The linear arboricity of -regular graphs
The Loebl-Komlós-Sós conjecture for trees of diameter 5 and for certain caterpillars.
The maximum multiplicity and the two largest multiplicities of eigenvalues in a Hermitian matrix whose graph is a tree
The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree, M1, was understood fully (froma combinatorial perspective) by C.R. Johnson, A. Leal-Duarte (Linear Algebra and Multilinear Algebra 46 (1999) 139-144). Among the possible multiplicity lists for the eigenvalues of Hermitian matrices whose graph is a tree, we focus upon M2, the maximum value of the sum of the two largest multiplicities when the largest multiplicity is M1. Upper and lower bounds are given for M2. Using a combinatorial...
The monadic second-order logic of graphs III : tree-decompositions, minors and complexity issues
The Morse landscape of a riemannian disk
We study upper bounds on the length functional along contractions of loops in Riemannian disks of bounded diameter and circumference. By constructing metrics adapted to imbedded trees of increasing complexity, we reduce the nonexistence of such upper bounds to the study of a topological invariant of imbedded finite trees. This invariant is related to the complexity of the binary representation of integers. It is also related to lower bounds on the number of points in level sets of a real-valued...
The number of archiral trees.
The number of -matchings in almost every tree is a zero residue.
The number of isomorphism classes of spanning trees of a graph
The Number of Labeled Dissections of a k-Ball.
The number of labelled -arch graphs.
The number of spanning trees and chains of graphs.
The number of spanning trees of double graphs
The outer-distance of nodes in random trees.
The packing and covering of the complete graph. I: The forests of order five.
The path is the tree with smallest greatest Laplacian eigenvalue
The P-Laplacian spectral radius of weighted trees with a degree sequence and a weight set.
The realization of input-output maps using bialgebras.
The Reconstruction of a Connected Graph from its Spanning Trees
The relation between the number of leaves of a tree and its diameter
Let denote the minimum possible number of leaves in a tree of order and diameter Lesniak (1975) gave the lower bound for When is even, But when is odd, is smaller than in general. For example, while In this note, we determine using new ideas. We also consider the converse problem and determine the minimum possible diameter of a tree with given order and number of leaves.
The Rooted Tree Embedding Problem into Points in the Plane.