Related papers: Trees with Given Stability Number and Minimum Numb…

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 $v$. Miklavi\v{c} and \v{S}parl (arXiv:2011.01635v1) define the distance-unbalancedness of…

We generalize the results from [X.-D. Zhang, X.-P. Lv, Y.-H. Chen, \textit{Ordering trees by the Laplacian coefficients}, Linear Algebra Appl. (2009), doi:10.1016/j.laa.2009.04.018] on the partial ordering of trees with given diameter. For…

The Euclidean Steiner problem is the problem of finding a set $St$, with the shortest length, such that $St \cup A$ is connected, where $A$ is a given set in a Euclidean space. The solutions $St$ to the Steiner problem will be called…

Let $T$ be a tree on $n$ vertices. We can regard the edges of $T$ as transpositions of the vertex set; their product (in any order) is a cyclic permutation. All possible cyclic permutations arise (each exactly once) if and only if the tree…

We study the number of random records in an arbitrary split tree (or equivalently, the number of random cuttings required to eliminate the tree). We show that a classical limit theorem for convergence of sums of triangular arrays to…

A subset of vertices is a {\it maximum independent set} if no two of the vertices are adjacent and the subset has maximum cardinality. A subset of vertices is called a {\it maximum dissociation set} if it induces a subgraph with vertex…

We determine the maximum distance between any two of the center, centroid, and subtree core among trees with a given order. Corresponding results are obtained for trees with given maximum degree and also for trees with given diameter. The…

Let $A$ be a (not necessarily unital) separable non-elementary simple amenable C*-algebra whose tracial basis may not have finite covering dimension and may not be compact but satisfies certain condition (C). We show that $A$ is ${\cal…

We prove properties of extremal graphs of girth 5 and order 20 <=v <= 32. In each case we identify the possible minimum and maximum degrees, and in some cases prove the existence of (non-trivial) embedded stars. These proofs allow for…

For a uniform random labelled tree, we find the limiting distribution of tree parameters which are stable (in some sense) with respect to local perturbations of the tree structure. The proof is based on the martingale central limit theorem…

We determine the tree which maximizes the distance between characteristic set and subtree core over all trees on $n$ vertices. The asymptotic nature of this distance is also discussed. The problem of extremizing the distance between…

Tensor-multi-scalar theories (TMST) are among the most natural generalizations of Einstein's theory, they are mathematically self-consistent and free from pathologies. They pass through all the known observations but contrary to standard…

The minimum linear arrangement problem on a network consists of finding the minimum sum of edge lengths that can be achieved when the vertices are arranged linearly. Although there are algorithms to solve this problem on trees in polynomial…

A vertex subset of a graph is called a distance-$k$ independent set if the distance between any two of its distinct vertices is at least $k + 1$. For all $n,k \geq 1$, we determine the minimum possible number of inclusion-wise maximal…

The Steiner $k$-eccentricity of a vertex in graph $G$ is the maximum Steiner distance over all $k$-subsets containing the vertex. %Some general properties of the Steiner 3-eccentricity of trees are given. Let $\mathbb{T}_n$ be the set of…

Motivated by online recommendation systems, we study a family of random forests. The vertices of the forest are labeled by integers. Each non-positive integer $i\le 0$ is the root of a tree. Vertices labeled by positive integers $n \ge 1$…

A spanning tree of a graph $G$ is a connected acyclic spanning subgraph of $G$. We consider enumeration of spanning trees when $G$ is a $2$-tree, meaning that $G$ is obtained from one edge by iteratively adding a vertex whose neighborhood…

Let $G$ be a simple graph. A dissociation set of $G$ is defined as a set of vertices that induces a subgraph in which every vertex has a degree of at most 1. A dissociation set is maximal if it is not contained as a proper subset in any…

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…

We study fragmentation of a random recursive tree into a forest by repeated removal of nodes. The initial tree consists of N nodes and it is generated by sequential addition of nodes with each new node attaching to a randomly-selected…