Random walk labelings of perfect trees and other graphs

We investigate the number of permutations that occur in random labellings of trees. This is a generalisation of the number of subpermutations occurring in a random permutation. It also generalises some recent results on the number of inversions in randomly labelled trees (Cai et al. in Combin Probab Comput 28(3):335–364, 2019). We consider complete binary trees as well as random split trees a large class of random trees of logarithmic height introduced by Devroye (SIAM J Comput 28(2):409–432, 1998. https://doi.org/10.1137/s0097539795283954). Split trees consist of nodes (bags) which can contain balls and are generated by a random trickle down process of balls through the nodes. For complete binary trees we show that asymptotically the cumulants of the number of occurrences of a fixed permutation in the random node labelling have explicit formulas. Our other main theorem is to show that for a random split tree, with probability tending to one as the number of balls increases, the cumulants of the number of occurrences are asymptotically an explicit parameter of the split tree. For the proof of the second theorem we show some results on the number of embeddings of digraphs into split trees which may be of independent interest.

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.

This paper deals with compact label-based representations for trees. Consider an n-node undirected connected graph G with a predefined numbering on the ports of each node. The all-ports tree labeling ${\mathcal L}_{all}$ gives each node v of G a label containing the port numbers of all the tree edges incident to v. The upward tree labeling ${\mathcal L}_{up}$ labels each node v by the number of the port leading from v to its parent in the tree. Our measure of interest is the worst case and total length of the labels used by the scheme, denoted Mup(T) and Sup(T) for ${\mathcal L}_{up}$ and Mall(T) and Sall(T) for ${\mathcal L}_{all}$. The problem studied in this paper is the following: Given a graph G and a predefined port labeling for it, with the ports of each node v numbered by 0,...,deg(v) – 1, select a rooted spanning tree for G minimizing (one of) these measures. We show that the problem is polynomial for Mup(T), Sup(T) and Sall(T) but NP-hard for Mall(T) (even for 3-regular planar graphs). We show that for every graph G and port numbering there exists a spanning tree T for which Sup(T) = O(n log log n). We give a tight bound of O(n) in the cases of complete graphs with arbitrary labeling and arbitrary graphs with symmetric port assignments. We conclude by discussing some applications for our tree representation schemes.

We consider random labelings of finite graphs conditioned on a small fixed number of peaks. We introduce a continuum framework where a combinatorial graph is associated with a metric graph and edges are identified with intervals. Next we consider a sequence of partitions of the edges of the metric graph with the partition size going to zero. As the mesh of the subdivision goes to zero, the conditioned random labelings converge, in a suitable sense, to a deterministic function which evolves as an increasing process of subsets of the metric graph that grows at rate one while maximizing an appropriate notion of entropy. We call such functions floodings. We present a number of qualitative and quantitative properties of floodings and some explicit examples.

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.

<abstract><p>Graph labeling is a source of valuable mathematical models for an extensive range of applications in technologies (communication networks, cryptography, astronomy, data security, various coding theory problems). An edge $ \; \delta - $ graceful labeling of a graph $ G $ with $ p\; $ vertices and $ q\; $ edges, for any positive integer $ \; \delta $, is a bijective $ \; f\; $ from the set of edge $ \; E(G)\; $ to the set of positive integers $ \; \{ \delta, \; 2 \delta, \; 3 \delta, \; \cdots\; , \; q\delta\; \} $ such that all the vertex labels $ \; f^{\ast} [V(G)] $, given by: $ f^{\ast}(u) = (\sum\nolimits_{uv \in E(G)} f(uv)\; )\; mod\; (\delta \; k) $, where $ k = max (p, q) $, are pairwise distinct. In this paper, we show the existence of an edge $ \; \delta- $ graceful labeling, for any positive integer $ \; \delta $, for the following graphs: the splitting graphs of the cycle, fan, and crown, the shadow graphs of the path, cycle, and fan graph, the middle graphs and the total graphs of the path, cycle, and crown. Finally, we display the existence of an edge $ \; \delta- $ graceful labeling, for the twig and snail graphs.</p></abstract>

Arbitrage transactions have played a significant role in allowing Tesla to exploit the minor differences in price between its similar and identical assets in two or more markets. In this case, as an arbitrage trader, Tesla undertakes the responsibility of purchasing assets in one market and selling them out at the same period to pocket any differences between the two prices. In this paper, there are two main parts of arbitrage transactions. The first part is three methods that can be utilized to analyze the options prices. These methods are linear regression and random Walk Theory, and Black Scholes. Their functionalities, influences, and theory are demonstrated in this paper. When using Black Scholes to calculate desired options prices and figure it in linear regression, assuming time to the expiration date, volatility, and risk-free rate are the same. The estimated trend and outcome are stated and described in this paper. The second part is about arbitrage transactions based on option prices. The circumstances of arbitrage transactions used by Tesla are analyzed and the utilized method called fundraising is demonstrated, and its functionality is analyzed with data and facts to prove the significance of arbitrage transactions. The paper also points out it is possible to use black scholes to estimate option prices in linear regression graphs and random walk theory. Call and put options prices are related to stock prices. They all shared a similar trend when time flows. The true value of this research is that people do not need to simply use a single method to estimate values, but to combine them, use them together to better understand the estimation of prices.

All the graphs in this paper are connected graphs. Let $G=(V,E)$ where $V(G)$ is a set of vertex from graph $G$ while $E(G)$ is a set of edge from graph $G$. A bijection function $f: V \rightarrow \{1,2,3,...,\lvert V(G)\rvert\}$ the associated weight of an edge $uv \in E(G)$ under $f$ is $W_f{(uv)}=f(u)+f(v)$. A path $P$ in a vertex-labeled graph $G$ is said to be a rainbow path if for every two edges $uv$, $u'v' \in E(P)$, there is $w_f{(uv)}\neq w_f{u'v'}$. If for every two vertices $u$ and $v$ of $G$, there is a rainbow $u$-$v$ path, then $f$ is called a rainbow antimagic labeling of $G$. A graph $G$ is rainbow antimagic if $G$ has a rainbow antimagic labeling. The minimum number of color needed to make $G$ rainbow connected, called rainbow antimagic connection number, denoted by rac(G). In this paper, we will analyze the rainbow antimagic coloring on comb product of path graph.

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

Graph labeling is the concept of assigning identifiers to the nodes in a graph. The concept, being introduced in the late 1960s, has flourished much over time, and numerous papers have been written based on multiple labeling concepts. Graceful labeling, a particularly unique concept of graph labeling, was introduced by Rosa in 1967, which follows the idea that every vertex of a graph with $e$ edges can be assigned a unique label from the set { 0, 1, 2, …, $e$ }, and the labels of the edges will be assigned with the absolute difference between the labels of its adjoining edges, where every edge will have a unique label. Much research has been conducted to prove more and more classes are graceful, which itself has significant application in numerous fields, like road mapping, resource distribution, radio astronomy, cryptography, traffic routing, X-ray crystallography. In this paper, we introduce a class of graph, created from a collection of path graph and C4 cycle graph, terming it as P - n Legged C 4 - p’ graphs, and provide enough reference to prove that the derived class is graceful. We also provide mathematical expressions for every vertex in the graph, whilst considering multiple cases of path and chain length.

Graph labeling is the assignment of integers to edges or vertices or both depending on some factors which is consequential in the field of graph theory. In this paper, anti-magic graph labeling of two graphs, middle graph and total graph of path graph have been mainly focused. Modified proofs of theorems on anti-magic graph labeling have been introduced here. Both of the theorems are applicable for strong anti-magic graph. The correctness of the proofs has been demonstrated through various examples.

In this paper, the concept of intuitionistic fuzzy labelling (IFL) has been introduced. Intuitionistic fuzzy bi-magic labelling (IFBL) and intuitionistic fuzzy anti-magic labelling (IFAL) have been discussed for cycle, star and path graph. Intuitionistic fuzzy sets will be given both degree of membership and degree of non-membership, which are independent of each other to the extent, that their sum is not greater than one. Various types of arcs have been analysed and blank arc has been introduced and also discussed for intuitionistic fuzzy labelling. We also discussed about intuitionistic fuzzy bridge and intuitionistic fuzzy cut-vertices with examples and their properties are studied.

This paper focus on the dynamic and quickly developing subject of graph Domination study. In this paper, Co-Secure Domination and Square diffrence Labeling for Middle graph of Paths and Regular Spider Graphs has been determined. The findings can be applied on protection strategy for a various communication and interconnection networks, where the routing servers as the network’s vertices and the connection between them as its edges, protecting each node in addition to the guard nodes. This ensures that the guard nodes remain secure as well, and the network as a whole is protected.

We introduce the notion of total distance irregular labeling, called the local total distance irregular labeling. All edges and vertices are labeled with positive integers 1 to k such that the weight calculated at the vertices induces a vertex coloring if two adjacent vertices has different weight. The weight of a vertex $u\in V(G)$ is defined as the sum of the labels of all vertices adjacent and edges incident to $u$ (distance $1$ from $u$). The minimum cardinality of the largest label over all such irregular assignment is called the local total distance irregularity strength, denoted by $tdis_l(G)$. In this paper, we established the lower bound of the local total distance irregularity strength of graphs $G$ and determined exact values of some classes of graphs namely path, cycle, star, bipartite complete, fan and sun graph.

This study investigates the existence of discrete and strong discrete labeling for several classes of path and star-related graphs, contributing to the broader field of graph labeling theory. A graph is said to admit a discrete labeling if its vertices can be assigned binary labels (0 and 1), such that each edge receives a label determined by the exclusive OR (XOR) of the labels of its incident vertices, subject to specified constraints. We first establish that the super subdivision of a path <img src=image/13441705_01.gif> by the complete bipartite graph <img src=image/13441705_02.gif> is discrete, demonstrating that replacing each edge of <img src=image/13441705_01.gif> with <img src=image/13441705_02.gif> preserves the discrete labeling property. Additionally, we prove that the square of a path <img src=image/13441705_01.gif>, obtained by connecting vertices at a distance of at most two, also admits a discrete labeling. Further, we examine the shadow graph <img src=image/13441705_04.gif>, constructed by duplicating <img src=image/13441705_01.gif> and connecting each vertex to its copy's neighbors, and show that it is also discrete. Another variant, the graph <img src=image/13441705_03.gif> formed by switching one pendant vertex in <img src=image/13441705_01.gif>, is similarly proven to support discrete labeling. Beyond paths, we explore star-related graphs, confirming that the shadow graph of a star <img src=image/13441705_05.gif> and the shadow graph of a bi-star <img src=image/13441705_06.gif> both permit discrete labeling. Extending the concept, we introduce strong discrete labeling, which imposes the stricter requirement by adding one more constraint to discrete labeling. Under this stronger condition, we prove that the star graph <img src=image/13441705_07.gif> admits a strong discrete labeling if and only if m is even, highlighting a parity-based constraint absent in standard discrete labeling. These findings expand the known families of graphs with discrete and strong discrete labeling, offering new insights into structural properties that facilitate such labeling. Potential applications include coding theory, network design, and algorithm optimization, while open questions remain regarding the extensibility of these results to other graph operations and more complex graph classes.