For a graph $G$, and two distinct vertices $u$ and $v$ of $G$, let $n_G(u,v)$ be the number of vertices of $G$ that are closer in $G$ to $u$ than to $v$. Miklavič and Šparl (arXiv:2011.01635v1) define the distance-unbalancedness of $G$ as the sum of $|n_G(u,v)-n_G(v,u)|$ over all unordered pairs of distinct vertices $u$ and $v$ of $G$. Confirming one of their conjectures, we show that the stars minimize the distance-unbalancedness among all trees of a fixed order.
In this paper we propose and study a new structural invariant for graphs, called distance-unbalanced\-ness, as a measure of how much a graph is (un)balanced in terms of distances. Explicit formulas are presented for several classes of well-known graphs. Distance-unbalancedness of trees is also studied. A few conjectures are stated and some open problems are proposed.
For a graph $G$, and two distinct vertices $u$ and $v$ of $G$, let $n_G(u,v)$ be the number of vertices of $G$ that are closer in $G$ to $u$ than to $v$. Miklavič and Šparl (arXiv:2011.01635v1) define the distance-unbalancedness ${\rm uB}(G)$ of $G$ as the sum of $|n_G(u,v)-n_G(v,u)|$ over all unordered pairs of distinct vertices $u$ and $v$ of $G$. For positive integers $n$ up to $15$, they determine the trees $T$ of fixed order $n$ with the smallest and the largest values of ${\rm uB}(T)$, respectively. While the smallest value is achieved by the star $K_{1,n-1}$ for these $n$, which we then…
Weighted Szeged index is a recently introduced extension of the well-known Szeged index. In this paper, we present a new tool to analyze and characterize minimum weighted Szeged index trees. We exhibit the best trees with up to 81 vertices and use this information, together with our results, to propose various conjectures on the structure of minimum weighted Szeged index trees.
The nullity of a graph is the multiplicity of the eigenvalue zero in its adjacency spectrum. In this paper, we give a closed formula for the minimum and maximum nullity among trees with the same degree sequence, using the notion of matching number and annihilation number. Algorithms for constructing such minimum-nullity and maximum-nullity trees are described.
We give a necessary and sufficient condition for the maximum multiplicity of a root of the matching polynomial of a tree to be equal to the minimum number of vertex disjoint paths needed to cover it.
The main purpose of the paper is to develop an approach to evaluation or estimation of the spanning tree congestion of planar graphs. This approach is used to evaluate the spanning tree congestion of triangular grids.
Let $T$ be a rooted tree, and $V(T)$ its set of vertices. A subset $X$ of $V(T)$ is called an infima closed set of $T$ if for any two vertices $u,v\in X$, the first common ancestor of $u$ and $v$ is also in $X$. This paper determines the trees with minimum number of infima closed sets among all rooted trees of given order, thereby answering a question of Klazar. It is shown that these trees are essentially complete binary trees, with the exception of vertices at the last levels. Moreover, an asymptotic estimate for the minimum number of infima closed sets in a tree with $n$ vertices is also pr…
A vertex cover of a graph $G = (V, E)$ is a set $X \subseteq V$ such that each edge of $G$ is incident to at least one vertex of $X$. A dominating set $D \subseteq V$ is a total dominating set of $G$ if the subgraph induced by $D$ has no isolated vertices. A $(γ_t-τ)$-set of $G$ is a minimum vertex cover which is also a minimum total dominating set. In this article we give a constructive characterization of trees having a $(γ_t-τ)$-set.
The minimum stretch spaning tree problem for a grah G is to find a spaning tree T of G such as that the maximum distance in T between two adjacent vertices is minimized. The minimum value of this optimization problem gives rise to a grpah invariant σ T(G) called the tree stretch of G. The problem has been studied in the algorithmic aspects, such as NP-hardness and fixed-parameter solvability. This paper presents the exact values σ T(G) of hamming graphs Kn1 * Kn2 * ... * Knd and the higer-dimensional grids Pn1 * Pn2 * ... * Pnd.
The chromatic sum of a graph is the smallest sum of colors among all proper colorings with natural numbers. The strength is the minimum number of colors needed to achieve the chromatic sum. We construct for each positive integer k a tree with strength k that has maximum degree only 2k-2. The result is best possible.
The minimum rank of a graph is the minimum possible rank of a real matrix whose zero-nonzero pattern is described by the graph. The current algorithms can compute efficiently the minimum rank of undirected trees. This paper provides an algorithm to compute in polynomial time the minimum rank of directed trees allowing loops.
Substantial efforts have been made to compute or estimate the minimum number $c(G)$ of cycles needed to partition the edges of an Eulerian graph. We give an equivalent characterization of Eulerian graphs of treewidth $2$ and with maximum degree $4$. This characterization enables us to present a linear time algorithm for the computation of $c(G)$ for all $G$ in this class.
Let $G=$ $\left( V,E\right) $ be a simple graph.\ A non-empty set $S \subseteq V$ is called a global offensive alliance if $S$ is a dominating set and for every vertex $v$ in $V-S$, at least half of the vertices from the closed neighborhood of $v$ are in $S$. The global offensive alliance number is the minimum cardinality of a global offensive alliance in $G$. In this paper, we give a constructive characterization of trees having a unique minimum global offensive alliance.
We propose the following conjecture: For every fixed $α\in [0,\frac 13)$, each graph of minimum degree at least $(1+α)\frac k2$ and maximum degree at least $2(1-α)k$ contains each tree with $k$ edges as a subgraph. Our main result is an approximate version of the conjecture for bounded degree trees and large dense host graphs. We also show that our conjecture is asymptotically best possible. The proof of the approximate result relies on a second result, which we believe to be interesting on its own. Namely, we can embed any bounded degree tree into host graphs of minimum/maximum degree asympto…
We prove that, if $m$ is sufficiently large, every graph on $m+1$ vertices that has a universal vertex and minimum degree at least $\lfloor \frac{2m}{3} \rfloor$ contains each tree $T$ with $m$ edges as a subgraph. Our result confirms, for large $m$, an important special case of a conjecture by Havet, Reed, Stein, and Wood. The present paper builds on the results of a companion paper in which we proved the statement for all trees having a vertex that is adjacent to many leaves.
The recently developed atom-bond sum-connection (ABS) index is a variation of the well-studied graph-based molecular descriptors connectivity (Randic), atom-bond connectivity, and sum-connectivity indices.We present the minimum ABS index of trees with a given number of pendent vertices in this paper, which provides a solution to the problem proposed by A. Ali, I. Gutman and I. Redzepovic, Atom-Bond Sum-Connectivity Index of Unicyclic Graphs and Some Applications, Electron. J. Math.(2023) 1-7
For a tree with the given sequence of vertex degrees the spectral radius of its terminal distance matrix is shown to be bounded from below by the the average row sum of the terminal distance matrix of the, so called, BFS-tree (also known as a greedy tree). This lower bound is typically not tight; nevertheless, since spectral radius of the terminal distance matrix of BFS-tree is a natural upper bound, the numeric simulation shows that relative gap between the upper and the lower bound does not exceed 3% (we also make a step towards justifying this fact analytically.) Therefore, the conjecture t…
In this paper and a companion paper, we prove that, if $m$ is sufficiently large, every graph on $m+1$ vertices that has a universal vertex and minimum degree at least $\lfloor \frac{2m}{3} \rfloor$ contains each tree $T$ with $m$ edges as a subgraph. Our result confirms, for large $m$, an important special case of a recent conjecture by Havet, Reed, Stein, and Wood. The present paper already contains an approximate version of the result.
Trees with labelled leaves and with all other vertices of degree three play an important role in systematic biology and other areas of classification. A classical combinatorial result ensures that such trees can be uniquely reconstructed from the distances between the leaves (when the edges are given any strictly positive lengths). Moreover, a linear number of these pairwise distance values suffices to determine both the tree and its edge lengths. A natural set of pairs of leaves is provided by any `triplet cover' of the tree (based on the fact that each non-leaf vertex is the median vertex of…
The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices. Sam Spiro [The Wiener index of signed graphs, Appl. Math. Comput., 416(2022)126755] recently introduced the Wiener index for a signed graph and conjectured that the path $P_n$ with alternating signs has the minimum Wiener index among all signed trees with $n$ vertices. By constructing an infinite family of counterexamples, we prove that the conjecture is false whenever $n$ is at least 30.
We study the minimum spanning tree problem on the complete graph $K_n$ where an edge $e$ has a weight $W_e$ and a cost $C_e$, each of which is an independent copy of the random variable $U^γ$ where $γ\leq 1$ and $U$ is the uniform $[0,1]$ random variable. There is also a constraint that the spanning tree $T$ must satisfy $C(T)\leq c_0$. We establish, for a range of values for $c_0,γ$, the asymptotic value of the optimum weight via the consideration of a dual problem.
With applications in distribution systems and communication networks, the minimum stretch spanning tree problem is to find a spanning tree T of a graph G such that the maximum distance in T between two adjacent vertices is minimized. The problem has been proved to be NP-hard and fixed-parameter polynomial algorithms have been obtained for some special classes of graphs. In this paper, we concentrate on the optimality characterizations for typical classes of graphs. We determine the exact optimality representations for Petersen graph, the complete k-partite graphs, split graphs, generalized con…
Let $G =(V_{G}, E_{G})$ be a simple connected graph with its vertex set $V_{G}$ and edge set $E_{G}$. The Mostar index $Mo(G)$ was defined as $Mo(G)=\sum\limits_{e=uv\in E(G)}|n_{u}-n_{v}|$, where $n_{u}$ (resp., $n_{v}$) is the number of vertices whose distance to vertex $u$ (resp., $v$) is smaller than the distance to vertex $v$ (resp., $u$). In this study, we determine the first three minimum Mostar indices of tree-like phenylenes and characterize all the tree-like phenylenes attaining these values. At last, we give some numerical examples and discussion.
We study the expected value of the length $L_n$ of the minimum spanning tree of the complete graph $K_n$ when each edge $e$ is given an independent uniform $[0,1]$ edge weight. We sharpen the result of Frieze \cite{F1} that $\lim_{n\to\infty}\E(L_n)=\z(3)$ and show that $\E(L_n)=\z(3)+\frac{c_1}{n}+\frac{c_2+o(1)}{n^{4/3}}$ where $c_1,c_2$ are explicitly defined constants.
Random spanning trees of a graph $G$ are governed by a corresponding probability mass distribution (or "law"), $μ$, defined on the set of all spanning trees of $G$. This paper addresses the problem of choosing $μ$ in order to utilize the edges as "fairly" as possible. This turns out to be equivalent to minimizing, with respect to $μ$, the expected overlap of two independent random spanning trees sampled with law $μ$. In the process, we introduce the notion of homogeneous graphs. These are graphs for which it is possible to choose a random spanning tree so that all edges have equal usage probab…
A tree-based network $\mathcal N$ on $X$ is universal if every rooted binary phylogenetic $X$-tree is a base tree for $\mathcal N$. Hayamizu and, independently, Zhang constructively showed that, for all positive integers $n$, there exists an universal tree-based network on $n$ leaves. For all $n$, Hayamizu's construction contains $Θ(n!)$ reticulations, while Zhang's construction contains $Θ(n^2)$ reticulations. A simple counting argument shows that an universal tree-based network has $Ω(n\log n)$ reticulations. With this in mind, Hayamizu as well as Steel posed the problem of determining wheth…
For a connected graph $G$ on at least three vertices, the augmented Zagreb index (AZI) of $G$ is defined as $$AZI(G)=\sum_{uv\in E(G)}\left(\frac{d(u)d(v)}{d(u)+d(v)-2}\right)^{3},$$ being a topological index well-correlated with the formation heat of heptanes and octanes. A $k$-apex tree $G$ is a connected graph admitting a $k$-subset $X\subset V(G)$ such that $G-X$ is a tree, while $G-S$ is not a tree for any $S\subset V(G)$ of cardinality less than $k$. By investigating some structural properties of $k$-apex trees, we identify the graphs minimizing the AZI among all $k$-apex trees on $n$ ve…
We classify the trees on $n$ vertices with the maximum and the minimum number of certain generalized colorings, including conflict-free, odd, non-monochromatic, star, and star rainbow vertex colorings. We also extend a result of Cutler and Radcliffe on the maximum and minimum number of existence homomorphisms from a tree to a completely looped graph on $q$ vertices.
The \emph{spanning tree packing number} of a graph $G$ is the maximum number of edge-disjoint spanning trees contained in $G$. Let $k\geq 1$ be a fixed integer. Palmer and Spencer proved that in almost every random graph process, the hitting time for having $k$ edge-disjoint spanning trees equals the hitting time for having minimum degree $k$. In this paper, we prove that for any $p$ such that $(\log n+ω(1))/n\leq p\leq (1.1\log n)/n$, almost surely the random graph $G(n,p)$ satisfies that the spanning tree packing number is equal to the minimum degree. Note that this bound for $p$ will allow …
For a weighted outerplanar graph, the set of lex short cycles is known to be a minimum cycle basis [Inf. Process. Lett. 110 (2010) 970-974 ]. In this work, we show that the set of lex short cycles is a minimum cycle basis in weighted partial 2-trees (graphs of treewidth two) which is a superclass of outerplanar graphs.
Minimum Bisection denotes the NP-hard problem to partition the vertex set of a graph into two sets of equal sizes while minimizing the width of the bisection, which is defined as the number of edges between these two sets. We first consider this problem for trees and prove that the minimum bisection width of every tree $T$ on $n$ vertices satisfies $MinBis(T) \leq 8 n Δ(T) / diam(T)$. Second, we generalize this to arbitrary graphs with a given tree decomposition $(T,X)$ and give an upper bound on the minimum bisection width that depends on the structure of $(T,X)$. Moreover, we show that a bis…
Choi et. al (2011) introduced a minimum spanning tree (MST)-based method called CLGrouping, for constructing tree-structured probabilistic graphical models, a statistical framework that is commonly used for inferring phylogenetic trees. While CLGrouping works correctly if there is a unique MST, we observe an indeterminacy in the method in the case that there are multiple MSTs. In this work we remove this indeterminacy by introducing so-called vertex-ranked MSTs. We note that the effectiveness of CLGrouping is inversely related to the number of leaves in the MST. This motivates the problem of f…
We show that, in the graph spectrum of the normalized graph Laplacian on trees, the eigenvalue 1 and eigenvalues near 1 are strongly related to minimum vertex covers. In particular, for the eigenvalue 1, its multiplicity is related to the size of a minimum vertex cover, and zero entries of its eigenvectors correspond to vertices in minimum vertex covers; while for eigenvalues near 1, their distance to 1 can be estimated from minimum vertex covers; and for the largest eigenvalue smaller than 1, the sign graphs of its eigenvectors take vertices in a minimum vertex cover as representatives.
Minimum $k$-Section denotes the NP-hard problem to partition the vertex set of a graph into $k$ sets of sizes as equal as possible while minimizing the cut width, which is the number of edges between these sets. When $k$ is an input parameter and $n$ denotes the number of vertices, it is NP-hard to approximate the width of a minimum $k$-section within a factor of $n^c$ for any $c<1$, even when restricted to trees with constant diameter. Here, we show that every tree $T$ allows a $k$-section of width at most $(k-1) (2 + 16n / diam(T) ) Δ(T)$. This implies a polynomial-time constant-factor appro…
One of the goals of phylogenetic research is to find the species tree describing the evolutionary history of a set of species. But the trees derived from geneti data with the help of tree inference methods are gene trees that need not coincide with the species tree. This can for example happen when so-called deep coalescence events take place. It is also known that species trees can differ from their most likely gene trees. Therefore, as a means to find the species tree, it has been suggested to use subtrees of the gene trees, for example triples, and to puzzle them together in order to find t…
Assign i.i.d. standard exponential edge weights to the edges of the complete graph K_n, and let M_n be the resulting minimum spanning tree. We show that M_n converges in the local weak sense (also called Aldous-Steele or Benjamini-Schramm convergence), to a random infinite tree M. The tree M may be viewed as the component containing the root in the wired minimum spanning forest of the Poisson-weighted infinite tree (PWIT). We describe a Markov process construction of M starting from the invasion percolation cluster on the PWIT. We then show that M has cubic volume growth, up to lower order flu…
We attempt to shed new light on the notion of 'tree-like' metric spaces by focusing on an approach that does not use the four-point condition. Our key question is: Given metric space $M$ on $n$ points, when does a fully labelled positive-weighted tree $T$ exist on the same $n$ vertices that precisely realises $M$ using its shortest path metric? We prove that if a spanning tree representation, $T$, of $M$ exists, then it is isomorphic to the unique minimum spanning tree in the weighted complete graph associated with $M$, and we introduce a fourth-point condition that is necessary and sufficient…
Measures of tree balance play an important role in the analysis of phylogenetic trees. One of the oldest and most popular indices in this regard is the Colless index for rooted bifurcating trees, introduced by Colless (1982). While many of its statistical properties under different probabilistic models for phylogenetic trees have already been established, little is known about its minimum value and the trees that achieve it. In this manuscript, we fill this gap in the literature. To begin with, we derive both recursive and closed expressions for the minimum Colless index of a tree with $n$ lea…
We consider a family of local search algorithms for the minimum-weight spanning tree, indexed by a parameter $ρ$. One step of the local search corresponds to replacing a connected induced subgraph of the current candidate graph whose total weight is at most $ρ$ by the minimum spanning tree (MST) on the same vertex set. Fix a non-negative random variable $X$, and consider this local search problem on the complete graph $K_n$ with independent $X$-distributed edge weights. Under rather weak conditions on the distribution of $X$, we determine a threshold value $ρ^*$ such that the following holds. …
The geometric $δ$-minimum spanning tree problem ($δ$-MST) is the problem of finding a minimum spanning tree for a set of points in a normed vector space, such that no vertex in the tree has a degree which exceeds $δ$, and the sum of the lengths of the edges in the tree is minimum. The similarly defined geometric $δ$-minimum bottleneck spanning tree problem ($δ$-MBST), is the problem of finding a degree bounded spanning tree such that the length of the longest edge is minimum. For point sets that lie in the Euclidean plane, both of these problems have been shown to be NP-hard for certain specif…
Consider the minimum spanning tree (MST) of the complete graph with n vertices, when edges are assigned independent random weights. Endow this tree with the graph distance renormalized by n^{1/3} and with the uniform measure on its vertices. We show that the resulting space converges in distribution, as n tends to infinity, to a random measured metric space in the Gromov-Hausdorff-Prokhorov topology. We additionally show that the limit is a random binary R-tree and has Minkowski dimension 3 almost surely. In particular, its law is mutually singular with that of the Brownian continuum random tr…
In the complete graph on n vertices, when each edge has a weight which is an exponential random variable, Frieze proved that the minimum spanning tree has weight tending to zeta(3)=1/1^3+1/2^3+1/3^3+... as n goes to infinity. We consider spanning trees constrained to have depth bounded by k from a specified root. We prove that if k > log_2 log n+omega(1), where omega(1) is any function going to infinity with n, then the minimum bounded-depth spanning tree still has weight tending to zeta(3) as n -> infinity, and that if k < log_2 log n, then the weight is doubly-exponentially large in log_2 lo…
We derive tight bounds on the expected weights of several combinatorial optimization problems for random point sets of size $n$ distributed among the leaves of a balanced hierarchically separated tree. We consider {\it monochromatic} and {\it bichromatic} versions of the minimum matching, minimum spanning tree, and traveling salesman problems. We also present tight concentration results for the monochromatic problems.
This paper studies Minimum Spanning Trees under incomplete information for its vertices. We assume that no information is available on the precise placement of vertices so that it is only known that vertices belong to some neighborhoods that are second order cone representable and distances are measured with a $\ell_q$-norm. Two mixed integer non linear mathematical programming formulations are presented, based on alternative representations of subtour elimination constraints. A solution scheme is also proposed, resulting from a reformulation suitable for a Benders-like decomposition, which is…
We investigate the problem of simultaneously dominating all spanning trees of a given graph. We prove that on 2-connected graphs, a subset of the vertices dominates all spanning trees of the graph if and only if it is a vertex cover. Using this fact we present an exact algorithm that finds a simultaneous dominating set of minimum size using an oracle for finding a minimum vertex cover. The algorithm can be implemented to run in polynomial time on several graph classes, such as bipartite or chordal graphs. We prove that there is no polynomial time algorithm that finds a minimum simultaneous dom…
