Bilangan Terhubung Titik Pelangi pada Graf Hasil Operasi Korona Graf Prisma (P_(m,2)) dan Graf Lintasan (P_3)

Rainbow vertex-connection number is the minimum k-coloring on the vertex graph G and is denoted by rvc(G). Besides, the rainbow-vertex connection number can be applied to some special graphs, such as prism graph and path graph. Graph operation is a method used to create a new graph by combining two graphs. Therefore, this research uses corona product operation to form rainbow-vertex connection number at the graph resulting from corona product operation of prism graph and path graph (Pm,2 P3) (P3 Pm,2). The results of this study obtain that the theorem of rainbow vertex-connection number at the graph resulting from corona product operation of prism graph and path graph (Pm,2 P3) (P3 Pm,2) for 3 = m = 7 are rvc (G) = 2m rvc (G) = 2.

Determination of Fractional Chromatic Numbers in the Operation of Adding Two Different Graphs

The development of graph theory has provided many new pieces of knowledge, one of them is graph color. Where the application is spread in various fields such as the coding index theory. Fractional coloring is multiple coloring at points with different colors where the adjoining point has a different color. The operation in the graph is known as the sum operation. Point coloring can be applied to graphs where the result of operations is from several special graphs. In this case, the graph summation results of the path graph and the cycle graph will produce the same fractional chromatic number as the sum of the fractional chromatic numbers of each graph before it is operated.

The Partition Dimension of a Path Graph

Resolving partition is part of graph theory. This article, explains about resolving partition of the path graph, with. Given a connected graph and is a subset of writen . Suppose there is , then the distance between and is denoted in the form . There is an ordered set of -partitions of, writen then the representation of with respect tois the The set of partitions ofis called a resolving partition if the representation of each to is different. The minimum cardinality of the solving-partition to is called the partition dimension of G which is denoted by . Before getting the partition dimension of a path graph, the first step is to look for resolving partition of the graph. Some resolving partitions of path graph, with , and are obtained. Then, the partition dimension of the path graph which is the minimum cardinality of resolving partition, namely pd (Pn)=2Resolving partition is part of graph theory. This article, explains about resolving partition of the path graph, with. Given a connected graph and is a subset of writen . Suppose there is , then the distance between and is denoted in the form . There is an ordered set of -partitions of, writen then the representation of with respect tois the The set of partitions ofis called a resolving partition if the representation of each to is different. The minimum cardinality of the solving-partition to is called the partition dimension of G which is denoted by . Before getting the partition dimension of a path graph, the first step is to look for resolving partition of the graph. Some resolving partitionsof path graph, with , and are obtained. Then, the partition dimension of the path graph which is the minimum cardinality of resolving partition, namely.

On the Locating Chromatic Number of Barbell Shadow Path Graph

The locating-chromatic number was introduced by Chartrand in 2002. The locating chromatic number of a graph is a combined concept between the coloring and partition dimension of a graph. The locating chromatic number of a graph is defined as the cardinality of the minimum color classes of the graph. In this paper, we discuss about the locating-chromatic number of shadow path graph and barbell graph containing shadow graph.

Graph Connectivity in Log Steps Using Label Propagation

The fastest deterministic algorithms for connected components take logarithmic time and perform superlinear work on a Parallel Random Access Machine (PRAM). These algorithms maintain a spanning forest by merging and compressing trees, which requires pointer-chasing operations that increase memory access latency and are limited to shared-memory systems. Many of these PRAM algorithms are also very complicated to implement. Another popular method is “leader-contraction” where the challenge is to select a constant fraction of leaders that are adjacent to a constant fraction of non-leaders with high probability, but this can require adding more edges than were in the original graph. Instead we investigate label propagation because it is deterministic, easy to implement, and does not rely on pointer-chasing. Label propagation exchanges representative labels within a component using simple graph traversal, but it is inherently difficult to complete in a sublinear number of steps. We are able to overcome the problems with label propagation for graph connectivity. We introduce a surprisingly simple framework for deterministic, undirected graph connectivity using label propagation that is easily adaptable to many computational models. It achieves logarithmic convergence independently of the number of processors and without increasing the edge count. We employ a novel method of propagating directed edges in alternating direction while performing minimum reduction on vertex labels. We present new algorithms in PRAM, Stream, and MapReduce. Given a simple, undirected graph [Formula: see text] with [Formula: see text] vertices, [Formula: see text] edges, our approach takes O(m) work each step, but we can only prove logarithmic convergence on a path graph. It was conjectured by Liu and Tarjan (2019) to take [Formula: see text] steps or possibly [Formula: see text] steps. Our experiments on a range of difficult graphs also suggest logarithmic convergence. We leave the proof of convergence as an open problem.

Paths of Length Three are $K_{r+1}$-Turán-Good

The generalized Turán problem ex$(n,T,F)$ is to determine the maximal number of copies of a graph $T$ that can exist in an $F$-free graph on $n$ vertices. Recently, Gerbner and Palmer noted that the solution to the generalized Turán problem is often the original Turán graph. They gave the name "$F$-Turán-good" to graphs $T$ for which, for large enough $n$, the solution to the generalized Turán problem is realized by a Turán graph. They prove that the path graph on two edges, $P_2$, is $K_{r+1}$-Turán-good for all $r \ge 3$, but they conjecture that the same result should hold for all $P_\ell$. In this paper, using arguments based in flag algebras, we prove that the path on three edges, $P_3$, is also $K_{r+1}$-Turán-good for all $r \ge 3$.

Metric Dimension on Path-Related Graphs

Graph theory has a large number of applications in the fields of computer networking, robotics, Loran or sonar models, medical networks, electrical networking, facility location problems, navigation problems etc. It also plays an important role in studying the properties of chemical structures. In the field of telecommunication networks such as CCTV cameras, fiber optics, and cable networking, the metric dimension has a vital role. Metric dimension can help us in minimizing cost, labour, and time in the above discussed networks and in making them more efficient. Resolvability also has applications in tricky games, processing of maps or images, pattern recognitions, and robot navigation. We defined some new graphs and named them s − middle graphs, s -total graphs, symmetrical planar pyramid graph, reflection symmetrical planar pyramid graph, middle tower path graph, and reflection middle tower path graph. In the recent study, metric dimension of these path-related graphs is computed.

On a conjecture of Laplacian energy of trees

Let [Formula: see text] be a simple graph with [Formula: see text] vertices, [Formula: see text] edges having Laplacian eigenvalues [Formula: see text]. The Laplacian energy LE[Formula: see text] is defined as LE[Formula: see text], where [Formula: see text] is the average degree of [Formula: see text]. Radenković and Gutman conjectured that among all trees of order [Formula: see text], the path graph [Formula: see text] has the smallest Laplacian energy. Let [Formula: see text] be the family of trees of order [Formula: see text] having diameter [Formula: see text]. In this paper, we show that Laplacian energy of any tree [Formula: see text] is greater than the Laplacian energy of [Formula: see text], thereby proving the conjecture for all trees of diameter [Formula: see text]. We also show the truth of conjecture for all trees with number of non-pendent vertices at most [Formula: see text]. Further, we give some sufficient conditions for the conjecture to hold for a tree of order [Formula: see text].

A Graceful Labeling of Square of Path Graph with Quadratic Complexity Algorithm

A method for relaxed graceful labeling of P2n graphs is presented together with an algorithm designed for labeling these graphs. Graceful labeling is achieved by relaxing the range to 2m and perform the labeling using an algorithm with quadratic complexity (O(n2)). The algorithm can be used for labeling any P2n graph with n ≥ 3, as far as the machine can handle the size of the problem.

Automatic Generation of Meta-Path Graph for Concept Recommendation in MOOCs

In MOOCs, generally speaking, curriculum designing, course selection, and knowledge concept recommendation are the three major steps that systematically instruct users to learn. This paper focuses on the knowledge concept recommendation in MOOCs, which recommends related topics to users to facilitate their online study. The existing approaches only consider the historical behaviors of users, but ignore various kinds of auxiliary information, which are also critical for user embedding. In addition, traditional recommendation models only consider the immediate user response to the recommended items, and do not explicitly consider the long-term interests of users. To deal with the above issues, this paper proposes AGMKRec, a novel reinforced concept recommendation model with a heterogeneous information network. We first clarify the concept recommendation in MOOCs as a reinforcement learning problem to offer a personalized and dynamic knowledge concept label list to users. To consider more auxiliary information of users, we construct a heterogeneous information network among users, courses, and concepts, and use a meta-path-based method which can automatically identify useful meta-paths and multi-hop connections to learn a new graph structure for learning effective node representations on a graph. Comprehensive experiments and analyses on a real-world dataset collected from XuetangX show that our proposed model outperforms some state-of-the-art methods.