Toeplitz Graphs Research Articles

On the Constant Partition Dimension of Some Generalized Families of Toeplitz Graph

On the Constant Partition Dimension of Some Generalized Families of Toeplitz Graph

Decomposition of kth Order Slant Toeplitz Operators

In this paper, we establish the block matrix decomposition of kth order slant Toeplitz operators. We also establish some relations between the compressions of kth order slant Toeplitz and kth order slant Hankel operators on H2. In the last section, we introduce the notion of kth order slant Toeplitz graphs.

On Sharp Bounds of Local Fractional Metric Dimension for Certain Symmetrical Algebraic Structure Graphs

The smallest set of vertices needed to differentiate or categorize every other vertex in a graph is referred to as the graph’s metric dimension. Finding the class of graphs for a particular given metric dimension is an NP-hard problem. This concept has applications in many different domains, including graph theory, network architecture, and facility location problems. A graph G with order n is known as a Toeplitz graph over the subset S of consecutive collections of integers from one to n, and two vertices will be adjacent to each other if their absolute difference is a member of S. A graph G(Zn) is called a zero-divisor graph over the zero divisors of a commutative ring Zn, in which two vertices will be adjacent to each other if their product will leave the remainder zero under modulo n. Since the local fractional metric dimension problem is NP-hard, it is computationally difficult to identify an optimal solution or to precisely determine the minimal size of a local resolving set; in the worst case, the process takes exponential time. Different upper bound sequences of local fractional metric dimension are suggested in this article, along with a comparison analysis for certain families of Toeplitz and zero-divisor graphs. Furthermore, we note that the analyzed local fractional metric dimension upper bounds fall into three metric families: constant, limited, and unbounded.

On Degree-Based Topological Indices of Toeplitz Graphs

In this research paper, the determination of the orientation of a linear interpolation in an interconnected graph can be achieved by measuring its distance from a group of sonar stations strategically positioned within the graph. The study utilizes the metric dimension of toeplitz graphs. Several indices play a crucial role in analyzing motivating activities within such complex structures. The indices covered in this study include the general connectivity index of the toeplitz graph, zagreb indices, symmetric division degree index and randic indices, among others.

Asymptotic spectra of large (grid) graphs with a uniform local structure, Part II: Numerical applications

In the current work we are concerned with sequences of graphs having a grid geometry, with a uniform local structure in a bounded domain Ω⊂Rd, d≥1. When Ω=[0,1], such graphs include the standard Toeplitz graphs and, for Ω=[0,1]d, the considered class includes d-level Toeplitz graphs. In the general case, the underlying sequence of adjacency matrices has a canonical eigenvalue distribution, in the Weyl sense, and it has been shown in the theoretical part of this work that we can associate to it a symbol f. The knowledge of the symbol and of its basic analytical features provides key information on the eigenvalue structure in terms of localization, spectral gap, clustering, and global distribution. In the present paper, many different applications are discussed and various numerical examples are presented in order to underline the practical use of the developed theory. Tests and applications are mainly obtained from the approximation of differential operators via numerical schemes such as Finite Differences, Finite Elements, and Isogeometric Analysis. Moreover, we show that more applications can be taken into account, since the results presented here can be applied as well to study the spectral properties of adjacency matrices and Laplacian operators of general large graphs and networks, whenever the involved matrices enjoy a uniform local structure.

Similar d– even vertex odd mean labeling of diverse graphs

An injective function F where, F : V(G) → {0,2,4, ..., 2b + 2d - 2} is said to be d-even vertex odd mean labeling (d-EVOML) of the graph G(a, b) when the induced mapping F* : E(G) → {1,3, ..., 2b - 1} given by: F*(lw) = F(l)+F(w)/2, is a bijective function. A d - even vertex odd mean graph is a graph which allows even vertex odd mean labelling. In this study, we identify the lowest value of d for which the graphs: Y-tree, star graph, Px ʘ Kh, crown graph Rh, and rooted product Ph◊C4 have a d- even vertex odd mean labeling. Furthermore, we find the minimum number d for which the graphs: cycle graph Ch when h ≡ 2 mod 4, dragon graph Px(Ch) when h ≡ 2 mod 4, x ≥ 1, prism graph Ph, and Toeplitz graphs Th(1, 3), Th(1, 5) and Th(1, 3, 5) have a similar d- even vertex odd mean labeling. In the end, we establish that no odd cycle Ch is an even vertex odd mean graph for all d.

Minimal Doubly Resolving Sets of Certain Families of Toeplitz Graph

The doubly resolving sets are a natural tool to identify where diffusion occurs in a complicated network. Many real-world phenomena, such as rumour spreading on social networks, the spread of infectious diseases, and the spread of the virus on the internet, may be modelled using information diffusion in networks. It is obviously impractical to monitor every node due to cost and overhead limits because there are too many nodes in the network, some of which may be unable or unwilling to send information about their state. As a result, the source localization problem is to find the number of nodes in the network that best explains the observed diffusion. This problem can be successfully solved by using its relationship with the well-studied related minimal doubly resolving set problem, which minimizes the number of observers required for accurate detection. This paper aims to investigate the minimal doubly resolving set for certain families of Toeplitz graph Tn(1, t), for t ≥ 2 and n ≥ t + 2. We come to the conclusion that for Tn(1, 2), the metric and double metric dimensions are equal and for Tn(1, 4), the double metric dimension is exactly one more than the metric dimension. Also, the double metric dimension for Tn(1, 3) is equal to the metric dimension for n = 5, 6, 7 and one greater than the metric dimension for n ≥ 8.

Computing the partition dimension of certain families of Toeplitz graph.

Let G = (V(G), E(G)) be a graph with no loops, numerous edges, and only one component, which is made up of the vertex set V(G) and the edge set E(G). The distance d(u, v) between two vertices u, v that belong to the vertex set of H is the shortest path between them. A k-ordered partition of vertices is defined as β = {β1, β2, …, βk}. If all distances d(v, βk) are finite for all vertices v ∈ V, then the k-tuple (d(v, β1), d(v, β2), …, d(v, βk)) represents vertex v in terms of β, and is represented by r(v|β). If every vertex has a different presentation, the k-partition β is a resolving partition. The partition dimension of G, indicated by pd(G), is the minimal k for which there is a resolving k-partition of V(G). The partition dimension of Toeplitz graphs formed by two and three generators is constant, as shown in the following paper. The resolving set allows obtaining a unique representation for computer structures. In particular, they are used in pharmaceutical research for discovering patterns common to a variety of drugs. The above definitions are based on the hypothesis of chemical graph theory and it is a customary depiction of chemical compounds in form of graph structures, where the node and edge represent the atom and bond types, respectively.

Structural properties of Toeplitz graphs

In this paper, we study structural properties of Toeplitz graphs. We characterize Kq-free Toeplitz graphs for an integer q≥3 and give equivalent conditions for a Toeplitz graph Gn〈t1,t2,…,tk〉 with t1<⋯<tk and n≥tk−1+tk being chordal and equivalent conditions for a Toeplitz graph Gn〈t1,t2〉 being perfect. Then we compute the edge clique cover number and the vertex clique cover number of a chordal Toeplitz graph. Finally, we characterize the degree sequence (d1,d2,…,dn) of a Toeplitz graph with n vertices and show that a Toeplitz graph is a regular graph if and only if it is a circulant graph.

Metric Dimension of Some Generalized Families of Toeplitz Graphs

The metric dimension of a graph G is the selection of the minimum possible number of vertices such that each vertex of the graph G is distinctively defined by its vector of distances to the set of selected vertices. It was proved that the problem of determining the metric dimension of a graph is NP-hard. In this paper, the metric dimension of Toeplitz graphs with two and three generators is discussed and the exact values are found. Also, two conjectures about the exact metric dimension of Toeplitz graphs are given.

Some Resolving Parameters in a Class of Cayley Graphs

Resolving parameters are a fundamental area of combinatorics with applications not only to many branches of combinatorics but also to other sciences. In this study, we construct a class of Toeplitz graphs and will be denoted by T2n(W) so that they are Cayley graphs. First, we review some of the features of this class of graphs. In fact, this class of graphs is vertex transitive, and by calculating the spectrum of the adjacency matrix related with them, we show that this class of graphs cannot be edge transitive. Moreover, we show that this class of graphs cannot be distance regular, and because of the difficulty of the computing resolving parameters of a class of graphs which are not distance regular, we regard this as justification for our focus on some resolving parameters. In particular, we determine the minimal resolving set, doubly resolving set, and strong metric dimension for this class of graphs.

Encoding Labelled p-Riordan Graphs by Words and Pattern-Avoiding Permutations

The notion of a p-Riordan graph generalizes that of a Riordan graph, which, in turn, generalizes the notions of a Pascal graph and a Toeplitz graph. In this paper we introduce the notion of a p-Riordan word, and show how to encode p-Riordan graphs by p-Riordan words. For special important cases of Riordan graphs (the case $$p=2$$ ) and oriented Riordan graphs (the case $$p=3$$ ) we provide alternative encodings in terms of pattern-avoiding permutations and certain balanced words, respectively. As a bi-product of our studies, we provide an alternative proof of a known enumerative result on closed walks in the cube.

Asymptotic Spectra of Large (Grid) Graphs with a Uniform Local Structure (Part I): Theory

We are mainly concerned with sequences of graphs having a grid geometry,with a uniform local structure in a bounded domain Ω⊂Rd,d≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\Omega} {\\subset} \\mathbb{R}^{d}, d \\geq 1$$\\end{document}. When Ω=[0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega = [0, 1]$$\\end{document}, such graphs include the standard Toeplitz graphs and, for Ω=[0,1]d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega = [0, 1]^{d}$$\\end{document},the considered class includes d-level Toeplitz graphs. In the general case, the underlyingsequence of adjacency matrices has a canonical eigenvalue distribution, inthe Weyl sense, and we show that we can associate to it a symbol f\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak{f}$$\\end{document}. The knowledgeof the symbol and of its basic analytical features provides many information onthe eigenvalue structure, of localization, spectral gap, clustering, and distributiontype.Few generalizations are also considered in connection with the notion of generalizedlocally Toeplitz sequences and applications are discussed, stemming e.g.from the approximation of differential operators via numerical schemes. Nevertheless,more applications can be taken into account, since the results presentedhere can be applied as well to study the spectral properties of adjacency matricesand Laplacian operators of general large graphs and networks.

Counting independent sets in Riordan graphs

The notion of a Riordan graph was introduced recently, and it is a far-reaching generalization of the well-known Pascal graphs and Toeplitz graphs. However, apart from a certain subclass of Toeplitz graphs, nothing was known on independent sets in Riordan graphs.In this paper, we give exact enumeration and lower and upper bounds for the number of independent sets for various classes of Riordan graphs. Remarkably, we offer a variety of methods to solve the problems that range from the structural decomposition theorem to methods in combinatorics on words. Some of our results are valid for any graph.

Exponents of primitive directed Toeplitz graphs

ABSTRACT Given disjoint non-empty subsets S and T of , a digraph D with the vertex set is called a directed Toeplitz graph provided the arc occurs if and only if or . We investigate strong connectivity and primitivity of directed Toeplitz graphs. We prove that any primitive directed Toeplitz graph with vertices has exponent at least 3 and for each , there is a primitive directed Toeplitz graph of order n which has exponent 3. By Wielandt's result, we know that for a primitive digraph D of order n. We characterize the primitive directed Toeplitz graph, for which the upper bound it attained.

The Property of Hamiltonian Connectedness in Toeplitz Graphs

A spanning path in a graph G is called a Hamiltonian path. To determine which graphs possess such paths is an NP-complete problem. A graph G is called Hamiltonian-connected if any two vertices of G are connected by a Hamiltonian path. We consider here the family of Toeplitz graphs. About them, it is known only for n=3 that Tnp,q is Hamiltonian-connected, while some particular cases of Tnp,q,r for p=1 and q=2,3,4 have also been investigated regarding Hamiltonian connectedness. Here, we prove that the nonbipartite Toeplitz graph Tn1,q,r is Hamiltonian-connected for all 1&lt;q&lt;r&lt;n and n≥5r−2.

Word-representability of Toeplitz graphs

Distinct letters x and y alternate in a word w if after deleting in w all letters but the copies of x and y we either obtain a word of the form xyxy⋯ (of even or odd length) or a word of the form yxyx⋯ (of even or odd length). A graph G=(V,E) is word-representable if there exists a word w over the alphabet V such that letters x and y alternate in w if and only if xy is an edge in E.In this paper we initiate the study of word-representable Toeplitz graphs, which are Riordan graphs of the Appell type. We prove that several general classes of Toeplitz graphs are word-representable, and we also provide a way to construct non-word-representable Toeplitz graphs. Our work not only merges the theories of Riordan matrices and word-representable graphs via the notion of a Riordan graph, but also it provides the first systematic study of word-representability of graphs defined via patterns in adjacency matrices. Moreover, our paper introduces the notion of an infinite word-representable Riordan graph and gives several general examples of such graphs. It is the first time in the literature when the word-representability of infinite graphs is discussed.

Riordan graphs I: Structural properties

In this paper, we use the theory of Riordan matrices to introduce the notion of a Riordan graph. The Riordan graphs are a far-reaching generalization of the well known and well studied Pascal graphs and Toeplitz graphs, and also some other families of graphs. The Riordan graphs are proved to have a number of interesting (fractal) properties, which can be useful in creating computer networks with certain desirable features, or in obtaining useful information when designing algorithms to compute values of graph invariants. The main focus in this paper is the study of structural properties of families of Riordan graphs obtained from infinite Riordan graphs, which includes a fundamental decomposition theorem and certain conditions on Riordan graphs to have an Eulerian trail/cycle or a Hamiltonian cycle. We will study spectral properties of the Riordan graphs in a follow up paper.

Riordan graphs II: Spectral properties

The authors of this paper have used the theory of Riordan matrices to introduce the notion of a Riordan graph in [3]. Riordan graphs are proved to have a number of interesting (fractal) properties, and they are a far-reaching generalization of the well known and well studied Pascal graphs and Toeplitz graphs, and also some other families of graphs. The main focus in [3] is the study of structural properties of families of Riordan graphs obtained from certain infinite Riordan graphs.In this paper, we use a number of results in [3] to study spectral properties of Riordan graphs. Our studies include, but are not limited to the spectral graph invariants for Riordan graphs such as the adjacency eigenvalues, (signless) Laplacian eigenvalues, nullity, positive and negative inertia indices, and rank. We also study determinants of Riordan graphs, in particular, giving results about determinants of Catalan graphs.

Computing Metric Dimension of Certain Families of Toeplitz Graphs

The position of a moving point in a connected graph can be identified by computing the distance from the point to a set of sonar stations which have been appropriately situated in the graph. Let Q = {q 1 , q 2 , ... , q k } be an ordered set of vertices of a graph G and a is any vertex in G, then the code/representation of a w.r.t Q is the k-tuple (r(a, q 1 ), r(a, q 2 ), ... , r(a, q k )), denoted by r(a|Q). If the different vertices of G have the different representations w.r.t Q, then Q is known as a resolving set/locating set. A resolving/locating set having the least number of vertices is the basis for G and the number of vertices in the basis is called metric dimension of G and it is represented as dim(G). In this paper, the metric dimension of Toeplitz graphs generated by two and three parameters denoted by T n 〈1, t〉 and T n 〈1, 2, t〉, respectively is discussed and proved that it is constant.