Ary Trees Research Articles

The Coherence and Properties Analysis of Balanced $2^{p}$-Ary Tree Networks

The Coherence and Properties Analysis of Balanced $2^{p}$-Ary Tree Networks

On Gibbs measures of the Ising Model on (k, m)-Ary Trees

On Gibbs measures of the Ising Model on <i>(k, m)</i>-Ary Trees

A Note on Acyclic Token Sliding Reconfiguration Graphs of Independent Sets

We continue the study of Token Sliding (reconfiguration) graphs of independent sets initiated by the authors in an earlier paper [Graphs Comb. 39.3, 59, 2023]. Two of the topics in that paper were to study which graphs \(G\) are Token Sliding graphs and which properties of a graph are inherited by a Token Sliding graph. In this paper, we continue this study specializing in the case of when \(G\) and/or its Token Sliding graph \(\mathsf{TS}_k(G)\) is a tree or forest, where \(k\) is the size of the independent sets considered. We consider two problems. The first is to find necessary and sufficient conditions on \(G\) for \(\mathsf{TS}_k(G)\) to be a forest. The second is to find necessary and sufficient conditions for a tree or forest to be a Token Sliding graph. For the first problem, we give a forbidden subgraph characterization for the cases of \(k=2,3\). For the second problem, we show that for every \(k\)-ary tree \(T\) there is a graph \(G\) for which \(\mathsf{TS}_{k+1}(G)\) is isomorphic to \(T\). A number of other results are given along with a join operation that aids in the construction of \(\mathsf{TS}_k\)-graphs.

Computing Edge Irregularity Strength of Star and Banana Trees Using Algorithmic Approach

After the Chartrand definition of graph labeling, since 1988 many graph families have been labeled through mathematical techniques. A basic approach in those labelings was to find a pattern among the labels and then prove them using sequences and series formulae. In 2018, Asim applied computer-based algorithms to overcome this limitation and label such families where mathematical solutions were either not available or the solution was not optimum. Asim et al. in 2018 introduced the algorithmic solution in the area of edge irregular labeling for computing a better upper-bound of the complete graph \(es(K_n)\) and a tight upper-bound for the complete \(m\)-ary tree \({es(T}_{m,h})\) using computer-based experiments. Later on, more problems like complete bipartite and circulant graphs were solved using the same technique. Algorithmic solutions opened a new horizon for researchers to customize these algorithms for other types of labeling and for more complex graphs. In this article, to compute edge irregular \(k\)-labeling of star \(S_{m,n}\) and banana tree \({BT}_{m,n}\), new algorithms are designed, and results are obtained by executing them on computers. To validate the results of computer-based experiments with mathematical theorems, inductive reasoning is adopted. Tabulated results are analyzed using the law of double inequality and it is concluded that both families of trees observe the property of edge irregularity strength and are tight for \(\left\lceil \frac{|V|}{2} \right\rceil\)-labeling.

Regularity of the edge ideals of perfect [ν,h]-ary trees and some unicyclic graphs

We compute the Castelnuovo-Mumford regularity of the quotient rings of edge ideals of perfect [ν,h]-ary trees and some unicyclic graphs.

A Note to Non-adaptive Broadcasting

Broadcasting is a fundamental problem in the information dissemination area. In classical broadcasting, a message must be sent from one network member to all other members as rapidly as feasible. Although it has been demonstrated that this problem is NP-Hard for arbitrary graphs, it has several applications in various fields. As a result, the universal lists model, replicating real-world restrictions like the memory limits of nodes in large networks, is introduced as a branch of this problem in the literature. In the universal lists model, each node is equipped with a fixed list and has to follow the list regardless of the originator. In this study, we focus on the non-adaptive branch of universal lists broadcasting. In this regard, we establish the optimal broadcast time of [Formula: see text]-ary trees and binomial trees under the non-adaptive model and provide an upper bound for complete bipartite graphs. We also improved a general upper bound for trees under the same model and showed that our upper bound cannot be improved in general.

Analysis of -ary tree algorithms with successive interference cancellation

AbstractWe calculate the mean throughput, number of collisions, successes, and idle slots for random tree algorithms with successive interference cancellation. Except for the case of the throughput for the binary tree, all the results are new. We furthermore disprove the claim that only the binary tree maximizes throughput. Our method works with many observables and can be used as a blueprint for further analysis.

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A k-plane tree is a plane tree whose vertices are assigned labels between 1 and k in such a way that the sum of the labels along any edge is no greater than k+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k+1$$\\end{document}. These trees are known to be related to (k+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(k+1)$$\\end{document}-ary trees, and they are counted by a generalised version of the Catalan numbers. We prove a surprisingly simple refined counting formula, where we count trees with a prescribed number of labels of each kind. Several corollaries are derived from this formula, and an analogous theorem is proven for k-noncrossing trees, a similarly defined family of labelled noncrossing trees that are related to (2k+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(2k+1)$$\\end{document}-ary trees.

A Stochastic Combustion Model with Thresholds on Trees

Place one active particle at the root of a graph and a Poisson-distributed number of dormant particles at the other vertices. Active particles perform simple random walk. Once the number of visits to a site reaches a random threshold, any dormant particles there become active. For this process on infinite $d$-ary trees, we show that total root visits undergoes a phase transition.

Relations Between Vertex–Edge Degree Based Topological Indices and Mve-Polynomial of r−Regular Simple Graph

One of the more exciting polynomials among the newly presented graph algebraic polynomials is the M−Polynomial, which is a standard method for calculating degree−based topological indices. In this paper, we define the Mve−polynomials based on vertex edge degree and derive various vertex–edge degree based topological indices from them. Thus, for any graph, we provide some relationships between vertex–edge degree topological indices. Also, we discuss the general Mve−polynomial of r−regular simple graph. Finally, we computed the Mve−polynomial of the 2−ary tree graph.

The characterization of lucky edge coloring in graphs

The lucky edge coloring of graph [Formula: see text] is a proper edge coloring which is induced by a vertex coloring such that each edge is labeled by the sum of its vertices. The least integer [Formula: see text] for which [Formula: see text] has a lucky edge coloring in the set [Formula: see text] is called lucky number, denoted by [Formula: see text]. The lucky numbers were already calculated for a large number of graphs, but not yet for trees. In this paper, we provide the characterization of lucky edge coloring and calculate the lucky number for graphs which can be regarded as complete [Formula: see text]-ary trees.

Self-similar abelian groups and their centralizers

We extend results on transitive self-similar abelian subgroups of the group of automorphisms \mathcal{A}_m of an m -ary tree \mathcal{T}_m by Brunner and Sidki to the general case where the permutation group induced on the first level of the tree, has s\geq 1 orbits. We prove that such a group A embeds in a self-similar abelian group A^* which is also a maximal abelian subgroup of \mathcal{A}_m . The construction of A^{*} is based on the definition of a free monoid \Delta of rank s of partial diagonal monomorphisms of \mathcal{A}_m . Precisely, A^{*} = \overline{\Delta(B(A))} , where B(A) denotes the product of the projections of A in its action on the different s orbits of maximal subtrees of \mathcal{T}_m , and bar denotes the topological closure. Furthermore, we prove that if A is non-trivial, then A^{*} = C_{\mathcal{A}_m} (\Delta(A)) , the centralizer of \Delta(A) in \mathcal{A}_m . When A is a torsion self-similar abelian group, it is shown that it is necessarily of finite exponent. Moreover, we extend recent constructions of self-similar free abelian groups of infinite enumerable rank to examples of such groups which are also \Delta -invariant for s=2 . In the final section, we introduce for m=ns \geq 2 , a generalized adding machine a , an automorphism of \mathcal{T}_{m} , and show that its centralizer in \mathcal{A}_{m} to be a split extension of \langle a \rangle^{*} by \mathcal{A}_s . We also describe important \mathbb{Z}_n [\mathcal{A}_s] submodules of \langle a\rangle^{*} .

Some algebraic invariants of the edge ideals of perfect $ [h, d] $-ary trees and some unicyclic graphs

&lt;abstract&gt;&lt;p&gt;This article is mainly concerned with computations of some algebraic invariants of quotient rings of edge ideals of perfect $ [h, d] $-ary trees and unicyclic graphs. We compute exact values of depth and Stanley depth and consequently projective dimension for above mentioned quotient rings, except for the one special case of unicyclic graph for which best possible bounds of Stanley depth are given.&lt;/p&gt;&lt;/abstract&gt;

The Gessel correspondence and the partial [formula omitted]-positivity of the Eulerian polynomials on multiset Stirling permutations

Pondering upon the grammatical labeling of 0–1–2 increasing plane trees, we come to the realization that the grammatical labels play a role as records of chopped off leaves of the original increasing binary trees. While such an understanding is purely psychological, it does give rise to an efficient apparatus to tackle the partial γ-positivity of the Eulerian polynomials on multiset Stirling permutations, as long as we bear in mind the combinatorial meanings of the labels x and y in the Gessel representation of a k-Stirling permutation by means of an increasing (k+1)-ary tree. More precisely, we introduce a Foata–Strehl action on the Gessel trees resulting in an interpretation of the partial γ-coefficients of the aforementioned Eulerian polynomials, different from the ones found by Lin–Ma–Zhang and Yan–Huang–Yang. In particular, our strategy can be adapted to deal with the partial γ-coefficients of the second order Eulerian polynomials, which in turn can be readily converted to the combinatorial formulation due to Ma–Ma–Yeh in connection with certain statistics of Stirling permutations.

The Swendsen–Wang dynamics on trees

AbstractThe Swendsen–Wang algorithm is a sophisticated, widely‐used Markov chain for sampling from the Gibbs distribution for the ferromagnetic Ising and Potts models. This chain has proved difficult to analyze, due in part to its global nature. We present optimal bounds on the convergence rate of the Swendsen–Wang algorithm for the complete ‐ary tree. Our bounds extend to the non‐uniqueness region and apply to all boundary conditions. We show that the spatial mixing conditions known asvariance mixingandentropy mixingimply spectral gap and mixing time, respectively, for the Swendsen–Wang dynamics on the ‐ary tree. We also show that these bounds are asymptotically optimal. As a consequence, we establish mixing for the Swendsen–Wang dynamics forallboundary conditions throughout (and beyond) the tree uniqueness region. Our proofs feature a novel spectral view of the variance mixing condition and utilize recent work on block factorization of entropy.

Semi‐uniform deployment of mobile robots in perfect ℓ$$ \ell $$‐ary trees

SummaryIn this paper, we consider the problem of semi‐uniform deployment for mobile robots in perfect ‐ary trees. This problem requires robots to spread in the tree so that, for some positive integer and some fixed integer , each node of depth is occupied by a robot. Robots have an infinite visibility range but are opaque, and each robot can emit a light color visible to itself and other robots, taken from a set of colors, at each time step. Then, we clarify the solvability of the semi‐uniform deployment problem, focusing on the number of available light colors. First, we consider robots with . In this setting, we show that there is no collision‐free algorithm to solve the problem. Next, relax the number of available light colors, that is, we consider robots with . In this setting, we propose a collision‐free algorithm that can solve the problem. From these results, we can show that the semi‐uniform deployment problem can be solved when , and our proposed algorithm is optimal with respect to the number of used light colors (i.e., 2).

The generalized microscopic image reconstruction problem

This paper presents and studies a generalization of the microscopic image reconstruction problem ( MIR ) introduced by Frosini and Nivat (2007) and Nivat (2002). Consider a specimen for inspection, represented as a collection of points typically organized on a grid in the plane. Assume each point x has an associated physical value ℓ x , which we would like to determine. However, it might be that obtaining these values precisely (by what we call a surgical probe ) is difficult, risky, or impossible. The alternative is to employ aggregate measuring techniques (such as EM, CT, US or MRI), whereby each measurement is taken over a larger window, and the exact values at each point are subsequently extracted by computational methods. In this paper, we extend the MIR framework in a number of ways. First, we consider a generalized setting where the inspected object is represented by an arbitrary graph G , and the vector ℓ ∈ R n assigns a value ℓ v to each node v . A probe centred at a node v will capture a window encompassing its entire neighbourhood N [ v ] , i.e., the outcome of a probe centred at v is P v = ∑ w ∈ N [ v ] ℓ w . We give a criterion for the graphs for which the extended MIR problem can be solved by extracting the vector ℓ from the collection of probes, P = { P v ∣ v ∈ V } . We then consider cases where such reconstruction is impossible (namely, graphs G for which the probe vector P is inconclusive, in the sense that there may be more than one vector ℓ yielding P ). Assume that surgical probes are technically available, yet are expensive or risky, and must be used sparingly. We show that in such cases, it may still be possible to achieve reconstruction based on a combination of a collection of ordinary (aggregate) probes together with a suitable set of surgical probes. We aim at identifying the minimum number of surgical probes necessary for a unique reconstruction, depending on the graph topology. This is referred to as the Minimum Surgical Probing problem (MSP). Besides providing a solution for the above problems for arbitrary graphs, we also explore the range of possible behaviours of the Minimum Surgical Probing problem by determining the number of surgical probes necessary in certain specific graph families, such as perfect k -ary trees, paths, cycles, grids, tori, tubes and hypercubes.

Hopf algebras of parking functions and decorated planar trees

We construct three new combinatorial Hopf algebras based on the Loday-Ronco operations on planar binary trees. The first and second algebras are defined on planar trees and labeled planar trees extending the Loday-Ronco and Malvenuto-Reutenauer Hopf algebras respectively. We show that the latter is bidendriform which implies that it is also free, cofree, and self-dual. The third algebra involves a new visualization of parking functions as decorated binary trees; it is also bidendriform, free, cofree, and self-dual, and therefore abstractly isomorphic to the algebra PQSym of Novelli and Thibon.We define partial orders on the objects indexing each of these three Hopf algebras, one of which, when restricting to (m+1)-ary trees, coarsens the m-Tamari order of Bergeron and Préville-Ratelle. We show that multiplication of dual fundamental basis elements is given by intervals in each of these orders.Finally, we use an axiomatized version of the techniques of Aguiar and Sottile on the Malvenuto-Reutenauer Hopf algebra to define a monomial basis on each of our Hopf algebras, and to show that comultiplication is cofree on the monomial elements. This in particular, implies the cofreeness of the Hopf algebra on planar trees. We also find explicit positive formulas for the multiplication on monomial basis and a cancellation-free and grouping-free formula for the antipode of monomial elements.

“Pass the Buck” on a Complete Binary Tree

Summary We employ the stochastic abacus to compute winning probabilities at each level of the game “Pass the Buck” on a complete binary tree with the starting vertex being the root of the tree. The derivation is also generalized to play on a complete -ary trees.

The continuous-time quantum walk on some graphs based on the view of quantum probability

In this paper, continuous-time quantum walk is discussed based on the view of quantum probability, i.e. the quantum decomposition of the adjacency matrix A of graph. Regard adjacency matrix A as Hamiltonian which is a real symmetric matrix with elements 0 or 1, so we regard [Formula: see text] as an unbiased evolution operator, which is related to the calculation of probability amplitude. Combining the quantum decomposition and spectral distribution [Formula: see text] of adjacency matrix A, we calculate the probability amplitude reaching each stratum in continuous-time quantum walk on complete bipartite graphs, finite two-dimensional lattices, binary tree, [Formula: see text]-ary tree and [Formula: see text]-fold star power [Formula: see text]. Of course, this method is also suitable for studying some other graphs, such as growing graphs, hypercube graphs and so on, in addition, the applicability of this method is also explained.