Related papers: Some sharp bounds on the average Steiner (k, l)-ec…

In this paper, we address the problem of packing large trees in $G_{n,p}$. In particular, we prove the following result. Suppose that $T_1, \dotsc, T_N$ are $n$-vertex trees, each of which has maximum degree at most $(np)^{1/6} / (\log…

The treewidth is a structural parameter that measures the tree-likeness of a graph. Many algorithmic and combinatorial results are expressed in terms of the treewidth. In this paper, we study the treewidth of outer $k$-planar graphs, that…

A formula based on a vertex contributions of the Steiner $k$-Wiener index is induced by a newly introduced $k$-Steiner betweenness centrality, which measures the number of $k$-Steiner trees that include a particular vertex as a non-terminal…

In this paper, we study some spanning trees with bounded degree and leaf degree from eigenvalues. For any integer $k\geq2$, a $k$-tree is a spanning tree in which every vertex has degree no more than $k$. Let $T$ be a spanning tree of a…

Let $T$ be a finite tree graph, $T^N$ be the Cartesian power graph of $T$, and $d^N$ be the graph distance metric on $T^N$. Also let \[ \mathbb S_r^N(x) := \{v \in T^N: d^N(x,v) = r\} \] be the sphere of radius $r$ centered at $x$ and $M$…

In this paper, we investigate the structures of an extremal tree which has the minimal number of subtrees in the set of all trees with the given degree sequence of a tree. In particular, the extremal trees must be caterpillar and but in…

A recent upper bound by Le and Solomon [STOC '23] has established that every $n$-node graph has a $(1+\varepsilon)(2k-1)$-spanner with lightness $O(\varepsilon^{-1} n^{1/k})$. This bound is optimal up to its dependence on $\varepsilon$; the…

We derive the asymptotic distribution of the total length $L_n$ of a $\operatorname {Beta}(2-\alpha,\alpha)$-coalescent tree for $1<\alpha<2$, starting from $n$ individuals. There are two regimes: If $\alpha\le1/2(1+\sqrt{5})$, then $L_n$…

On trees of fixed order, we show a direct relation between Kemeny's constant and Wiener index, and provide a new formula of Kemeny's constant from the relation with a combinatorial interpretation. Moreover, the relation simplifies proofs of…

The problem of determining the largest number $f(n;\bar{\kappa}\leq \ell)$ of edges for graphs with $n$ vertices and maximal local connectivity at most $\ell$ was considered by Bollob\'{a}s. Li et al. studied the largest number…

Motivated by the work of Lov\'asz and Szegedy on the convergence and limits of dense graph sequences, we investigate the convergence and limits of finite trees with respect to sampling in normalized distance. Based on separable real trees,…

We introduce generalizations of Aldous' Brownian Continuous Random Tree as scaling limits for multicritical models of discrete trees. These discrete models involve trees with fine-tuned vertex-dependent weights ensuring a k-th root…

We prove that if an $n$-vertex graph $G$ is non-extremal and $T$ is a bounded degree tree on $n$ vertices, then $T\subset G$ even when the minimum degree of $G$ is less than $n/2$ by a linear term. We avoid the use of the Regularity lemma,…

Let $G$ be a molecular graph. The total-eccentricity index of graph $G$ is defined as the sum of eccentricities of all vertices of $G$. %In [R. Farooq, M.A. Malik, J. Rada, Extremal graphs with respect to total-eccentricity index, 2017,…

Generalized gamma distributions arise as limits in many settings involving random graphs, walks, trees, and branching processes. Pek\"oz, R\"ollin, and Ross (2016, arXiv:1309.4183 [math.PR]) exploited characterizing distributional fixed…

As well known the rotation distance D(S,T) between two binary trees S, T of n vertices is the minimum number of rotations of pairs of vertices to transform S into T. We introduce the new operation of chain rotation on a tree, involving two…

For a vertex $u$ of a tree $T$, the leaf (internal, respectively) status of $u$ is the sum of the distances from $u$ to all leaves (internal vertices, respectively) of $T$. The minimum (maximum, respectively) leaf status of a tree $T$ is…

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…

Generalizing work from the 1970s on the determinants of distance hypermatrices of trees, we consider the hyperdeterminants of order-$k$ Steiner distance hypermatrices of trees on $n$ vertices. We show that they can be nearly diagonalized as…

This study is dedicated to precise distributional analyses of the height of non-plane unlabelled binary trees ("Otter trees"), when trees of a given size are taken with equal likelihood. The height of a rooted tree of size $n$ is proved to…