A true Tree Calculus is being developed to make a joint study of the two statistics “eoc” (end of minimal chain) and “pom” (parent of maximum leaf) on the set of secant trees. Their joint distribution restricted to the set {eoc-pom ≤ 1} is shown to satisfy two partial difference equation systems, to be symmetric and to be expressed in the form of an explicit three-variable generating function.

In this paper we consider duplexes, which are sets with two associative binary operations. Dimonoids in the sense of Loday are examples of duplexes. The set of all permutations carries a structure of a duplex. Our main result asserts that it is a free duplex with an explicitly described set of generators. The proof uses a construction of the free duplex with one generator by planary trees.

The topology and combinatorial structure of the Mandelbrot set ${ℳ}^d$ (of degree d ≥ 2) can be studied using symbolic dynamics. Each parameter is mapped to a kneading sequence, or equivalently, an internal address; but not every such sequence is realized by a parameter in ${ℳ}^d$. Thus the abstract Mandelbrot set is a subspace of a larger, partially ordered symbol space, ${\Lambda }^d$. In this paper we find an algorithm to construct “visible trees” from symbolic sequences which works whether or not the sequence is realized....

A caterpillar is a tree with the property that after deleting all its vertices of degree 1 a simple path is obtained. The signed 2-domination number ${\gamma }_{\mathrm{s}}^2\left(G\right)$ and the signed total 2-domination number ${\gamma }_{\mathrm{st}}^2\left(G\right)$ of a graph $G$ are variants of the signed domination number ${\gamma }_{\mathrm{s}}\left(G\right)$ and the signed total domination number ${\gamma }_{\mathrm{st}}\left(G\right)$. Their values for caterpillars are studied.

By a ternary system we mean an ordered pair $(W,R)$, where $W$ is a finite nonempty set and $R\subseteq W\times W\times W$. By a signpost system we mean a ternary system $(W,R)$ satisfying the following conditions for all $x,y,z\in W$: if $(x,y,z)\in R$, then $(y,x,x)\in R$ and $(y,x,z)\notin R$; if $x\ne y$, then there exists $t\in W$ such that $(x,t,y)\in R$. In this paper, a signpost system is used as a common description of a connected graph and a spanning tree of the graph. By a ct-pair we mean an ordered pair $(G,T)$, where $G$ is a connected graph and $T$ is a spanning tree of $G$. If $(G,T)$ is a ct-pair, then by the guide to...

En este trabajo se estudia el problema de la representación de un conjunto mediante árboles aditivos, en el sentido de hallar una formalización que permita abordar el mismo desde la perspectiva general de los métodos geométricos de representación del análisis multivariante.

An infinite family of T-factorizations of complete graphs $K_{2n}$, where 2n = 56k and k is a positive integer, in which the set of vertices of T can be split into two subsets of the same cardinality such that degree sums of vertices in both subsets are not equal, is presented. The existence of such T-factorizations provides a negative answer to the problem posed by Kubesa.

Vertex-degree parity in large implicit “exchange graphs” implies some EP theorems asserting the existence of a second object without evidently providing a polytime algorithm for finding a second object.

We observe that a lobster with diameter at least five has a unique path $H=x_0,x_1,...,x_m$ with the property that besides the adjacencies in $H$ both $x_0$ and $x_m$ are adjacent to the centers of at least one $K_{1,s}$, where $s>0$, and each $x_i$, $1\le i\le m-1$, is adjacent at most to the centers of some $K_{1,s}$, where $s\ge 0$. This path $H$ is called the central path of the lobster. We call $K_{1,s}$ an even branch if $s$ is nonzero even, an odd branch if $s$ is odd and a pendant branch if $s=0$. In the existing literature only some specific classes of lobsters have been found...

Using isometric embedding of metric trees into Banach spaces, this paper will investigate barycenters, type and cotype, and various measures of compactness of metric trees. A metric tree (T, d) is a metric space such that between any two of its points there is a unique arc that is isometric to an interval in ℝ. We begin our investigation by examining isometric embeddings of metric trees into Banach spaces. We then investigate the possible images x₀ = π((x₁ + ... + xₙ)/n), where π is a contractive...