Non-isomorphic Trees Research Articles

Deletion-contraction for a unified Laplacian and applications

We define a graph Laplacian with vertex weights in addition to the more classical edge weights, which unifies the combinatorial Laplacian and the normalised Laplacian. Moreover, we give a combinatorial interpretation for the coefficients of the weighted Laplacian characteristic polynomial in terms of weighted spanning forests and use this to prove a deletion-contraction relation. We prove various interlacing theorems relating to deletion and contraction, as well as to rectangular tilings, drawing on the work of Brooks, Smith, Stone and Tutte on square tilings. Additionally, we show that the weighted Laplacian also satisfies a vertex analogue of deletion-contraction. We give applications of weighted Laplacian eigenvalues to sparse cuts, independent sets and graph colouring, and establish new cases of a problem of Stanley on distinguishing nonisomorphic trees.

A rooted variant of Stanley's chromatic symmetric function

Richard Stanley defined the chromatic symmetric function XG of a graph G and asked whether there are non-isomorphic trees T and U with XT=XU. We study variants of the chromatic symmetric function for rooted graphs, where we require the root vertex to either use or avoid a specified color. We present combinatorial identities and recursions satisfied by these rooted chromatic polynomials, explain their relation to pointed chromatic functions and rooted U-polynomials, and prove three main theorems. First, for all non-empty connected graphs G, Stanley's polynomial XG(x1,…,xN) is irreducible in Q[x1,…,xN] for all large enough N. The same result holds for our rooted variant where the root node must avoid a specified color. We prove irreducibility by a novel combinatorial application of Eisenstein's Criterion. Second, we prove the rooted version of Stanley's Conjecture: two rooted trees are isomorphic as rooted graphs if and only if their rooted chromatic polynomials are equal. In fact, we prove that a one-variable specialization of the rooted chromatic polynomial (obtained by setting x0=x1=q, x2=x3=1, and xn=0 for n>3) already distinguishes rooted trees. Third, we answer a question of Pawlowski by providing a combinatorial interpretation of the monomial expansion of pointed chromatic functions.

Ranking trees based on global centrality measures

Trees, or connected graphs with no cycles, are a commonly studied combinatorial family. When many natural metrics on networks are applied to the set of all trees of a fixed order, the extremal values are realized at the star and path graphs. In this paper, we prove inequalities for several global centrality measures, such as global closeness and betweenness centralities, and for graphical Stirling numbers of the first kind that interpolate these extremes. Moreover, we provide two algorithms that allow us to traverse the space of non-isomorphic trees of a fixed order, one towards the star graph of the same order and the other towards the path. Furthermore, we investigate the relationship between these global centrality measures on the one hand and the (n−2)nd Stirling numbers of the first kind for small trees on the other hand, demonstrating a strong association between them, in particular with respect to the hierarchical structures obtained from applying our two interpolating algorithms. Based on our observations from these small trees, we prove general bounds that relate the (n−2)nd Stirling numbers of the first kind of trees of order n to these global centrality measures. Finally, we provide two related approaches to totally order the set of all non-isomorphic trees of fixed order. We show that the totally ordering obtained from one of these approaches is consistent with the hierarchical structure obtained from our two tree interpolation algorithms in addition to being one of the features to use for predicting the (n−2)nd Stirling numbers of the first kind for small trees.

Виды предгеометрий ациклических теорий

The article is devoted to the description of types of pregeometries with an algebraic closure operator for acyclic theories. In these theories we describe conditions of violation of the exchange property for a pregeometry. Taking into account these conditions, we introduce new concepts that do not rely on the exchange property: 𝑎- pregeometry, 𝑎-modularity, etc. The dependence conditions for an 𝑎-modular and 𝑎- locally finite 𝑎-pregeometry on the number of non-isomorphic trees and special points are established. Sufficient conditions of dependence for a 𝑎-local finite 𝑎-pregeometry on the vertices of the 𝑎-type are established, too.

A Relation on Trees and the Topological Indices Based on Subgraph

A topological index reflects the physical, chemical and structural properties of a molecule, and its study has an important role in molecular topology, chemical graph theory and mathematical chemistry. It is a natural problem to characterize non-isomorphic graphs with the same topological index value. By introducing a relation on trees with respect to edge division vectors, denoted by $\langle\mathcal{T}_n, \preceq \rangle$, in this paper we give some results for the relation order in $\langle\mathcal{T}_n, \preceq \rangle$, it allows us to compare the size of the topological index value without relying on the specific forms of them, and naturally we can determine which trees have the same topological index value. Based on these results we characterize some classes of trees that are uniquely determined by their edge division vectors and construct infinite classes of non-isomorphic trees with the same topological index value, particularly such trees of order no more than $10$ are completely determined.

A Modification of the Frechet Distance for Nonisomorphic Trees

The paper presents a modification of the Frechet distance for nonisomorphic trees. While the classical Frechet distance between nonisomorphic trees is undefined, a new measure called similarity of a tree to a reference tree is given that is defined for a wider class of trees. A polynomial-time algorithm is given to determine whether one tree’s similarity to another is less than a given number.

On the smallest trees with the same restricted [formula omitted]-polynomial and the rooted [formula omitted]-polynomial

In this article, we construct explicit examples of pairs of non-isomorphic trees with the same restricted U-polynomial for every k; by this we mean that the polynomials agree on terms with degree at most k+1. The main tool for this construction is a generalization of the U-polynomial to rooted graphs, which we introduce and study in this article. Most notably we show that rooted trees can be reconstructed from its rooted U-polynomial.

An Efficient Algorithm to Count Tree-Like Graphs with a Given Number of Vertices and Self-Loops.

Graph enumeration with given constraints is an interesting problem considered to be one of the fundamental problems in graph theory, with many applications in natural sciences and engineering such as bio-informatics and computational chemistry. For any two integers and , we propose a method to count all non-isomorphic trees with n vertices, self-loops, and no multi-edges based on dynamic programming. To achieve this goal, we count the number of non-isomorphic rooted trees with n vertices, self-loops and no multi-edges, in time and space, since every tree can be uniquely viewed as a rooted tree by either regarding its unicentroid as the root, or in the case of bicentroid, by introducing a virtual vertex on the bicentroid and assuming the virtual vertex to be the root. By this result, we get a lower bound and an upper bound on the number of tree-like polymer topologies of chemical compounds with any “cycle rank”.

A few more trees the chromatic symmetric function can distinguish

A well-known open problem in graph theory asks whether Stanley's chromatic symmetric function, a generalization of the chromatic polynomial of a graph, distinguishes between any two non-isomorphic trees. Previous work has proven the conjecture for a class of trees called spiders. This paper generalizes the class of spiders to $n$-spiders, where normal spiders correspond to $n = 1$, and verifies the conjecture for $n = 2$.

The smallest positive eigenvalue of graphs under perturbation

The effect on the smallest positive eigenvalue of a bipartite graph is studied when the graph is perturbed by attaching a pendant vertex at one of its vertices. Let $${\widehat{T}}(v)$$ be the graph obtained by attaching a pendant at vertex v of T. We characterize the vertices v such that the smallest positive eigenvalue of $${\widehat{T}}(v)$$ is equal or greater than that of T. As an application, we obtain the pairs of nonisomorphic noncospectral trees having the same smallest positive eigenvalue.

On graphs with the same restricted U-polynomial and the U-polynomial for rooted graphs

In this abstract, we construct explicitly, for every k, pairs of non-isomorphic trees with the same restricted U-polynomial; by this we mean that the polynomials agree on terms with degree at most k. The construction is done purely in algebraic terms, after introducing and studying a generalization of the U-polynomial to rooted graphs.

On the Shapley Value of Unrooted Phylogenetic Trees.

The Shapley value, a solution concept from cooperative game theory, has recently been considered for both unrooted and rooted phylogenetic trees. Here, we focus on the Shapley value of unrooted trees and first revisit the so-called split counts of a phylogenetic tree and the Shapley transformation matrix that allows for the calculation of the Shapley value from the edge lengths of a tree. We show that non-isomorphic trees may have permutation-equivalent Shapley transformation matrices and permutation-equivalent null spaces. This implies that estimating the split counts associated with a tree or the Shapley values of its leaves does not suffice to reconstruct the correct tree topology. We then turn to the use of the Shapley value as a prioritization criterion in biodiversity conservation and compare it to a greedy solution concept. Here, we show that for certain phylogenetic trees, the Shapley value may fail as a prioritization criterion, meaning that the diversity spanned by the top k species (ranked by their Shapley values) cannot approximate the total diversity of all n species.

Invariant subsets of scattered trees and the tree alternative property of Bonato and Tardif

A tree is scattered if it does not contain a subdivision of the complete binary tree as a subtree. We show that every scattered tree contains a vertex, an edge, or a set of at most two ends preserved by every embedding of T. This extends results of Halin, Polat and Sabidussi. Calling two trees equimorphic if each embeds in the other, we then prove that either every tree that is equimorphic to a scattered tree T is isomorphic to T, or there are infinitely many pairwise non-isomorphic trees which are equimorphic to T. This proves the tree alternative conjecture of Bonato and Tardif for scattered trees, and a conjecture of Tyomkyn for locally finite scattered trees.

On trees with the same restricted [formula omitted]-polynomial and the Prouhet–Tarry–Escott problem

This paper focuses on the well-known problem due to Stanley of whether two non-isomorphic trees can have the same U-polynomial (or, equivalently, the same chromatic symmetric function). We consider the Uk-polynomial, which is a restricted version of U-polynomial, and construct, for any given k, non-isomorphic trees with the same Uk-polynomial. These trees are constructed by encoding solutions of the Prouhet–Tarry–Escott problem. As a consequence, we find a new class of trees that are distinguished by the U-polynomial up to isomorphism.

Extractable Common Randomness From Gaussian Trees: Topological and Algebraic Perspectives

In this paper, we study both topological and algebraic properties of unrooted Gaussian trees in order to characterize their security performance. Such performance is measured by the corresponding potential in extracting common randomness from a given tree, which is further determined by max–min and min–max conditional mutual information (CMI) values, subject to the order of selecting variables from the tree by legitimate nodes Alice and Bob, and an eavesdropper Eve, respectively. A new operation is proposed to transform a Gaussian tree into another, and also to order different Gaussian trees. Through such operation we construct several equivalent classes of Gaussian trees. Each class includes multiple Gaussian trees that can be partially ordered based on the associated max–min or min–max CMI metric, and thus, we can find the most secure and the least secure trees in each partially ordered set (poset). The union of all posets generates all possible non-isomorphic trees of the given number of variables. Then, we assign a particular polynomial to each Gaussian tree, and show that such polynomial can determine the relative security performance of the Gaussian tree with respect to other trees within the same class. In the end, based on a generalized integer partition method, we propose a novel approach to efficiently enumerate the most secure structures of all posets.

Indistinguishable Trees and Graphs

We show that a number of graph invariants are, even combined, insufficient to distinguish between non-isomorphic trees or general graphs. Among these are: the spectrum of eigenvalues (equivalently, the characteristic polynomial), the number of independent sets of all sizes or the number of connected subgraphs of all sizes. We therefore extend the classical theorem of Schwenk that almost every tree has a cospectral mate, and we provide an answer to a question of Jamison on average subtree orders of trees. The simple construction that we apply for this purpose is based on finding graphs with two distinguished vertices (called pseudosimilar) that do not belong to the same orbit but whose removal yields isomorphic graphs.

Some remarks on inverse Wiener index problem

In 1995, Gutman and Yeh (1995) [3] conjectured that for every large enough integer w there exists a tree with Wiener index equal to w. The conjecture has been solved by Wang and Yu (2006) [7] and independently by Wagner (2006) [6]. We present an algorithm with a constant number of operations to construct a tree with a given Wiener index. Moreover, we show that there exist 2Ω(w4) non-isomorphic trees with Wiener index w.

Optimal solutions for the balanced minimum evolution problem

Phylogenies are trees representing the evolutionary relationships of a set of species (called taxa). Phylogenies find application in several scientific areas ranging from medical research to drug discovery, epidemiology, systematics and population dynamics. In these applications the available information is usually restricted to the leaves of a phylogeny and is represented by molecular data extracted from the species analyzed. On the contrary, the information about the phylogeny itself is generally missing and must be determined by solving an optimization problem, called the phylogeny estimation problem (PEP), whose versions depend on the criterion used to select a phylogeny among plausible alternatives.In this paper, we investigate one of the most significant versions of the PEP, called the balanced minimum evolution problem. We propose an exact algorithm based on the enumeration of non-isomorphic trees and the subsequent solution of quadratic assignment problems. Furthermore, by exploiting the underlying parallelism of the algorithm, we present a parallel version of the algorithm which shows a linear speed-up with respect to the sequential version. Extensive computational results prove the effectiveness of the proposed algorithms.

Spiders are status unique in trees

The status of a vertex x in a graph is the sum of the distances between x and all other vertices. The status sequence of a graph is the list of the statuses of all vertices arranged in nondecreasing order. It is well known that non-isomorphic trees may have the same status sequence. A graph G is said to be status unique in a family F of graphs if G is a member of F and G is uniquely determined in F by its status sequence. The main result of this paper is that every spider is status unique in the family of all trees. Some conjectures about status unique graphs are proposed at the end of the paper.

Changing the heights of automorphism towers by forcing with Souslin trees over L

AbstractWe prove that there are groups in the constructible universe whose automorphism towers are highly malleable by forcing. This is a consequence of the fact that, under a suitable diamond hypothesis, there are sufficiently many highly rigid non-isomorphic Souslin trees whose isomorphism relation can be precisely controlled by forcing.