Caterpillar Trees Research Articles

On the Randić energy of caterpillar graphs

On the Randić energy of caterpillar graphs

On the quartet distance given partial information

AbstractLet be an arbitrary phylogenetic tree with leaves. It is well known that the average quartet distance between two assignments of taxa to the leaves of is . However, a longstanding conjecture of Bandelt and Dress asserts that is also themaximumquartet distance between two assignments. While Alon, Naves, and Sudakov have shown this indeed holds for caterpillar trees, the general case of the conjecture is still unresolved. A natural extension is when partial information is given: the two assignments are known to coincide on a given subset of taxa. The partial information setting is biologically relevant as the location of some taxa (species) in the phylogenetic tree may be known, and for other taxa it might not be known. What can we then say about the average and maximum quartet distance in this more general setting? Surprisingly, even determining theaveragequartet distance becomes a nontrivial task in the partial information setting and determining the maximum quartet distance is even more challenging, as these turn out to be dependent on the structure of . In this paper we prove nontrivial asymptotic bounds that are sometimes tight for the average quartet distance in the partial information setting. We also show that the Bandelt and Dress conjecture does not generally hold under the partial information setting. Specifically, we prove that there are cases where the average and maximum quartet distance substantially differ.

On Fault-Tolerant Partition Dimension of Homogeneous Caterpillar Graphs

Metric-related parameters in graph theory have several applications in robotics, navigation, and chemical strata. An important such parameter is the partition dimension of graphs that plays an important role in engineering, computer science, and chemistry. In the context of chemical and pharmaceutical engineering, these parameters are used for unique representation of chemical compounds and their structural analysis. The structure of benzenoid hydrocarbon molecules is represented in the form of caterpillar trees and studied for various attributes including UV absorption spectrum, molecular susceptibility, anisotropy, and heat of atomization. Several classes of trees have been studied for partition dimension; however, in this regard, the advanced variant, the fault-tolerant partition dimension, remains to be explored. In this paper, we computed fault-tolerant partition dimension for homogeneous caterpillars C p ; 1 , C p ; 2 , and C p ; 3 for p ≥ 5 , p ≥ 3 , and p ≥ 4 , respectively, and it is found to be constant. Further numerical examples and an application are furnished to elaborate the accuracy and significance of the work.

Asymmetric circular graph with Hosoya index and negative continued fractions

It has been known that the Hosoya index of caterpillar graph can be calculated as the numerator of the simple continued fraction. Recently in [MATCH Commun. Math. Comput. Chem. 2020, 84 (2), 399-428], the author introduces a more general graph called caterpillar-bond graph and shows that its Hosoya index can be calculated as the numerator of the general continued fraction.
 In this paper, we show how the Hosoya index of the graph with non-uniform ring structure can be calculated from the negative continued fraction. We also give the relation between some radial graphs and multidimensional continued fractions in the sense of the Hosoya index.

Predictive Sampling Method for Spread Models in Networks

This paper proposes a novel, exploration-based network sampling algorithm called caterpillar quota walk sampling (CQWS) inspired by the caterpillar tree. Network sampling identifies a subset of nodes and edges from a network, creating an induced graph. Beginning from an initial node, exploration-based sampling algorithms grow the induced set by traversing and tracking unvisited neighboring nodes from the original network. Tunable and trainable parameters allow CQWS to maximize the sum of the degrees of the induced graph from multiple trials when sampling dense networks. A network spread model renders effective use in various applications, including tracking the spread of epidemics, visualizing information transmissions through social media, and cell-to-cell spread of neurodegenerative diseases. CQWS generates a spread model as its sample by visiting the highest-degree neighbors of previously visited nodes. For each previously visited node, a top proportion of the highest-degree neighbors fulfills a quota and branches into a new caterpillar tree. Sampling more high-degree nodes constitutes an objective among various applications. Many exploration-based sampling algorithms suffer drawbacks that limit the sum of degrees of visited nodes and thus the number of high-degree nodes visited. Furthermore, a strategy may not be adaptable to volatile degree frequencies throughout the original network architecture, which influences how deep into the original network an algorithm could sample. This paper analyzes CQWS in comparison to four other exploration-based network in tackling these two problems by sampling sparse and dense randomly generated networks.

Several Topological Indices of Random Caterpillars

In chemical graph theory, caterpillar trees have been an appealing model to represent the molecular structures of benzenoid hydrocarbon. Meanwhile, topological index has been thought of as a powerful tool for modeling quantitative structure-property relationship and quantitative structure-activity between molecules in chemical compounds. In this article, we consider a class of caterpillar trees that are incorporated with randomness, called random caterpillars, and investigate several popular topological indices of this random class, including Zagreb index, Randić index and Wiener index, etc. Especially, a central limit theorem is developed for the asymptotic distribution of the Zagreb index of random caterpillars.

Bilangan Kromatik Modular Pada Beberapa Subkelas Graf

Let G be a graph, modular k-coloring, k > 2 on graph G without isolated vertex is a vertex coloring on graph G with elements in the set of integers modulo k, Zk satisfying the properties for every two neighboring vertex in G, the number of colors (v) from their different neigbors in Zk. The modular chromatic number mc(G) in G is the minimum k integer where there is modular k-coloring on graph G. In this article describes modular chromatic numbers on Star Graph (Sn), Caterpillar Graph , Fan Graph (Fn), Helm Graph (Hn) and Triangular Book Graph (Btn).
 Keywords: Modular coloring, Modular chromatic number

The total irregularity strength of caterpillars with odd number of internal vertices of degree three

Given a graph G consisting of vertex set V and edget set E, repectively. Assume G is simple, connected, and the edges do not have direction. A function λ that maps V ∪ E into a set of κ-integers is named a totally irregular total k-labelling if no vertices have the same weight and also the edges of G get distinct weights. We call the minimum number k for which G has totally irregular total k-labelling as total irregularity strength of G, ts(G). In this article, we construct labels of vertices and edges of caterpillar graphs which have q internal vertices of degree 3 where q is 5,7, and 9. We obtain the exact values of ts in the following: n + 2 if the caterpillars have q=5 internal vertices, n + 3 for q=7, and n + 4 for q=9.

Graceful Labelings of Caterpillar Trees: A Proof Without Words

Graceful Labelings of Caterpillar Trees: A Proof Without Words

Super total graceful labeling of some trees

A graph labeling is an assignment of integers to the edges, vertices, or both of a graph so that it meets to certain conditions. A graph labeling is called total labeling if labeling is given to edges and vertices. Let G be a simple and finite graph having vertex set V(G), edge set E(G), number of vertices p, and number of edges q. A super total graceful labeling of G is a bijection f from V(G) ∪ E(G) to the set {1,2,3,….,p + q} such that f(uv) = |f(u) – f(v)| for every uv ∈ E(G) and f(E(G)) = {1,2,3,…, q}. A graph that admits a super total graceful labeling is called a super total graceful graph. In this paper we show that star graph K 1,n , spider graph SP(1 n , 2 m ), and caterpillar graph P 3⨀nK 1 are super total graceful graphs.

The complement metric dimension of particular tree

Let G be a connected graph with vertex set V(G) and edge set E(G). The distance between vertices u and v in G is denoted by d(u,v), which serves as the shortest path length from u to v. Let W = {whw2,…,wk} ⊆ V(G) be an ordered set, and v is a vertex in G. The representation of v with respect to W is an ordered set k-tuple, r(v|W) = (d(v,w1),d(v,w2),…,d(wk)). The set Wis called a complement resolving set for G if there are two vertices u,v⊆V(G)\\W, such that r(u|W)=r(v|W). A complement basis of G is the complement resolving set containing maximum cardinality. The number of vertices in a complement basis of G is called complement metric dimension of G, which is denoted by (G). In this paper, we examined complement metric dimension of particular tree graphs such as caterpillar graph (Cmn), firecrackers graph (Fmn), and banana tree graph (Bm,n). We got = m(n+1)-2 for m>1 and n>2, = m(n+2)-2 for m>1 and n>2, and = m(n+1)-1 if m>1 and n>2.

Bounds of the sum of edge lengths in linear arrangements of trees

A fundamental problem in network science is the normalization of the topological or physical distance between vertices, which requires understanding the range of variation of the unnormalized distances. Here we investigate the limits of the variation of the physical distance in linear arrangements of the vertices of trees. In particular, we investigate various problems of the sum of edge lengths in trees of a fixed size: the minimum and the maximum value of the sum for specific trees, the minimum and the maximum in classes of trees (bistar trees and caterpillar trees) and finally the minimum and the maximum for any tree. We establish some foundations for research on optimality scores for spatial networks in one dimension.

Complexity issues of perfect secure domination in graphs

Let G = (V, E) be a simple, undirected and connected graph. A dominating set S is called a secure dominating set if for each u ∈ V \ S, there exists v ∈ S such that (u, v) ∈ E and (S \{v}) ∪{u} is a dominating set of G. If further the vertex v ∈ S is unique, then S is called a perfect secure dominating set (PSDS). The perfect secure domination number γps(G) is the minimum cardinality of a perfect secure dominating set of G. Given a graph G and a positive integer k, the perfect secure domination (PSDOM) problem is to check whether G has a PSDS of size at most k. In this paper, we prove that PSDOM problem is NP-complete for split graphs, star convex bipartite graphs, comb convex bipartite graphs, planar graphs and dually chordal graphs. We propose a linear time algorithm to solve the PSDOM problem in caterpillar trees and also show that this problem is linear time solvable for bounded tree-width graphs and threshold graphs, a subclass of split graphs. Finally, we show that the domination and perfect secure domination problems are not equivalent in computational complexity aspects.

Values and bounds for depth and Stanley depth of some classes of edge ideals

<abstract> In this paper we study depth and Stanley depth of the quotient rings of the edge ideals associated with the corona product of some classes of graphs with arbitrary non-trivial connected graph $ G $. These classes include caterpillar, firecracker and some newly defined unicyclic graphs. We compute formulae for the values of depth that depend on the depth of the quotient ring of the edge ideal $ I(G) $. We also compute values of depth and Stanley depth of the quotient rings associated with some classes of edge ideals of caterpillar graphs and prove that both of these invariants are equal for these classes of graphs. </abstract>

Flow number of signed Halin graphs

The flow number of a signed graph (G,Σ) is the smallest positive integer k such that (G,Σ) admits a nowhere-zero integer k-flow. In 1983, Bouchet (JCTB) conjectured that every flow-admissible signed graph has flow number at most 6. This conjecture remains open for general signed graphs even for signed planar graphs. A Halin graph is a plane graph consisting of a tree without vertices of degree two and a circuit connecting all leaves of the tree. In this paper, we prove that every flow-admissible signed Halin graph has flow number at most 5, and determine the flow numbers of signed Halin graphs with a (3,1)-caterpillar tree as its characteristic tree.

Porous exponential domination number of some graphs

AbstractLet G be a graph and S ⊆ V(G). If for all v ∈ V(G), then S is a porous exponential dominating set for G, where d(u, v) is the distance between vertices u and v. The smallest cardinality of a porous exponential dominating set is the porous exponential domination number of G and is denoted by . In this article, we examine porous exponential domination number of some shadow graphs and trees such as comet, double comet, double star, binomial tree, and generalized caterpillar graphs.

Totally irregular total labeling of some caterpillar graphs

Assume that G ( V,E ) is a graph with V and E as its vertex and edge sets, respectively. We have G is simple, connected, and undirected. Given a function λ from a union of V and E into a set of k -integers from 1 until k . We call the function λ as a totally irregular total k -labeling if the set of weights of vertices and edges consists of different numbers. For any u ∈ V , we have a weight wt ( u )=λ( u )+ ∑ {uy ∈ E} λ( uy ). Also, it is defined a weight wt ( e )= λ( u )+ λ( uv ) + λ( v ) for each e = uv ∈ E . A minimum k used in k -total labeling λ is named as a total irregularity strength of G , symbolized by ts( G ). We discuss results on ts of some caterpillar graphs in this paper. The results are ts (S {p,2,2,q} ) = ⌈ (p+q-1)/2 ⌉ for p , q greater than or equal to 3, while ts(S {p,2,2,2,p} ) = ⌈(2p-1)/2 ⌉, p ≥ 4.

On Kemeny's constant for trees with fixed order and diameter

ABSTRACT Kemeny's constant of a connected graph G is a measure of the expected transit time for the random walk associated with G. In the current work, we consider the case when G is a tree and, in this setting, we provide lower and upper bounds for in terms of the order n and diameter δ of G by using two different techniques. The lower bound is given as Kemeny's constant of a particular caterpillar tree and, as a consequence, it is sharp. The upper bound is found via induction, by repeatedly removing pendent vertices from G. By considering a specific family of trees – the broom-stars – we show that the upper bound is asymptotically sharp.

Extremal hitting times of trees with some given parameters

Given a connected graph with the hitting time is defined as the expected number of steps that a simple random walk takes to go from x to y. A hitting-time-based invariant, called the ZZ index, was first proposed by Zhu and Zhang [The hitting time of random walk on unicyclic graphs. Linear Multilinear Algebra. doi:10.1080/03081087.2019.1611732] recently. It was defined to be In this paper, some extremal problems on the ZZ index of trees with some given parameters are considered. Firstly, sharp upper and lower bounds on are determined, respectively, for trees with given diameter, matching number, given vertex bipartition and given pendant numbers. Secondly, ordering the n-vertex caterpillar trees with respect to their ZZ indices are established.

Bounded degree complexes of forests

Given an arbitrary sequence of non-negative integers λ→=(λ1,…,λn) and a graph G with vertex set {v1,…,vn}, the bounded degree complex, denoted as BDλ→(G), is a simplicial complex whose faces are the subsets H⊆E(G) such that for each i∈{1,…,n}, the degree of vertex vi in the induced subgraph G[H] is at most λi. When λi=k for all i, the bounded degree complex BDλ→(G) is called the k-matching complex, denoted as Mk(G).In this article, we determine the homotopy type of bounded degree complexes of forests. In particular, we show that, for all k≥1, the k-matching complexes of caterpillar graphs are either contractible or homotopy equivalent to a wedge of spheres, thereby proving a conjecture of Vega (2019, Conjecture 7.3). We also give a closed form formula for the homotopy type of the bounded degree complexes of those caterpillar graphs in which every non-leaf vertex is adjacent to at least one leaf vertex.