Related papers: Three remarks on Matula numbers

We investigate the tree gonality of a genus-$g$ metric graph, defined as the minimum degree of a tropical morphism from any tropical modification of the metric graph to a metric tree. We give a combinatorial constructive proof that this…

In their paper, Bounds on the Number of Edges in Hypertrees, G.Y. Katona and P.G.N. Szab\'o introduced a new, natural definition of hypertrees in $k$-uniform hypergraphs and gave lower and upper bounds on the number of edges. They also…

In this paper we present a polynomial time algorithm to compute the local zeta function Z(s,f) attached to a polynomial f(x) in Z[x] (in one variable, with splitting field Q) and a prime p. The algorithm reduces in polynomial time the…

The height $H(n)$ of $n$ is the least integer $i$ such that the $i$-th iterate of Euler's totient function $\varphi^{(i)}(n)$ equals $1$. H. N. Shapiro showed that this $H$ is almost completely additive. Building on the fact that this…

Let $K$ be a field, complete with respect to a discrete non-archimedian valuation and let $k$ be the residue field. Consider a system $F$ of $n$ polynomial equations in $K\vars$. Our first result is a reformulation of the classical Hensel's…

Let's denote a complete $m$-ary rooted tree graph of height $n$ as $G$. In scope of this paper we prove the certain relations between the properties of $G$ and the expectation and variance of the distribution of lengths of strings,…

The edges of the characteristic imset polytope, $\operatorname{CIM}_p$, were recently shown to have strong connections to causal discovery as many algorithms could be interpreted as greedy restricted edge-walks, even though only a strict…

Topological phylogenetic trees can be assigned edge weights in several natural ways, highlighting different aspects of the tree. Here the rooted triple and quartet metrizations are introduced, and applied to formulate novel fast methods of…

Let $\sigma$ be the scattering relation on a compact Riemannian manifold $M$ with non-necessarily convex boundary, that maps initial points of geodesic rays on the boundary and initial directions to the outgoing point on the boundary and…

We consider unicellular maps, or polygon gluings, of fixed genus. A few years ago the first author gave a recursive bijection transforming unicellular maps into trees, explaining the presence of Catalan numbers in counting formulas for…

In this paper, we find computational formulae for generalized characteristic polynomials of graph bundles. We show that the number of spanning trees in a graph is the partial derivative (at (0,1)) of the generalized characteristic…

In this paper, we prove that the self-dual morphological hierarchical structure computed on a n-D gray-level wellcomposed image u by the algorithm of G{\'e}raud et al. [1] is exactly the mathematical structure defined to be the tree of…

We consider finite trees with edges labeled by letters on a finite alphabet $\varSigma$. Each pair of nodes defines a unique labeled path whose trace is a word of the free monoid $\varSigma^*$. The set of all such words defines the language…

We explain how to obtain the set of solutions of a multivariate polynomial equation modulo a power of a prime number. These solutions are determined by a tree, called the trunk, which makes it possible to reconstruct all solutions. We apply…

In this paper we continue the study of permutations avoiding the vincular pattern $1-32-4$ by constructing a generating tree with a single label for these permutations. This construction finally provides a clearer explanation of why a…

For a pair consisting of a gene tree and a species tree, the ancestral configurations at an internal node of the species tree are the distinct sets of gene lineages that can be present at that node. Ancestral configurations appear in…

Let $n$ and $k$ be positive integers, and $f_n(k)$ (resp. $g_n(k)$) be the number of unital subrings (resp. unital irreducible subrings) of $\mathbb{Z}^n$ of index $k$. The numbers $f_n(k)$ are coefficients of certain zeta functions of…

An independent set in a graph is a set of pairwise non-adjacent vertices. The independence number $\alpha{(G)}$ is the size of a maximum independent set in the graph $G$. The independence polynomial of a graph is the generating function for…

In this paper, we examine roots of graph polynomials where those roots can be considered as structural graph measures. More precisely, we prove analytical results for the roots of certain modified graph polynomials and also discuss…

Let I be an arbitrary ideal in C[[x,y]]. We use the Newton algorithm to compute by induction the motivic zeta function of the ideal, yielding only few poles, associated to the faces of the successive Newton polygons. We associate a minimal…