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This article explores the notions of$\\mathcal {F}$-transitivity and topological$\\mathcal {F}$-recurrence for backward shift operators on weighted$\\ell ^p$-spaces and$c_0$-spaces on directed trees, where$\\mathcal {F}$represents a Furstenberg family of subsets of$\\mathbb {N}_0$. In particular, we establish the equivalence between recurrence and hypercyclicity of these operators on unrooted directed trees. For rooted directed trees, a backward shift operator is hypercyclic if and only if it possesses an orbit of a bounded subset that is weakly dense.

Let$Sz(G),Sz^*(G)$and$W(G)$be the Szeged index, revised Szeged index and Wiener index of a graph$G.$In this paper, the graphs with the fourth, fifth, sixth and seventh largest Wiener indices among all unicyclic graphs of order$n\\geqslant 10$are characterized; as well the graphs with the first, second, third, and fourth largest Wiener indices among all bicyclic graphs are identified. Based on these results, further relation on the quotients between the (revised) Szeged index and the Wiener index are studied. Sharp lower bound on$Sz(G)/W(G)$is determined for all connected graphs each of which contains at least one non-complete block. As well the connected graph with the second smallest value on$Sz^*(G)/W(G)$is identified for$G$containing at least one cycle. Comment: 25 pages, 5 figures

A set$S$of vertices in a graph$G$is a$2$ -dominating set if every vertex of$G$not in$S$is adjacent to at least two vertices in$S$ , and$S$is a$2$ -independent set if every vertex in$S$is adjacent to at most one vertex of$S$ . The$2$ -domination number$\\gamma_2(G)$is the minimum cardinality of a$2$ -dominating set in$G$ , and the$2$ -independence number$\\alpha_2(G)$is the maximum cardinality of a$2$ -independent set in$G$ . Chellali and Meddah [ıt Trees with equal$2$ -domination and$2$ -independence numbers, Discussiones Mathematicae Graph Theory 32 (2012), 263--270] provided a constructive characterization of trees with equal$2$ -domination and$2$ -independence numbers. Their characterization is in terms of global properties of a tree, and involves properties of minimum$2$ -dominating and maximum$2$ -independent sets in the tree at each stage of the construction. We provide a constructive characterization that relies only on local properties of the tree at each stage of the construction. Comment: 17 pages, 4 figures

Given a set$Y$of decreasing plane trees and a permutation$\\pi$ , how many trees in$Y$have$\\pi$as their postorder? Using combinatorial and geometric constructions, we provide a method for answering this question for certain sets$Y$and all permutations$\\pi$ . We then provide applications of our results to the study of the deterministic stack-sorting algorithm. Comment: 15 pages, 4 figures

Let k ≥ 2 be an integer. A tree T is called a k -tree if d T ( v ) ≤ k for each v ∈ V ( T ) ; that is, the maximum degree of a k -tree is at most k . A k -tree T is a spanning k -tree if T is a spanning subgraph of a connected graph G . Let λ 1 ( D ( G ) ) denote the distance spectral radius in G , where D ( G ) denotes the distance matrix of G . In this paper, we verify an upper bound for λ 1 ( D ( G ) ) in a connected graph G to guarantee the existence of a spanning k -tree in G .

When matching n pairs of socks, drawn randomly one at a time, there is a question of how big the pile of unmatched socks is expected to get. Each permutation of socks drawn has a corresponding Dyck path, with the total number of Dyck paths equaling the n-th Catalan number. However, some discussions failed to take into account that not every Dyck path is equally likely in the process of sorting socks. In this paper we will take the probabilities of the Dyck paths into account, and find a method of finding the expected maximum size of the unmatched sock pile. We also find the first two terms of the asymptotic series for this maximum, and give a conjecture on the third term.

Chung defined the pebbling move of a graph which involves choosing a vertex with at least two pebbles, discarding those two pebbles from that vertex and adding one pebble to a nearby vertex. The 2-target pebbling number of the vertices in graph a is the least number ) has the characteristic that for every configuration of ) pebbles on , it is possible to move a pebble to and simultaneously by a sequence of pebbling moves. The 2-target pebbling number of graph , denoted by ), is the maximum ) over all pairs of the vertices in . In this paper, we discuss the 2-target pebbling number for some standard graphs.

Stanley lists the class of Dyck paths where all returns to the axis are of odd length as one of the many objects enumerated by (shifted) Catalan numbers. By the standard bijection in this context, these special Dyck paths correspond to a class of rooted plane trees, so-called Catalan-Stanley trees. This paper investigates a deterministic growth procedure for these trees by which any Catalan-Stanley tree can be grown from the tree of size one after some number of rounds; a parameter that will be referred to as the age of the tree. Asymptotic analyses are carried out for the age of a random Catalan-Stanley tree of given size as well as for the \"speed\" of the growth process by comparing the size of a given tree to the size of its ancestors.

Let H = ( V , E ) be a hypergraph. A subset S ⊆ V is called F -isolating of H if the induced subhypergraph H [ V \\ N [ S ] ] contains no any member in F as a subhypergraph. The F -isolation number of H is the minimum cardinality of an F -isolating set of H , denoted by ι ( H , F ) . A subset S ⊆ V is an isolating set of H if V \\ N [ S ] is an independent set of H . The cardinality of a minimum isolating set of H is called the isolation number of H , denoted by ι ( H ) . In this paper, we introduce the F -isolating set of hypergraphs and give some results about the F -isolation number of hypergraphs.

In this article sets of certain subgraphs of a graph are formalized in the Mizar system [ ], [ ], based on the formalization of graphs in [ ] briefly sketched in [ ]. The main result is the spanning subgraph theorem.