Vertices Of Graph Research Articles

Status Sombor Indices of Unitary Cayley Graph

Objectives: The objective of the study is to obtain the values of status Sombor indices for the unitary Cayley graph. Methods: The unitary Cayley graph 𝑋𝑛 has vertex set 𝑍𝑛 = {0, 1, 2, ...𝑛 − 1}. Vertices a, and b are adjacent, if 𝑔𝑐𝑑(𝑎 − 𝑏, 𝑛) = 1. The status 𝜎 (𝑢) of a vertex u in a graph, G is the sum of distances of all other vertices from u in 𝐺. In this paper, we obtain the values of the Status Sombor index, Modified Status Sombor index, and their corresponding exponentials and compute exact formulae for unitary Cayley graph 𝑋𝑛. Findings: Different values of status Sombor indices have been obtained for unitary Cayley graphs of various orders. Novelty: The values of status Sombor indices have been extensively studied by various researchers recently and most of the research has been done for different types of graphs in chemical graph theory. The present work reveals that the unitary Cayley graph is one of the graphs that is extensively studied in both the areas of group theory and graph theory. We derive the values of status Sombor indices that are dependent on the status and degree of the vertices of the unitary Cayley graph. Mathematics Subject Classification: 05C09, 05C25, 05C50. Keywords: Unitary Cayley graphs; Status Sombor index; Modified status Sombor index; Status Sombor exponential; Modified status Sombor exponential

A novel approach to Fuzzy mixed graph structure with application towards trade relations between countries

Many of the phenomena around us are a combination of directed and undirected relationships between different subjects, which will be more complex despite the existence of multiple relationships between objects. For example, in business relations between countries and social networks, communication is sometimes one-way or two-way. Checking and processing such information is managed in mixed graphs. Previous research has been based on the assumption that there is a general relationship between all vertices in a mixed graph. In this article, a new framework for fuzzy information management is introduced by combining fuzzy mixed graph and graph structure along with mentioning its properties. The concept of connectedness has been considered as one of the topics in this study. In this regard, some of their properties were examined by introducing some basic concepts such as paths, cycles, bridges, and cut vertices. Also, some attributes such as degree, neighborhood, order and size are defined. The results showed that although there existed a close relationship between the fuzzy mixed graph structure and the graph structure, the existence of multiple and distinct relationships between the vertices has created new definitions of concepts. This change can be seen especially in the degree of nodes and neighborhoods. Finally, its application in the field of trade relations between countries is presented.

The Minimum Number of Maximal Independent Sets in Twin-Free Graphs

The problem of determining the maximum number of maximal independent sets in certain graph classes dates back to a paper of Miller and Muller and a question of Erdős and Moser from the 1960s. The minimum was always considered to be less interesting due to simple examples such as stars. In this paper we show that the problem becomes interesting when restricted to twin-free graphs, where no two vertices have the same open neighbourhood. We consider the question for arbitrary graphs, bipartite graphs and trees. The minimum number of maximal independent sets turns out to be logarithmic in the number of vertices for arbitrary graphs, linear for bipartite graphs and exponential for trees. In the latter case, the minimum and the extremal graphs have been determined earlier by Taletskii and Malyshev, but we present a shorter proof.

Equitable List Coloring of Planar Graphs With Given Maximum Degree

ABSTRACTIf is a list assignment of colors to each vertex of an ‐vertex graph , then an equitable ‐coloring of is a proper coloring of vertices of from their lists such that no color is used more than times. A graph is equitably ‐choosable if it has an equitable ‐coloring for every ‐list assignment . In 2003, Kostochka, Pelsmajer, and West (KPW) conjectured that an analog of the famous Hajnal–Szemerédi Theorem on equitable coloring holds for equitable list coloring, namely, that for each positive integer every graph with maximum degree at most is equitably ‐choosable. The main result of this paper is that for each and each planar graph , a stronger statement holds: if the maximum degree of is at most , then is equitably ‐choosable. In fact, we prove the result for a broader class of graphs—the class of the graphs in which each bipartite subgraph with has at most edges. Together with some known results, this implies that the KPW Conjecture holds for all graphs in , in particular, for all planar graphs. We also introduce the new stronger notion of strongly equitable (SE, for short) list coloring and prove all bounds for this parameter. An advantage of this is that if a graph is SE ‐choosable, then it is both equitably ‐choosable and equitably ‐colorable, while neither of being equitably ‐choosable and equitably ‐colorable implies the other.

Thick embeddings of graphs into symmetric spaces via coarse geometry

We prove estimates for the optimal volume of thick embeddings of finite graphs into symmetric spaces, generalising results of Kolmogorov-Barzdin and Gromov-Guth for embeddings into Euclidean spaces. We distinguish two very different behaviours depending on the rank of the non-compact factor. For rank at least 2, we construct thick embeddings of N N -vertex graphs with volume C N ln ⁡ ( 1 + N ) CN\ln (1+N) and prove that this is optimal. For rank at most 1 1 we prove lower bounds of the form c N a cN^a for some (explicit) a > 1 a>1 which depends on the dimension of the Euclidean factor and the conformal dimension of the boundary of the non-compact factor. The main tool is a coarse geometric analogue of a thick embedding called a coarse wiring, with the key property that the minimal volume of a thick embedding is comparable to the “minimal volume” of a coarse wiring for symmetric spaces of dimension at least 3 3 . In the appendix it is proved that for each k ≥ 3 k\geq 3 every bounded degree graph admits a coarse wiring into R k \mathbb {R}^k with volume at most C N 1 + 1 k − 1 CN^{1+\frac {1}{k-1}} . As a corollary, the same upper bound holds for real hyperbolic space of dimension k + 1 k+1 and in both cases this result is optimal.

Threshold Protocol Game on Graphs with Magic Square-Generalization Labelings

Graphical games describe strategic interactions among a specified network of players. The threshold protocol game is a graphical game that models the adoption of a lesser-used product in a population when individuals benefit by using the same product. The threshold protocol game has historically been considered using infinite, simple graphs. In general, however, players might value some relationships more than others or may have different levels of influence in the graph. These traits are described by weights on graph edges or vertices, respectively. Relative comparisons on arbitrarily weighted graphs have been studied for a variety of graphical games. Alternatively, graph labelings are functions that assign values to the edges and vertices of graphs based on a particular set of rules. This work demonstrates that the outcome of the threshold protocol game can be characterized on a magic square-generalization labeled graph. There are a variety of graph labelings that generalize the concept of magic squares. In each, the labels on similar sets of graph elements sum to a constant. The constant sums of magic square-generalization labelings mean that each player experiences a constant level of influence without needing to specify the value of players relative to one another. The game outcome is compared across different types and features of labelings.

Rumus Umum Pelabelan L(2, 1) pada Graf Koprima Grup Bilangan Bulat Modulo m

Groups are nonempty sets equipped with binary operations and fulfill certain conditions. One example of a group is the group of integers modulo . A coprime graph of a group is a graph whose vertices are elements in and any two vertices in are directly connected if and only if the greatest common factor of the orders x and y are relatively prime. labeling has important applications in wireless telecommunication networks, where there are limited radio frequencies available and adjacent transmitters cannot use the same frequency. To overcome this problem, frequency assignment is done using the concept of labeling, where transmitters are symbolized as vertices connected by edges. labeling is a graph labeling technique that assigns non-negative integer labels to graph vertices with the rule that two directly connected vertices must have a minimum label difference of two, and two vertices that are two apart must have a minimum label difference of one. The purpose of this research is to find out the general formula for the labeling on the coprime graph of the group of integers modulo and the general formula for the minimum value of the largest label of the labeling. The steps taken in this research are to determine the coprime graph form of the group of integers modulo . Then label each vertex with a non-negative label. Then label each vertex with labeling rule. From the result of this research, it can be concluded that the coprime graph of integer group modulo has a labeling with .

Borel Vizing’s theorem for graphs of subexponential growth

We show that every Borel graph G G of subexponential growth has a Borel proper edge-coloring with Δ ( G ) + 1 \Delta (G) + 1 colors. We deduce this from a stronger result, namely that an n n -vertex (finite) graph G G of subexponential growth can be properly edge-colored using Δ ( G ) + 1 \Delta (G) + 1 colors by an O ( log ∗ ⁡ n ) O(\log ^\ast n) -round deterministic distributed algorithm in the LOCAL model, where the implied constants in the O ( ⋅ ) O(\cdot ) notation are determined by a bound on the growth rate of G G .

Faster algorithms for the alignment of sparse correlated Erdős–Rényi random graphs

Abstract The correlated Erdős–Rényi random graph ensemble is a probability law on pairs of graphs with n vertices, parametrized by their average degree λ and their correlation coefficient s. It can be used as a benchmark for the graph alignment problem, in which the labels of the vertices of one of the graphs are reshuffled by an unknown permutation; the goal is to infer this permutation and thus properly match the pairs of vertices in both graphs. A series of recent works has unveiled the role of Otter’s constant α (that controls the exponential rate of growth of the number of unlabeled rooted trees as a function of their sizes) in this problem: for s > α and large enough λ, it is possible to recover in a time polynomial in n a positive fraction of the hidden permutation. The exponent of this polynomial growth is, however, quite large and depends on the other parameters, which limits the range of applications of the algorithm. In this work, we present a family of faster algorithms for this task, show through numerical simulations that their accuracy is only slightly reduced with respect to the original one and conjecture that they undergo, in the large λ limit, phase transitions at modified Otter’s thresholds α ^ > α , with α ^ related to the enumeration of a restricted family of trees.

Dimensi Partisi Graf Hasil Amalgamasi Sisi Graf Roda dengan Graf Bintang

This study discusses the analysis of the partition dimension of the graph resulting from the edge amalgamation between the wheel graph ( Wn) and the star graph ( Sm), where the partition dimension is an important parameter in graph theory that serves to measure the minimum number of partitions required to distinguish every pair of vertices through a set of supporting vertices. The amalgamation process is carried out by merging one edge of the wheel graph with one edge of the star graph, thus forming a new graph. This research employs theoretical and algorithmic approaches to calculate the partition dimension of the resulting amalgamated graph, focusing on the influence of the number of vertices in both constituent graphs on the changes in the partition dimension. The results show that pd(amal_s(Wn,Sm;v1v2,u1u2))=3 when 4n7 and m=3, and 4 when n=3 and 3≤m≤4, whereas if n≥8 and 3≤m≤⌊n/2⌋, then ⌊n/2⌋ is obtained.

ActiveReach: an active learning framework for approximate reachability query answering in large-scale graphs.

With graph reachability query, one can answer whether there exists a path between two query vertices in a given graph. The existing reachability query processing solutions use traditional reachability index structures and can only compute exact answers, which may take a long time to resolve in large graphs. In contrast, with an approximate reachability query, one can offer a compromise by enabling users to strike a trade-off between query time and the accuracy of the query result. In this study, we propose a framework, dubbed ActiveReach, for learning index structures to answer approximate reachability query. ActiveReach is a two-phase framework that focuses on embedding nodes in a reachability space. In the first phase, we leverage node attributes and positional information to create reachability-aware embeddings for each node. These embeddings are then used as nodes' attributes in the second phase. In the second phase, we incorporate the new attributes and include reachability information as labels in the training data to generate embeddings in a reachability space. In addition, computing reachability for all training data may not be practical. Therefore, selecting a subset of data to compute reachability effectively and enhance reachability prediction performance is challenging. ActiveReach addresses this challenge by employing an active learning approach in the second phase to selectively compute reachability for a subset of node pairs, thus learning the approximate reachability for the entire graph. Our extensive experimental study with various real attributed large-scale graphs demonstrates the effectiveness of each component of our framework.

Spanning trees for many different numbers of leaves

Let $G$ be a connected graph and $L(G)$ the set of all integers $k$ such that $G$ contains a spanning tree with exactly $k$ leaves. We show that for a connected graph $G$, the set $L(G)$ is contiguous. It follows from work of Chen, Ren, and Shan that every connected and locally connected $n$-vertex graph -- this includes triangulations -- has a spanning tree with at least $n/2 + 1$ leaves, so by a classic theorem of Whitney and our result, in any plane $4$-connected $n$-vertex triangulation one can find for any integer $k$ which is at least $2$ and at most $n/2 + 1$ a spanning tree with exactly $k$ leaves (and each of these trees can be constructed in polynomial time). We also prove that there exist infinitely many $n$ such that there is a plane $4$-connected $n$-vertex triangulation containing a spanning tree with $2n/3$ leaves, but no spanning tree with more than $2n/3$ leaves.

Distance Degree Sequence and Distance Neighborhood Degree Sequence of Vertices in Fuzzy Graphs and Some of Their Properties

Objectives: The objective of this study is to explore new types of sequences in fuzzy graphs namely distance degree sequence and distance neighborhood degree sequence of fuzzy graphs which may have various applications in the real world. Methods: In this paper, two unique sequences for each vertex of fuzzy graphs are given by means of its degree and neighborhood degree which are called as the distance degree sequence and distance neighborhood degree sequence of vertices of fuzzy graphs respectively. The n-tuple of the distance degree sequence of vertices of a fuzzy graph arranged in lexicographic order is called the distance degree sequence of the fuzzy graph and the n-tuple of distance neighborhood degree sequence of vertices of a fuzzy graph arranged in lexicographic order is called the distance neighborhood degree sequence of the fuzzy graph. Findings: The distance degree sequence and distance neighborhood degree sequence of fuzzy graphs are introduced. The properties of these sequences and the general forms of these sequences of various fuzzy graphs are given. The relationship between the distance degree sequence and distance neighborhood degree sequences is discussed. Novelty: The distance degree sequence of graphs has already been introduced. In fuzzy graph theory, the distance degree sequence and distance neighborhood degree sequence are introduced in this paper. Mathematics Subject Classification (202 0): 03E72, 05C72, 05C12. Keywords: Fuzzy graphs, Distance degree sequence, Distance neighborhood-degree sequence, Regular fuzzy graphs

On the automorphisms of the Drinfeld modular groups

Let [Formula: see text] be the ring of elements in an algebraic function field [Formula: see text] over [Formula: see text] which are integral outside a fixed place [Formula: see text]. In contrast to the classical modular group [Formula: see text] and the Bianchi groups, the Drinfeld modular group [Formula: see text] is not finitely generated and its automorphism group [Formula: see text] is uncountable. Except for the simplest case [Formula: see text] not much is known about the generators of [Formula: see text] or even its structure. We find a set of generators of [Formula: see text] for a new case. On the way, we show that every automorphism of [Formula: see text] acts on both the cusps and the elliptic points of [Formula: see text]. Generalizing a result of Reiner for [Formula: see text] we describe for each cusp an uncountable subgroup of [Formula: see text] whose action on [Formula: see text] is essentially defined on the stabilizer of that cusp. In the case where [Formula: see text] (the degree of [Formula: see text]) is [Formula: see text], the elliptic points are related to the isolated vertices of the quotient graph [Formula: see text] of the Bruhat–Tits tree. We construct an infinite group of automorphisms of [Formula: see text] which fully permutes the isolated vertices with cyclic stabilizer.

Polynomial Removal Lemma for Ordered Matchings

We prove that for every ordered matching $H$ on $t$ vertices, if an ordered $n$-vertex graph $G$ is $\varepsilon$-far from being $H$-free, then $G$ contains $\text{poly}(\varepsilon) n^t$ copies of $H$. This proves a special case of a conjecture of Tomon and the first author. We also generalize this statement to uniform hypergraphs.

GKE-TUNet: Geometry-Knowledge Embedded TransUNet Model for Retinal Vessel Segmentation Considering Anatomical Topology.

Automated retinal vessel segmentation is crucial for computer-aided clinical diagnosis and retinopathy screening. However, deep learning faces challenges in extracting complex intertwined structures and subtle small vessels from densely vascularized regions. To address these issues, we propose a novel segmentation model, called Geometry-Knowledge Embedded TransUNet (GKE-TUNet), which incorporates explicit embedding of topological features of retinal vessel anatomy. In the proposed GKE-TUNet model, a skeleton extraction network is pre-trained to extract the anatomical topology of retinal vessels from refined segmentation labels. During vessel segmentation, the dense skeleton graph is sampled as a graph of key-points and connections and is incorporated into the skip connection layer of TransUNet. The graph vertices are used as node features and correspond to positions in the low-level feature maps. The graph attention network (GAT) is used as the graph convolution backbone network to capture the shape semantics of vessels and the interaction of key locations along the topological direction. Finally, the node features obtained by graph convolution are read out as a sparse feature map based on their corresponding spatial coordinates. To address the problem of sparse feature maps, we employ convolution operators to fuse sparse feature maps with low-level dense feature maps. This fusion is weighted and connected to deep feature maps. Experimental results on the DRIVE, CHASE-DB1, and STARE datasets demonstrate the competitiveness of our proposed method compared to existing ones.

Pipe production cost management model based on graph theory

The pipe production process is a complex dynamical system with a significant number of production operations and interconnected cost localization centres. Purpose. To develop a mathematical model of the process of handling analytical information of a pipe enterprise using graph theory for the needs of cost management at each production stage. Methodology. Technological flow charts of the production job, where the list of mandatory activities and routing of the operating process are indicated, were used to organize cost accounting by operational centres of their localization with the help of computer modelling elements, namely the graph theory. The work used a complex of research methods, including analysis and scientific synthesis of scientific and technical information; theoretical studies; methods of mathematical and computer modelling, engineering developments. Findings. The cost accounting process is represented by a directed hypergraph. Each production operation is associated with a graph vertex. Each vertex is assigned to a series of interrelated marks: type of technological operation; weight factor; the value of direct costs for the operation; the value of indirect costs for the operation; the expense factor. The initial state of the hypergraph is specified by means of its initial marking, which corresponds to the set cipher of the batch of pipes and its weight. Further execution of the marked hypergraph is performed by running the allowed vertices. Marking of the graph ends if all the information from the technological flow charts of pipe products has been used. Originality. A model of cost management by operational centres is proposed which will allow dividing the prime cost of pipes which differ according to the production technology. It is recommended to rank pipe products within each technologically similar group for an economically feasible allocation of costs between batches of pipes. The proposed process model of processing analytical information of the pipe enterprise using graph theory will allow providing data governance as for the prime cost of each individual batch of pipe products, will contribute to the improvement in the order generation procedure taking into account the acquired information, and will allow making the appropriate adjustments when making management decisions. Practical value. Calculation of expenses for production operations allows one to identify the most cost-intensive of them. Relying on the results of the calculation, the enterprise management has the opportunity to manage costs at any stage of the production process. Moreover, the model allows determining the amount of consumed metal for each production operation, which is relevant for pipe enterprises. It is appropriate to focus further scientific developments in this direction on the possibility of managing non-manufacturing cost with the help of mathematical models in economics to optimize the supply and sales activities of the enterprise.

Game Chromatic Number of Tadpole Graph, Broom Graph, and Tribune Graph

Graph coloring game is one of application in graph theory. The goal in this article is determine game chromatic number of tadpole graph, broom graph, and tribune graph. The graphs are simple, connected, and undirected and thus eligible for playing graph coloring game. Given two players with the first player is called A and second player is called B coloring vertex of graph G with a set of colors C={c_1,c_2,c_3, ..., c_k}. A must to make sure that all vertex of G has colored and B must try to prevent A coloring of all vertex. The first step was taken by A as first player, two players take turns coloring vertex of graph G, with rule that every vertex have different color from the neighbourhood. If all vertex of graph G have been colored, then A win or B win if some vertex hasn’t colored. The smallest number of k if A has a strategy to win at graph G with k color, then k called game chromatic number which is denoted by χ_g (G). The strategy to win this game is coloring the biggest degree of vertex in graph first. The result obtained from this paper is A win the game with the strategy of first coloring the largest degrees of vertex. So, exact of game chromatic number of tadpole graph is 3, broom graph is 2 or 3 with several conditions, and tribune graph is 3 or 4 with several conditions.

Algorithms for the global domination problem

A dominating set D in a graph G is a subset of its vertices such that every its vertex that does not belong to set D is adjacent to at least one vertex from set D. A set of vertices of graph G is a global dominating set if it is a dominating set for both, graph G and its complement. The objective is to find a global dominating set with the minimum cardinality. Neither exact nor approximation algorithm existed for the problem known to be NP-hard. We show that it remains NP-hard for restricted types of graphs. At the same time, we specify some families of graphs for which the three heuristics, that we propose here, are optimal. Given the complexity status of the problem, our aim was the development of powerful heuristic algorithms that work well in practice for large-scaled instances. To measure the efficiency of our heuristics, we formulated the problem as an integer linear program (ILP) and also we developed an alternative implicit enumeration (IE) algorithm obtaining guaranteed optimal solutions for the existing benchmark instances with up to 8000 vertices. Remarkably, for 56.75% of these instances, at least one of our heuristics also created an optimal solution, where an average absolute error for the remaining instances was a single vertex. The average approximation ratio was 1.005, whereas for the largest benchmark instances with up to 25000 vertices our heuristics delivered solutions in less than 2 min.

Exact and parameterized algorithms for the independent cutset problem

The Independent Cutset problem asks whether there is a set of vertices in a given graph that is both independent and a cutset. This problem is ▪-complete even when the input graph is planar and has maximum degree five. We first present a O⁎(1.4423n)-time algorithm to compute a minimum independent cutset (if any). Since the property of having an independent cutset is MSO1-expressible, our main results are concerned with structural parameterizations for the problem considering parameters incomparable with clique-width. We present ▪-time algorithms under the following parameters: the dual of the maximum degree, the dual of the solution size, the size of a dominating set (where a dominating set is given as an additional input), the size of an odd cycle transversal, the distance to chordal graphs, and the distance to P5-free graphs. We close by introducing the notion of α-domination, which generalizes key ideas of this article.