Related papers: Characterizing Trees with Large Laplacian Energy

In this paper, we have developed a fully-dynamic algorithm for maintaining cardinality of maximum-matching in a tree using the construction of top-trees. The time complexities are as follows: 1. Initialization Time: $O(n(log(n)))$ to build…

Let $G$ be a connected graph and let $k$ be a positive integer. Let $T$ be a spanning tree of $G$. The leaf degree of a vertex $v\in V(T)$ is defined as the number of leaves adjacent to $v$ in $T$. The leaf degree of $T$ is the maximum leaf…

Given a rooted tree $T$ with vertices $u_1,u_2,\ldots,u_n$, the level matrix $L(T)$ of $T$ is the $n \times n$ matrix for which the $(i,j)$-th entry is the absolute difference of the distances from the root to $v_i$ and $v_j$. This matrix…

We revisit the \textsc{$k$-Secluded Tree} problem. Given a vertex-weighted undirected graph $G$, its objective is to find a maximum-weight induced subtree $T$ whose open neighborhood has size at most $k$. We present a fixed-parameter…

We calculate the exact number of contours of size $n$ containing a fixed vertex in $d$-ary trees and provide sharp estimates for this number for more general trees. We also obtain a characterization of the locally finite trees with…

Let $G$ be a simple graph with $n$ vertices and let $\mu_1 \geqslant \mu_2 \geqslant...\geqslant \mu_{n - 1} \geqslant \mu_n = 0$ be the eigenvalues of its Laplacian matrix. The Laplacian Estrada index of a graph $G$ is defined as $LEE (G)…

We study the spectral Tur\'an problem for trees. To avoid limiting our perspective to specific families of trees, we parametrize trees in terms of their unique bipartition. We say $T \in \mathcal{T}_{m,l+1}^{\delta}$ if $T$ is a tree of…

A subtree of a tree is any induced subgraph that is again a tree (i.e., connected). The mean subtree order of a tree is the average number of vertices of its subtrees. This invariant was first analyzed in the 1980s by Jamison. An intriguing…

The energy and iota energy of signed digraphs are respectively defined as the sum of absolute values of real parts and sum of absolute values of imaginary parts of its eigenvalues. Recently, Yang and Wang (2018) find the energy and iota…

I describe an `oct-tree' N-body code which randomly shifts, reorients, and resizes the root cell at each time step. Averaging over a plurality of root cell positions and orientations statistically restores translational and rotational…

The set of all permutations with $n$ symbols is a symmetric group denoted by $S_n$. A transposition tree, $T$, is a spanning tree over its $n$ vertices $V_T=${$1, 2, 3, \ldots n$} where the vertices are the positions of a permutation $\pi$…

The classical matrix-tree theorem relates the determinant of the combinatorial Laplacian on a graph to the number of spanning trees. We generalize this result to Laplacians on one- and two-dimensional vector bundles, giving a combinatorial…

An $n$-vertex tree $T$ is said to be $\textit{graceful}$ if there exists a bijective labelling $\phi:V(T)\to \{1,\ldots,n\}$ such that the edge-differences $\{|\phi(x)-\phi(y)| : xy\in E(T)\}$ are pairwise distinct. The longstanding…

We compute the magnitude (an isometric invariant of metric spaces) of compact $\mathbb{R}$-trees and show that it equals $1 + L/2$, where $L \in [0, \infty]$ denotes the total length. Although length is the only geometric invariant captured…

An out-tree $T$ is an oriented tree with only one vertex of in-degree zero. A vertex $x$ of $T$ is internal if its out-degree is positive. We design randomized and deterministic algorithms for deciding whether an input digraph contains a…

A tree with at most $k$ leaves is called a $k$-ended tree. A spanning 2-ended tree is a Hamilton path. A Hamilton cycle can be considered as a spanning 1-ended tree. The earliest result concerning spanning trees with few leaves states that…

Polytrees are a subclass of Bayesian networks that seek to capture the conditional dependencies between a set of $n$ variables as a directed forest and are motivated by their more efficient inference and improved interpretability. Since the…

The \textit{eccentricity matrix} $\mathcal{E}(G)$ of a connected graph $G$ is obtained from the distance matrix of $G$ by keeping the largest non-zero entries in each row and each column, and leaving zeros in the remaining ones. The…

The energy of a graph G is equal to the sum of absolute values of the eigenvalues of the adjacency matrix of G, whereas the Laplacian energy of a graph G is equal to the sum of the absolute value of the difference between the eigenvalues of…

A vertex of degree one is called an end-vertex, and an end-vertex of a tree is called a leaf. A tree with at most $k$ leaves is called a $k$-ended tree. For a positive integer $k$, let $t_k$ be the order of a largest $k$-ended tree. Let…