More on linear and metric tree maps

We consider linear and metric self-maps on vertex sets of finite combinatorial trees. Linear maps are maps which preserve intervals between pairs of vertices whereas metric maps are maps which do not increase distances between pairs of vertices. We obtain criteria for a given linear or a metric map to be a positive (negative) under some orientation of the edges in a tree, we characterize trees which admit maps with Markov graphs being paths and prove that the converse of any partial functional digraph is isomorphic to a Markov graph for some suitable map on a tree.

A Limit Theorem for the Six-length of Random Functional Graphs with a Fixed Degree Sequence

We obtain results on the limiting distribution of the six-length of a random functional graph, also called a functional digraph or random mapping, with given in-degree sequence. The six-length of a vertex $v\in V$ is defined from the associated mapping, $f:V\to V$, to be the maximum $i\in V$ such that the elements $v, f(v), \ldots, f^{i-1}(v)$ are all distinct. This has relevance to the study of algorithms for integer factorisation.

$\lambda$-Factorials of $n$

Recently, by the Riordan identity related to tree enumerations, \begin{align*} \sum_{k=0}^{n}\binom{n}{k}(k+1)!(n+1)^{n-k} = (n+1)^{n+1}, \end{align*} Sun and Xu have derived another analogous one, \begin{align*} \sum_{k=0}^{n}\binom{n}{k}D_{k+1}(n+1)^{n-k} = n^{n+1}, \end{align*} where $D_{k}$ is the number of permutations with no fixed points on $\{1,2,\dots, k\}$. In the paper, we utilize the $\lambda$-factorials of $n$, defined by Eriksen, Freij and W$\ddot{a}$stlund, to give a unified generalization of these two identities. We provide for it a combinatorial proof by the functional digraph theory and two algebraic proofs. Using the umbral representation of our generalized identity and Abel's binomial formula, we deduce several properties for $\lambda$-factorials of $n$ and establish interesting relations between the generating functions of general and exponential types for any sequence of numbers or polynomials.

ON THE MINKOWSKI DIMENSION OF FUNCTIONAL DIGRAPH

Functional digraphs are sometimes fractal sets. As a special kind of fractal sets, the dimension properties of the functional digraph are studied in this paper. Firstly, the proof of a Minkowski dimension theorem is discussed and a new proof is given. Secondly, according to this dimension theorem, the Minkowski dimensions of the digraphs of the sum, deviation, product and quotient of two functions are discussed. And the relations between these Minkowski dimensions and the Minkowski dimensions of the digraphs of the two functions are established. In the conclusion, the maximum Minkowski dimension of the two functional digraphs plays a decisive part in the Minkowski dimensions of the digraphs of the sum, deviation, product and quotient of two functions.