PELABELAN FUZZY PADA GRAF

Let G be a simple graph, let f : V(G)&#8594{1,2,...,|V(G)|} be a bijective mapping. The weight of v &#8712 V(G) is the sum of labels of all vertices adjacent to v . We say that f is a distance magic labeling of G if the weight of every vertex is the same constant k and we say that f is a handicap magic labeling of G if the weight of every vertex v is l + f(v) for some constant l. Graphs that allow such labelings are called distance magic or handicap, respectively. Distance magic and handicap labelings of regular graphs are used for scheduling incomplete tournaments. While distance magic labelings correspond to so called equalized tournaments, handicap labelings can be used to schedule incomplete tournaments that are more challenging to stronger teams or players, hence they increase competition and yield attractive schemes in which every games counts. We summarize known results on distance magic and handicap labelings and construct a new infinite class of 4-regular handicap graphs.

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.

In graph theory, the Randic index ( R ) is a topological graph invariant widely used as a physicochemical descriptor in the mathematical modeling of molecular structures. However, traditional molecular graphs fail to capture the heterogeneity of chemical bonds, since they treat all edges as uniform, ignoring variations in bond lengths and strengths. To overcome this limitation, we adopt a fuzzy graph structure, which provides a more realistic framework for representing molecular interactions. This dual perspective developing rigorous theoretical bounds for the Randic index on fuzzy graph classes and simultaneously applying them to real pharmaceutical compounds ensure both mathematical novelty and practical relevance. We derive analytical bounds for the Randic index over fuzzy versions of complete graphs, paths, stars, and cycles, as well as for graph operations such as Cartesian product, union, join, and composition, confirming the sharpness of the results in each case. To demonstrate the applicability, we compute the Randic index for fuzzy graph representations of Parkinson’s disease drugs (Levodopa, Procyclidine, Trihexyphenidyl, and Apomorphine). The findings indicate that the Randic index, within the fuzzy graph framework, reliably estimates key physicochemical properties such as polarizability, molar refractivity, surface tension, and molar volume, highlighting the strength of combining theoretical results with drug modeling applications.

The product fuzzy graph is an extension of the fuzzy graph definition by replacing the minimum operation with the product operation. This research is qualitative research with research stages consisting of determining open problems, constructing new definitions, constructing new properties as theorems, and verifying results in the form of mathematical proofs. The purpose of this research is to get the Cartesian product definition of the product fuzzy graph and its properties. The results showed that the Cartesian product of two product fuzzy graphs is a fuzzy graph, the Cartesian product of two strong product fuzzy graphs is a strong product fuzzy graph, and the resulting Cartesian product of two complete product fuzzy graphs is a strong product fuzzy graph. Thus, the novelty of this research is that new properties of product fuzzy graphs are obtained.

Types of arcs in a fuzzy graph

Let $G=(V, E)$ be a graph and let $(A,+)$ be an Abelian group with identity elemento. Let $f: V \rightarrow A$ be a vertex labeling and $f^*: E \rightarrow A$ be the induced labeling of $f$, defined by $f^*\left(v_1 v_2\right)=f\left(v_1\right)+f\left(v_2\right)$ for all $v_1 v_2 \in E$. Then $f^*$ again induces a labeling say $f^{* *}: V \rightarrow A$ defined by $f^{* *}(v)=\sum_{v v_1 \in E} f^*\left(v v_1\right)$. A graph $G=(V, E)$ is said to be an Induced $A$-Magic Graph (IAMG) if there exists a non zero labeling $f: V \rightarrow A$ such that $f \equiv f^{* *}$. The function $f$, so obtained is called an Induced $A$-Magic Labeling (IAML) of $G$ and a graph which has no such Induced Magic Labeling is called a Non-induced magic graph. In this paper we discuss the existence of Induced Magic Labeling of some special graphs like $P_n, C_n, K_n$ and $K_{m, n}$.

Let (V, μ, ρ) be a fuzzy graph. We now provide two popular ways of defining the distance between a pair of vertices. One way is to define the “distance” dis(x,y) between x and y as the length of the shortest strongest path between them. This “distance” is symmetric and is such that dis(x,x) = 0 since by our definition of a fuzzy graph, no path from x to x can have strength greater than μ(x), which is the strength of the path of length 0. However, it does not satisfy the triangle property, as we see from the following example. Let V = {u, v, x, y,z}, ρ(x, u) = ρ(u, v) = ρ(v, z) = 1 and ρ(x, y) = ρ(y, z) = 0.5. Here any path from x to y or from y to z has strength ≤ 1/2 since it must involve either edge (x,y) or edge (y, z). Thus the shortest strongest paths between them have length 1. On the other hand, there is a path from x to z, through u and v, that has length 3 and strength 1. Thus dis(x,z) = 3 > 1 + 1 = dis(x,y) + dis(y, z) in this case.KeywordsHamiltonian PathFuzzy SubsetFuzzy RelationConnected SubgraphFuzzy GraphThese 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.

This paper deals with the labeling of vertices and edges of a graph. Let G be a graph with vertex set V and edge set E, where |V| be the number of vertices and |E| edges of G. The two bijection methods for which we have designed algorithms are as follows. Initially A bijection λ 1:V∪E → {1, 2…|V| + |E|} is called a Vertex-Magic Total Labeling (VMTL) if there is a vertex magic constant vk such that the weight of vertex m is, λ 1(m) + Σ n εA (m) λ 1(mn) = vk, ∀ m εv Where A(m) is the set of vertices adjacent to x. In the similar fashion the bijection λ 2:V∪E → {1, 2…|V| + |E|} is called Edge-Magic Total Labeling (EMTL) if there is a edge magic constant ek such that the weight of an edge es(mn), λ 2(m) + λ 2(n) + λ 2 e(mn) = ek, ∀ e εE. A resultant Graph which consists of both VMTL and EMTL are said to be Total Magic Labeling (TML) for different vertex magic constant and edge magic constant values. Baker and Sawada proposed algorithms to find VMTLs on cycles and wheels. In this paper we enhanced these algorithms and also we proposed new algorithms to generate EMTLs and TMLs of cycles and wheels. We used the concept variations and sum set sequences to produce VMTLs and EMTLs on cycles and wheels. Also we designed modules to identify TML’s.KeywordsMagic labelingMagic labeling algorithmsMagic constantCyclesWheelsVertex Magic Total LabelingEdge Magic Total LabelingTotal Magic Labeling etc

A graph with a vertex set is said to be a prime graph if there exists a bijective mapping , where denotes the number of vertices in , such that for any two adjacent vertices and in have . Tensor Product graph is a way to combine (compose) two graphs into one larger and more complex graph. The result is a new graph that reflects the connection properties of the two original graphs, but in a very specific and more complex way than other graph operations. Therefore, this research aims to determine whether there is prime labeling in the class of graphs resulting from the Tensor Product of the path graph and the cycle graph . The research employed analytical and exploratory methods with a trial-and-error strategy to determine the labeling that possesses a prime property. The results of this study prove that two classes of the Tensor Product graph for , and graph , for are prime graph. This finding expands the results on classes of graphs that admit prime labeling and provides a basis for further research on graph labeling in other graph operations

Research in graph theory has lead to one of the important area called labeling of graphs. There are different types of labeling such as graceful labeling, magic labeling, edge-graceful labeling, prime labeling, radio labeling, harmonious labeling etc., In this paper, we investigate the existence of edge graceful labeling for the Corona graph Cn⊙K2¯ (n is odd n ≥ 3). Also we have defined a flower graph which does not admit an edge-graceful labeling.

We consider the problem of maintaining a maximum matching in a convex bipartite graph G = (V,E) under a set of update operations which includes insertions and deletions of vertices and edges. It is not hard to show that it is impossible to maintain an explicit representation of a maximum matching in sub-linear time per operation, even in the amortized sense. Despite this difficulty, we develop a data structure which maintains the set of vertices that participate in a maximum matching in O(log2 |V|) amortized time per update and reports the status of a vertex (matched or unmatched) in constant worst-case time. Our structure can report the mate of a matched vertex in the maximum matching in worst-case O(min{k log2 |V |+log |V|, |V| log |V|}) time, where k is the number of update operations since the last query for the same pair of vertices was made. In addition, we give an O(√|V| log2 |V|)-time amortized bound for this pair query.

L(2,1)-colorings and Irreducible No-hole Colorings of Cartesian Product of Graphs

Graph labeling is the term used to describe the procedure of giving unique identifiers. Graph attributing is a method for modeling real-world problems in industries such as circuit design, surveillance technology, and telecommunications networks. A graph is characterized as possessing cubic roots cordial labeling if each node is designated with either the value 1, δ or δ2 or another specified integer. The derived set containing the value 1 or another integer, the mathematical conditions η (k) − η (w)| ≤ 1 and | ξ (k) − ξ (w)| ≤ 1, k ≠ w and k, w ∈ {1, δ, δ2} where η (k) and ξ (k) are represents the aggregated vertices and edges assigned in k ∈ {1, δ, δ2}. Labeled graphs prove to be immensely beneficial across a variety of mathematical frameworks, including domains such as cyber security and radar code formulation. The application of cordial labeling has been demonstrated to provide significant advantages in the context of signal processing channels and challenges associated with specific code word design. The idea of cubic roots cordial labeling has attracted a lot of academic attention among various labeling techniques, unique structural characteristics and capacity to depict a fair flow of information within linked networks. The investigation of the behavior of cubic roots cordial labeling in relation to graph operations is currently receiving extensive scholarly examination, particularly concerning path graphs, despite its critical relevance. This paper explore the cordial labeling of cubic product graphs generated from the combination of two path graphs. The four fundamental graph operations on which we particularly focus on activities, we analyze the specific conditions to possess cubic roots cordial labeling.

A vertex coloring c : V(G) → ℕ of a non-trivial connected graph G is called a set coloring if NC(u) ≠ NC(v) for any pair of adjacent vertices u and v. Here, NC(x) denotes the set of colors assigned to vertices adjacent to x. The set chromatic number of G, denoted by χs (G), is defined as the fewest number of colors needed to construct a set coloring of G. In this paper, we study the set chromatic number in relation to two graph operations: join and comb prdocut. We determine the set chromatic number of wheels and the join of a bipartite graph and a cycle, the join of two cycles, the join of a complete graph and a bipartite graph, and the join of two bipartite graphs. Moreover, we determine the set chromatic number of the comb product of a complete graph with paths, cycles, and large star graphs.

Group decision-making (GDM) is crucial in various components of graph theory, management science, and operations research. In particular, in an intuitionistic fuzzy group decision-making problem, the experts communicate their preferences using intuitionistic fuzzy preference relations (IFPRs). This approach is a way that decision-makers rank or select the most desirable alternatives by gathering criteria-based information to estimate the best alternatives using a wider range of knowledge and experience. This article proposes a new statistical measure in a fuzzy environment when the data is ambiguous or unreliable to solve a decision-making problem. This study uses the variation coefficient measure combined with intuitionistic fuzzy graphs (IFG) and Laplacian energy (LE) to solve a GDM problem that utilizes intuitionistic fuzzy preference relations (IFPRs) to select a reliable alliance partner. Initially, the Laplacian energy determines the weight of individual standards, and the obtained weight average further estimates the overall criterion weight vector. We establish the authority criteria weights using the variation coefficient measure and then ultimately rank the alternatives for each criterion using the same measure. We examine four distinct companies Alpha, Beta, Delta, and Zeta to conduct a realistic GDM to choose which alliance partner would be ideal. We successfully implemented the suggested technique, determining that Alpha satisfies company standards and is ranked first among other companies. Moreover, this technique is useful for all kinds of Intuitionistic fuzzy group decision-making problems to select optimal ones.