable graph (D3EG)". This paper shows the existence and non-existence of D3EL of certain graphs, besides highlighting the applications of graph theory in functional materials and manufacturing. [ABSTRACT FROM AUTHOR]

Two spaces are said to be homotopy equivalent in classical homotopy theory if one space may be continuously deformed into the other. However, this theory disregards the discrete nature of graphs. Because of this, there is a discrete homotopy theory that distinguishes between a graph's vertices and edges. It's called A-homotopy theory, the purpose of this paper is to provide and investigate a novel notion of graph homotopy with an equivalence definition. The 3-cycle 퐶3 and 4-cycle 퐶4 are conteactible, that is, they are viewed as being identical to a single vertex. We define equivalence graph Homotopy proposing an alternate definition. [ABSTRACT FROM AUTHOR]

This paper introduces the hybrid of Potential Method (PM) and Graph Theory and Matrix Approach (GTMA) to surmount the issues in the existing hybrid of Analytic Hierarchy Process (AHP) and Graph Theory and Matrix Approach (GTMA) concerning the complexity of the calculation of the weight of the attributes when involving many attributes, no visualization of the relationship between attributes and complex calculation of ranking order of attributes for missing information. Subsequently, the implementation of the proposed method is utilized in the fuel selection problem in solving two issues namely complexity in pairwise comparison and missing information. Besides, the proposed method could surmount the complexity in pairwise comparison and deal well with missing information in providing the ranking order of alternatives. Finally, the results obtained indicate that the method could be utilized for ranking purposes in Multi-Criteria Decision Making (MCDM) problem. [ABSTRACT FROM AUTHOR]

A topological index is a numerical measure characterizing a whole structure of a particularly given graph. Moreover, in chemical graph theory, the Zagreb indices idea, which is based on the number of vertex degrees, was proposed. More Zagreb index variations were later created, and some of these indexes were recently produced. This paper outlines the Zagreb indices and Zagreb coindices with respect to the line graphs of Polyphenylene dendrimer subdivision graphs. Numerous creative families of Zagreb indices were also numerically simulated, comparing the two types of indices that can be relevant with respect to the quantitative structure-activity relationship (QSAR) as well as quantitative structure-property relationships (QSPR) research. [ABSTRACT FROM AUTHOR]

A soft robotic hand is a robotic end-effector that can pick and place objects with high flexibility to handle fragile objects and adapt to unstructured and irregular shapes. One promising approach to handling soft objects is combining the rigid structure of the tendon-driven robot hand with elastomeric or rubber elements. It is essential to elaborate the tendon-driven mechanisms either in the closed-loop or open-loop kinematic system o obtain proper behavior of the under-actuation robotic hand. More specifically, the functional representation of kinematic construction of related components like shafts, pulley, and tendons must be carefully considered. This paper presents a systematic review of some recent technological advances in the area of soft robotic grippers. To analyze the tendon-driven mechanism (TDM), the kinematic synthesis is performed in this paper using graph theory. After the displacement equations are generated, the input and output torque (forces) connections are further appropriately obtained. Finally, a proposed design is given for a soft 2 degree of freedom (DOF) robotic gripper through exploring the openhand Yale model T, which employs soft rubber as the connector to each rigid link. [ABSTRACT FROM AUTHOR]

The paper presents a study in which the author continues to develop PC software for the translation system to translate specialized texts from English into Russian and vice versa in the construction area. The basis for automatic analysis of a sentence and representation of its structure is graph theory. At the first stage, the sentence is represented by a linear graph. The article discusses ways of assigning features to each word in accordance with its belonging to different parts of speech for their subsequent processing by means of graph convolutional neural networks (GCN). The initial features of each element of the sentence are compared with obtained values , and using certain parameters for the compiled vectors, they are classified as parts of speech. The paper includes parts of the software program for the computer implementation of the proposed machine translation technology. [ABSTRACT FROM AUTHOR]

Material science has witnessed many applications of Mathematics involving Graph theory and Topology. In this paper some concepts of Graph theory and Topology has been linked together. The order of points of a topological space with topology as incidence topology generated by Comb graph and Helm graph has been presented in this paper. Also, the order of edges with respect to respective incidence topology of these special graphs have been studied. Various results based on connectedness and separation axioms of the incidence topology generated by these graphs have been discussed. Moreover, order of these nodes and edges with respect to the incidence topology has been established and generalized for these special graphs and supporting examples are illustrated along with the results. [ABSTRACT FROM AUTHOR]

In this paper the inclusion and exclusion approach was used to calculate the reliability expression for the Advanced Research Projects Agency Network (ARPANET). The policy entails the use of either a path group or a segment set. Additionally, the paper deals with a computationally efficient and easy-to-concept method for obtaining minimum path sets using the powers of a network connection matrix. The method is simple and computationally efficient, and it can be used on graphs with finite non-zero failure probabilities in both nodes and branches. The method makes use of Boolean algebra and graph theory principles. The method is demonstrated with an example. [ABSTRACT FROM AUTHOR]

Cryptography is a concept to protect information and communication from adversaries through the use of codes. Graph theory is ultimately the study of graphs. In this paper we are using cryptographic technique together with graph labeling to protect our message being hacked by any adversaries as well as safeguarding the communication. This paper proposes an enriched technique of coding a message based on Polybius cipher and permutation cipher for the regular graph using product edge labeling techniques. The resulting cipher text is forwarded to the receiver in a structure of cipher graph which admits our proposed edge labeling. The receiver, thus decrypts the edge labels using secret keys shared by the sender. Here we make use of symmetric key cryptosystem as we perform both encryption and decryption using the same key. The main advantage of symmetric cryptosystem is that it is effective and fast for a huge collection of data. [ABSTRACT FROM AUTHOR]

The aim of this paper is to calculate the Dominating set in the Famous Fractal Graph Von Koch Curve which is created by Hedge Von Koch. Domination is the most development area in Graph Theory. Its application plays a vital role in our natural lives. Here this paper analyzes the characteristics, construction, properties, nature, implementation, Iteration and uses of the famous Fractal Graph like Von Koch Curve. Here to explain how the Fractal Graph can be constructed and implemented. It shows that each Iteration of Koch Curve has some constant ratio for implementation of Vertices and Edges in all Iteration. By using Proof of Iterative methods, it evaluates the Dominating Set Cardinality of Von Koch Curve. The calculation of dominating sets shows the common ratio for all the Iteration of Von Koch Curve. Application of this calculation also discussed in our life. Also the algorithm of the calculation of Dominating Set has written using C Programme. [ABSTRACT FROM AUTHOR]

Domination has its origin in 1862 when C. F. De Jaenisch investigated the issue of identifying the smallest number of queens required to cover a chessboard. Around 1960 Berge and Ore started the mathematical exploration of domination theory in graphs. Let Z = (X, Y) denote any finite graph, and D represents any subset of vertex set X. A set of vertices D in a graph Z = (X, Y), is said to be a dominating set(DS) if the neighborhood of D is whole of Xz i. e., Nz[D] = Xz. The minimal cardinality of DS for Z is, therefore, the domination number (DN) denoted by γ(Z). In this paper we have reviewed literature on domination theory of graphs and then investigated and generalized some general results for DS and DN of the graphs such as diamond snake(Dm), banana tree(B(m, n)), coconut tree(CT(m, n)), and firecracker(F(m, n)). [ABSTRACT FROM AUTHOR]

In Graph theory, colouring is the best and most well-considered amplitude. The four-colour problem has contributed to great change in the world of mathematics. Graph colouring has become an essential of interest in graph theory. This often includes its disparate theoretical conclusions, its unconcluded issues and its several utilizations. Throughout this paper, the Conjecture Steinberg is studied with a counter example. Conjecture Steinberg states that "Each planar graph except cycles of length four or five is 3-colourable". Whereas an example by which there exists a planar graph with four or five cycle is 3-colourable is encountered for the Steinberg conjecture. [ABSTRACT FROM AUTHOR]

Graph rubbling and graph pebbling, known as graph moves in graph theory, are originated from number theory. Graph rubbling is an extension of graph pebbling. In graph pebbling, two pebbles from the vertices are removed, and one pebble from adjacent vertices will be added. Pebbles are thought of as fuel containers in graph theory. In graph pebbling, a vertex is said to be reachable if a pebble can be moved to that vertex using pebble moves. This paper addresses the extension of pebbling, which is called rubbling. Graph rubbling differs from Graph pebbling in the sense that graph rubbling possesses an additional step of adding a pebble on the vertices after the removal of a single pebble, each from the two adjacent vertices. If the adjacent vertices are similar, the rubbling becomes pebbling; however, if they are different, rubbling is called strict rubbling. The t–rubbling number of a graph G is the minimum number of n pebbles in a pebble distribution on the vertices of G with which we can make t–pebbles on any vertices with making them reachable. In this paper, we have extended the t pebbling, which is derived to find the t-rubbling number of Jahangir graphs and lamp graph. We derived the t-rubbling number for Jahangir graph for n = 3, n = 7, and n > 7, whereas for Lamp graph for n = 3, n = 4, n = 5, n = 6, n = 7, and n > 7. [ABSTRACT FROM AUTHOR]

The paper discusses approaches to the formation of an optimal program for improving scientific and technical activities (STA) efficiency of universities using modern approaches of organizational systems management, project management, and active systems theories. The successive stages of scientific and technical activity, which represent research and development work, design and development work, and product manufacturing based on their results, are considered. Graph theory methods are used to represent the structure of scientific and technical projects, modeling related processes, and to find optimal solutions in the field of scientific and technical activities of universities. The search for optimal solutions in the field of universities' STA is based on the application of dichotomous programming methods. The tasks of constructing the dependence of the project implementation effect on its resource availability are set. The described technology can be generalized to complex design structures. The paper describes the problem of coordinated program planning related to the presence of two sources of universities' scientific and technical activities formation – external orders and internal proposals. As the main issue of targeted and operational management of resources in the formation of the university's STA program, the main priority mechanisms for their distribution were discussed. [ABSTRACT FROM AUTHOR]

Graph theory can be said as one of the branches of applied mathematics that is currently developing. A problem can be explained in detail by simplifying it with graph theory. This paper concerns with graphs that are used in everyday life as a network system to describe parts of a structure in order to make it simpler and easier to understand. The network flow system is then applied in many disciplines, including applied mathematics, computer science, operations research, management, and engineering. [ABSTRACT FROM AUTHOR]

Concept of locating-chromatic number for a graph is one interesting concept in graph theory. This concept is married between two concept which old and new concept. That concept are coloring and dimension partition for a graph. The locating-chromatic number is development research in graph theory. Let G(V(G),E(G)) is a connected graph. Let ∏ = { S 1 , S 2 ,... , S k } and color code c π (v) = (d (v , S 1) , d (v , S 2) ,... , d (v , S k)) where d (v , S i) = min { d (v , x) | x ∈ s i } with 1≤i≤k. If every vertices of G have distinct color codes, then c is called a locating coloring of G. The minimum number of colors used in locating coloring of G is called locating chromatic number of G denoted by XL(G). In this paper we obtain the locating chromatic number for certain lobster graph Ln,m,1 especially for m=3,4,5. [ABSTRACT FROM AUTHOR]

The methodology of functional modeling provides visual and understandable means of describing the domain for a wide range of specialists. The paper proposes a formal language for describing a functional model based on graph theory. Within this language, each functional diagram is represented as a graph with marked edges. The nodes of this graph define the function blocks, the edges correspond to the arrows of the diagram. We develop rules for diagram bounds description with special nodes, the arrows positions using the system of roles, branching arrows as a set of special nodes and multiple edges. The hierarchy which links individual diagrams into a single model, is defined by the relation of decomposition on graphs. We form rules that connect the arrows of the parent and child functional diagrams in correctly constructed functional models. To verify these rules, a functional model verification algorithm has been developed. This algorithm is implemented by means of logic programming using the PROLOG language. The structure of the fact base for describing the relations of decomposition, nodes, and edges of graphs is proposed. A set of predicates is provided to verify the correctness of the functional model description. [ABSTRACT FROM AUTHOR]

In this paper, the relationship between the algebra and graph theory are been studied especially the zero divisor graph in the ring ZN and constitute graph of these element Γ(Zn). The graph obtained by the above relation are isomorphic to some certain graphs as a complete and complete bipartite graph. The number of some vertices of Γ(Zn) are been introduced. Also, the domination of the zero divisor graph to some certain graphs are been determined. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to Introduce a nano-topological space via graph theory which is depend on the neighborhood between the vertices. Some characterization of nano-topological space as nano closure , nano interior , nano kernel were studied At last , the relations among these concepts were studied and proved. [ABSTRACT FROM AUTHOR]

In topological spaces, which are branches of pure mathematics, are difficult to find, the significance of this paper is to find some in graph theory via some topological space concepts, we introduced some well-known concepts in topology and related them to a graph such as (graph interior, graph closure, graph exterior, graph boundary and graph limit point). We study a characterization and relationships between them and present some results and theorems about this notions. In this work includes all Species graphs. [ABSTRACT FROM AUTHOR]

For decades, the development of robots has been very significant along with technological development. Robots used are to help complete human tasks. Initially, researchers conducted a lot of research related to single-robot motion systems. However, under certain conditions, the single-robot cannot help complete human tasks and cannot work in large-scale environments, so a multi-robot system is developed. A Multi-robot system is a robot in an environment unknown to work cooperatively for a particular purpose. To estimate motion in a multi-robot motion system, one of the data assimilation methods can be used, namely Ensemble Kalman Filter. The Ensemble Kalman Filter used to estimate various problems shaped model strongly nonlinear system and has shown that able to solve a dynamical system nonlinear model in the state of being big. After estimation, this paper also controls the movement of the robot. The control design used is consensus control based on graph theory and control theory. In consensus control, the connected graphs are intended towards a specific goal, namely towards the point of agreement. The results of the implementation of Ensemble Kalman Filter and consensus control on a multi-robot motion system obtained worked very well with very small error. [ABSTRACT FROM AUTHOR]

One of the central algorithmic graph theory problems is the minimum dominating set (MDS) problem. The MDS has various implementations particularly in graph mining. Various fields such as engineering, physics, social and biological sciences are widely used graph theory.Within the most powerful naturally algorithms are inspired for optimization. The firefly algorithm (FFA) FFA had been successfully used for solving many problems in several fields. This paper introduces FFA for solving the MDS problem. FFA is a general purpose algorithm; however, the problem has to be presented in an appropriate manner to convenient the presented algorithm steps. Hence, binary presentation in this study has been implemented for the MDS problem. Overall, the effectiveness of the presented algorithm experimental results demonstrated over the MDS problem. [ABSTRACT FROM AUTHOR]

Graph theory and topology have applications in material science. In this paper, the order of elements of a topological space with incidence topology generated by Star graph has been studied. Various results based on connectedness and separation axioms of the incidence topology generated by this undirected graph have been discussed. Moreover, order of these nodes as well as order of edges with respect to the incidence topology have been established and generalized for star graph and sufficient examples are illustrated to support the results. [ABSTRACT FROM AUTHOR]

Material Science is a field that applies many aspects of Mathematics, including topology and graph theory. In this paper, the order of elements of a topological space with topology as incidence topology generated by Path graph has been studied. Various results based on connectedness and separation axioms of the incidence topology generated by path graph have been discussed. Moreover, order of these nodes as well as order of edges with respect to the incidence topology have been established and generalized for these graphs and sufficient examples are illustrated to support the results. [ABSTRACT FROM AUTHOR]

Combinatorial optimizations is subfield optimization which exists to find the optimality from finite set, and it has important applications in several fields. NP- hard problem is one of the important tools in combinatorial optimizations. We introduce classical application problems of maximum independent set in real daily life. We discuss known applications and different approaches. This list of several discussed applications are Matching Molecular Structures by Clique Detection, Macromolecular Docking, Integration of Genome Mapping Data, Comparative Modeling of Protein Structure, Covering Location Using Clique Partition, and coding theory and graph coloring. Also, this paper presents simple algorithms for NP-hard problem. [ABSTRACT FROM AUTHOR]

Graph theory is a branch of mathematics which is quit rich and interesting. The study of domination is one of the fastest growing area in graph. There are different types of domination such as connected domination, independent domination, total domination, paired domination, secure domination , etc. "The secure dominating set S of a graph G is a dominating set with the property that each vertex u∈V\ S is adjacent to a vertex v∈S such that (S\ v) Uu is a dominating set. The minimum cardinality of such a set is called the secure domination number and denoted by γs(G)". In this paper we discuss about the secure domination for some special graphs. [ABSTRACT FROM AUTHOR]

One of the important concepts of Graph Theory is Matching Theory. Several Concepts on Matching Theory has been dealt in [1, 2, 3]. The technique of maximum matching on directed graphs have been studied in [1]. In this paper, a new approach for finding maximum matching of an undirected dense graph based on largest geometric multiplicity of Eigen values using exact controllability network is studied. It will be interest to further study on their properties. [ABSTRACT FROM AUTHOR]

"In Graph theory, the study of domination is one of the fastest growing area in recent years. There are different types of domination such as connected domination, independent domination, total domination, paired domination, secure domination, fair domination, fair secure domination etc., A Fair dominating set (FD-set) in a graph G is a dominating set S such that all vertices in V-S are dominated by the equal number of vertices in S. The minimum cardinality of a Fair dominating set is the fair domination number and is denoted by γfd(G). In this paper we discuss in finding the fair domination number for some special graphs." [ABSTRACT FROM AUTHOR]

Chemical graph theory is an emerging field in research and attracts people mainly for its applications in Chemistry. Topological indices have been extensively used in this regard. A lot of topological indices have been introduced in the recent days. A recent study has shown that it is possible to even calculate the boiling point of a molecule using topological indices. This paper aimed at calculating the most trending degree based topological index for a family of graphs. [ABSTRACT FROM AUTHOR]

In this paper we discussed the prominence of Bipolar fuzzy graphs (BFG). Fuzzy set assigns a sequence of membership values to the elements of the universal set ranging from 0 to 1, whereas now our study about Bipolar fuzzy graphs whose membership degree range is [-1, 1]. The earnest efforts of the researchers are perceivable in the relevant establishment of the subject integrating coherent practicality and reality. When we assess the position of an object in space, we may have positive information expressed as a set of possible places and negative information expressed as a set of impossible places. This corresponds to the idea that the union of positive and negative information does not cover the whole space. Dominating sets have a vital function regarding the theory of fuzzy graphs. Traveling salesman problem, communication network, traffic route problem are largely discussed applications among the diverse applications dealing with the theory of dominations. In this paper we generalized Bipolar fuzzy graphs and explored various types of dominations on Bipolar fuzzy graphs such as Strong Dominations, Split and non-split dominations, Multiple dominations and some applications of Bipolar fuzzy graphs. Fuzzy graphs found an increasing number of applications in modeling real time systems where the information inherent in the system varies with different levels of precision. Bipolar fuzzy graphs can be used to model many problems in economics, operations research, etc; involving two similar, but opposite type of qualitative variables like success and failure, gain and loss etc. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new concept called semigraphs, combining the theory of semigraphs introduced by E. Sampath Kumar [2] and the theory of semiring, analogus to the theory of S-valued graphs introduced in [4], thus giving a free hand to researchers for further study. [ABSTRACT FROM AUTHOR]

Graph theory was first introduced by Leonhard Euler in 1736 and still being one of the mathematic's topics which is rapidly developing and can be used to simplify mathematic problems. There are many interesting topics in graph theory; one of them is graph labeling. There are many ways of labeling a graph, and one of them is graceful labeling. Let G(V, E) is a graph. The injective mapping F ∶ V → {0, 1,…, |E|} is called graceful if the weights of edge w(uv) = |f(u) - f(v)| are all different for every edge uv. There is a famous conjecture in graceful labeling. It is said that all trees are graceful. To prove this conjecture, then we must show that every tree is graceful. There are many research papers dealing with the special cases of trees. Many classes of trees have been proven graceful, and one of them is supercaterpillar. Previous research had proved that supercaterpillars with certain conditions are also graceful. In this paper, we generalize the concept of supercaterpillar, and show that the subclass of supercaterpillar graphs that has not been discussed earlier is also graceful, using an adjacency matrix for the construction. [ABSTRACT FROM AUTHOR]

In this paper we study a general failure that may occur with a certain probability, and its danger may in any way affect the health and functionality of an individual. We introduce the concept of value of risk and further important formulas to this related. Unlike previous papers we use a rather new concept with logarithmic dependence in risk calculation which is accurate and convenient in many situations. In the main part of this paper we derive important relations and results related to this value of risk. All this is illustrated by graphs and tables with important values of the appropriate quantities. [ABSTRACT FROM AUTHOR]

Current airport security management System is widely implemented ail around the world to ensure the safety of passengers, but it might not be an optimum one. This paper aims to seek a better security System, which can maximize security while minimize inconvenience to passengers. Firstly, we apply Petri net model to analyze the steps where the main bottlenecks lie. Based on average tokens and time transition, the most time-consuming steps of security process can be found. including inspection of passengers' identification and documents, preparing belongings to be scanned and the process for retrieving belongings back. Then, we develop a queuing model to figure out factors affecting those time-consuming steps. As for future improvement. the effective measures which can be taken include transferring current System as single-queuing and multi-served, intelligently predicting the number of security checkpoints supposed to be opened, building up green biological convenient lanes. Furthermore. to test the theoretical results. we apply some data to stimulate the model. And the stimulation results are consistent with what we have got through modeling. Finally, we apply our queuing model to a multi-cultural background. The result suggests that by quantifying and modifying the variance in wait time, the model can be applied to individuals with various habits customs and habits. Generally speaking, our paper considers multiple affecting factors, employs several models and does plenty of calculations, winch is practical and reliable for handling in reality. In addition, with more precise data available, we can further test and improve our models. [ABSTRACT FROM AUTHOR]