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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]

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]

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]

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]

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]

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]