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Motivated by the properties of the descent polynomials, which enumerate permutations of $S_n$ with a fixed descent set, we define descent polynomials for labeled rooted trees. We give recursive and explicit formulas for these polynomials…
The properties of energy levels in a family of classically pseudointegrable systems, the barrier billiards, are investigated. An extensive numerical study of nearest-neighbor spacing distributions, next-to-nearest spacing distributions,…
In a rooted tree, we call a vertex {\em balanced} if it is at equal distance from all its descendant leaves. We count balanced vertices in three different tree varieties. For decreasing binary trees, we can prove that the probability that a…
Here, we find the characteristics polynomial of normalized Laplacian of a tree. The coefficients of this polynomial are expressed by the higher order general Randi\'c indices for matching, whose values depend on the structure of the tree.…
Using a mapping of compact polymers on the Manhattan lattice to spanning trees, we calculate exactly the average number of bends at infinite temperature. We then find, in a high temperature approximation, the energy of the system as a…
In this paper, we study the energy of semigraphs and obtain some bounds, and show that one of the bounds is tight. We also study the spectra of the adjacency matrix of a special type of rooted 3-uniform semigraph and enumerate those…
We give a characterization of equilibrium measures for $p$-capacities on the boundary of an infinite tree of arbitrary (finite) local degree. For $p=2$, this provides, in the special case of trees, a converse to a theorem of Benjamini and…
We explore the relationship between polynomial functors and (rooted) trees. In the first part we use polynomial functors to derive a new convenient formalism for trees, and obtain a natural and conceptual construction of the category…
Given $n \geq 1$, we study the existence of a tree on $n$ vertices whose independence polynomial is symmetric and unimodal as well as the existence of a symmetric and unimodal independence polynomial of degree $n$ of a tree.
This paper investigates spectral properties of the deformed Laplacian matrix, which merges the Laplacian and signless Laplacian matrices of a graph through a one-parameter family of matrices. We present general results on the eigenvalues of…
We establish the various properties as well as diverse relations of the ascent and descent spectra for bounded linear operators. We specially focus on the theory of subspectrum. Furthermore, we construct a new concept of convergence for…
We study the Coxeter polynomials associated with certain star-like trees. In particular, we exhibit large Salem factors of these polynomials and give convergence properties of their dominant roots.
We consider random binary trees that appear as the output of certain standard algorithms for sorting and searching if the input is random. We introduce the subtree size metric on search trees and show that the resulting metric spaces…
The occurrence and the distribution of patterns of trees associated to natural numbers are investigated. Bounds from above and below are proven for certain natural quantities.
We study a class of rooted trees with a substitution type structure. These trees are not necessarily regular, but exhibit a lot of symmetries. We consider nearest neighbor operators which reflect the symmetries of the trees. The spectrum of…
In this paper we study the adjacency spectrum of families of finite rooted trees with regular branching properties. In particular, we show that in the case of constant branching, the eigenvalues are realized as the roots of a family of…
We study the problem of enumerating Braess edges for Kemeny's constant in trees. We obtain bounds and asympotic results for the number of Braess edges in some families of trees.
In the first part of this paper, the main concern is with smoothness properties of the boundary of the pseudospectrum of a matrix polynomial. In the second part, results are obtained concerning the number of connected components of…
Over some types of trees with a given number of vertices, which trees minimize or maximize the total number of subtrees or leaf containing subtrees are studied. Here are some of the main results:\ (1)\, Sharp upper bound on the total number…
The solution of some equations involving functional derivatives is given as a series indexed by planar binary trees. The terms of the series are given by an explicit recursive formula. Some algebraic properties of these series are…