Hanoi Graphs Research Articles

On the restricted Hanoi graphs

On the restricted Hanoi graphs

A tour of general Hanoi graphs

The Tower of Hanoi puzzle has fascinated researchers in mathematics and theoretical computer science for over a hundred years. Many variants of the classical puzzle have been posed, such as allowing more than 3 pegs, and adding restrictions on the possible moves between the pegs. It is natural to view the pegs and allowed moves by means of a graph. The pegs are represented by the vertices of the graph, and the allowed moves by the edges. For each n, this graph yields a graph on the set of all legal configurations of n disks. Thus, the questions about the original puzzle may be reformulated as questions about connectivity and shortest paths in the graph of all configurations. Moreover, it was shown that classical Hanoi graphs are related to several interesting and useful structures such as the Sierpiński gasket and Gray codes, and thus several properties of these graphs were studied, including Hamiltonicity and planarity. In this paper we study these properties, and several others – chromatic number, clique number, and girth – for general Hanoi graphs.

The ice model on the three-dimensional Hanoi graph

We study the ice model (with Boltzmann factors equal to one) on the 3D Hanoi graph H(n) at stage n. We derive recursive equations for functions that provide the number of configurations for each type of restricted arrow diagram. The total number of configurations I(n) can be computed exactly using these functions. The entropy per site in the thermodynamic limit is for H(n). This is comparable to the entropy per site for the square lattice (Lieb 1967 Phys. Rev. Lett. 18 692–4 and Lieb E H 1967 Phys. Rev. 162 162–72).

Analysis of Hanoi graph by using topological indices

One of the main challenges in molecular science is to model a chemical complex and predict its thermochemical properties. Various researchers have developed a number of hypothetical techniques in this regard, and one of them is focused on the topological indices. A topological index is a number associated with a molecule’s structural graph that is considered to be capable of predicting certain chemical and physical properties of molecules. Degree-based topological indices are one of the groups of topological indices that are crucial in chemical graph theory. Shannon’s entropy is a fundamental factor of a subclass of topological indices that measures the structural information of the graphs. In this study, we examine the entropy based on topological indices such as the first and second Zagreb indices, the general Randić index, the harmonic and sum-connectivity index of Hanoi graph. Moreover, we determine a number of multiplicative invariants, and different polynomials for Hanoi graphs with graphical illustrations.

HAUSDORFF DIMENSIONS OF FLOWER NETWORKS AND HANOI GRAPHS

Zeng and Xi introduced the Hausdorff dimension of a family of networks and investigated the dimensions of touching networks. In this paper, using the self-similarity and induction we obtain the Hausdorff dimension of flower networks and Hanoi graphs, which are not touching networks.

Enumeration of maximum matchings in the Hanoi graphs using matching polynomials

In this paper, we consider Tower of Hanoi graphs and study their matching properties. Explicit system of recurrences is derived for the matching polynomials of these graphs and their appropriate truncated variants. Consequently, we obtain exact formula for the numbers of maximum matchings in Hanoi graphs using matching polynomials, which is a new approach for old one problem.

On the treewidth of Hanoi graphs

The objective of the well-known Tower of Hanoi puzzle is to move a set of discs one at a time from one of a set of pegs to another, while keeping the discs sorted on each peg. We propose an adversarial variation in which the first player forbids a set of states in the puzzle, and the second player must then convert one randomly-selected state to another without passing through forbidden states. Analyzing this version raises the question of the treewidth of Hanoi graphs. We find this number exactly for three-peg puzzles and provide nearly-tight asymptotic bounds for larger numbers of pegs.

Independence polynomial of the Sierpinski gasket graph and the tower of Hanoi graph

Independence polynomial of the Sierpinski gasket graph and the tower of Hanoi graph

Degree sequence of the generalized Sierpiński graph

Sierpinski graphs are studied in fractal theory and have applications in diverse areas including dynamic systems, chemistry, psychology, probability, and computer science. Polymer networks and WK-recursive networks can be modeled by generalized Sierpinski graphs. The degree sequence of (ordinary) Sierpi\'nski graphs and Hanoi graphs (and some of their topological indices) are determined in the literature. The number of leaves (vertices of degree one) of the generalized Sierpinski graph $S(T, t)$ of any tree $T$ was determined in 2017 and in terms of $t$, $|V(T)|$, and the number of leaves of the base graph $T$. In this paper, we generalize these results. More precisely, for every simple graph $G$ of order $n$, we completely determine the degree sequence of the generalized Sierpinski graph $S(G,t)$ of $G$ in terms of $n$, $t$ and the degree sequence of $G$. By using it, we determine the exact value of the general first Zagreb index of $S(G,t)$ in terms of the same parameters of $G$.

Computing Molecular Topological Descriptors of Polymeric Networks Modeled by Sierpinski Networks

Sierpinski networks constitute an extensively studied class of graphs of fractal nature applicable in topology, mathematics of Tower of Hanoi, computer science, and Chemical graph theory. Recently the properties of Sierpinski networks have been studied like physico-chemical properties, thermodynamic properties, chemical activity, biological activity, metric dimension and degree based topological indices. In this paper, we further extend the study of Sierpinski networks and computed Zagreb types molecular descriptors (molecular topological indices) and their variants.

Some Two-Vertex Resistances of the Three-Towers Hanoi Graph Formed by a Fractal Graph

The resistance distance between two vertices of a simple connected graph G is equal to the resistance between two equivalent points on an electrical network, constructed to correspond to G, with each edge being replaced by a unit resistor. In this study, we apply the principle of substitution to three-towers Hanoi graph to procure an equivalent network that enables us to only use the series and parallel principles and the delta-wye transformation to acquire the some two-vertex resistance distances of three-towers Hanoi graph.

The sigma chromatic number of the Sierpiński gasket graphs and the Hanoi graphs

A vertex coloring c : V(G) → ℕ of a non-trivial connected graph G is called a sigma coloring if σ(u) ≠ σ(v) for any pair of adjacent vertices u and v. Here, σ(x) denotes the sum of the colors assigned to vertices adjacent to x. The sigma chromatic number of G, denoted by σ(G), is defined as the fewest number of colors needed to construct a sigma coloring of G. In this paper, we determine the sigma chromatic numbers of the Sierpiński gasket graphs and the Hanoi graphs. Moreover, we prove the uniqueness of the sigma coloring for Sierpiński gasket graphs.

Study of dimer–monomer on the generalized Hanoi graph

We study the number of dimer–monomers $$M_d(n)$$ on the Hanoi graphs $$H_d(n)$$ at stage n with dimension d equal to 3 and 4. The entropy per site is defined as $$z_{H_d}=\lim _{v \rightarrow \infty } \ln M_d(n)/v$$, where v is the number of vertices on $$H_d(n)$$. We obtain the lower and upper bounds of the entropy per site, and the convergence of these bounds approaches to zero rapidly when the calculated stage increases. The numerical values of $$z_{H_d}$$ for $$d=3, 4$$ are evaluated to more than a hundred digits correct. Using the results with d less than or equal to 4, we predict the general form of the lower and upper bounds for $$z_{H_d}$$ with arbitrary d.

$M$-Polynomial and topological indices of Hanoi graph and generalized wheel graph

$M$-Polynomial and topological indices of Hanoi graph and generalized wheel graph

Laceability in Hanoi Graphs

The topological structure of an interconnection network can be modelled by a connected, simple and undirected graph \(G=(V,E)\) where \(V\) represents the set of processors and \(E\) represents the set of communication links. Interconnection networks are used to interconnect the processors of data centres and cluster computers. The study of Hamiltonicity and the related areas such as Hamiltonian laceability and Hamiltonian connectedness has lot of significance in computer networks. A network(graph) is Hamiltonian connected if it contains a Hamiltonian path between two distinct nodes (vertices). In this paper we shall study the laceability properties associated with Hanoi graphs \(H_n\). To be more specific we shall explore Hamiltonian \(t^*\) connectedness of Hanoi graphs \(H_n\) for \(n\ge3\).

Dimer coverings on the Tower of Hanoi graph

We present the number of dimer coverings Nd(n) on the Tower of Hanoi graph THd(n) at n stage with dimension 2 [Formula: see text][Formula: see text]d[Formula: see text][Formula: see text] 5. When the number of vertices v(n) is even, Nd(n) gives the number of close-packed dimers; when the number of vertices is odd, it is impossible to have a close-packed configurations and one of the outmost vertices is allowed to be unoccupied. We define the entropy of absorption of diatomic molecules per vertex as S[Formula: see text][Formula: see text]=[Formula: see text][Formula: see text] Nd(n)/v(n), that can be shown exactly for TH2, while its lower and upper bounds can be derived in terms of the results at a certain n for THd(n) with 3 [Formula: see text][Formula: see text]d[Formula: see text][Formula: see text] 5. We find that the difference between the lower and upper bounds converges rapidly to zero as n increases, such that the value of S[Formula: see text] with d[Formula: see text]=[Formula: see text]3 and 5 can be calculated with at least 100 correct digits.

Anti-Ramsey number of Hanoi graphs

Anti-Ramsey number of Hanoi graphs

Algorithm for finding the Wiener indices of some families of graphs

In this paper we provide algorithms for finding the Wiener indices of SM family of Graphs and Hanoi Graphs. The Hanoi Graph is related to the popular Tower of Hanoi puzzle. SM family of Graphs consists of SM sum graphs, SM balancing graphs, its complement graphs and its subgraphs. The SM sum graph is associated with the relationship between the powers of 2 and positive integers. The SM balancing graphs are formed by using the fact that all positive integers can be written as a linear combination of powers of 3. These are systematically arranged graphs. Also the topological indices like Wiener indices are more important in life science and computer science. The values of Wiener indices need to be calculated for different values of n. A normal ordinary method will be difficult for large values of n. So we are deriving algorithmic method to find the Wiener indices and then the roots of the HOSOYA polynomial in the cases of these graphs.

Tower of Hanoi: Exploring Multiple Representations

Carefully designed tasks enable preservice teachers to explore this puzzle through concrete, pictorial, numerical, symbolic, and graphical representations and engage in explicit and recursive reasoning, deal with counting problems, create Hanoi graphs, and develop mathematical thinking.

Power domination in Kn\"odel graphs and Hanoi graphs

In this paper, we study the power domination problem in Knodel graphs W-Delta,W-2 nu and Hanoi graphs H-p(n). We determine the power domination number of W-3,W-2 nu and provide an upper bound for the power domination number of W-r+1,W-2r+1 for r >= 3. We also compute the k-power domination number and the k-propagation radius of H-p(2).