图的广义Zig-Zag积的性质

Zig-zag chains and metric equivalences between ultrametric spaces

We study the classification of ultrametric spaces based on their small scale geometry (uniform homeomorphism), large scale geometry (coarse equivalence) and both (all scale uniform equivalences). We prove that these equivalences can be characterized with parallel constructions using a combinatoric tool called common zig-zag chain.

锯齿形 Eulerian 多项式

Zig-zag Eulerian polynomials

本文利用松弛版 $P$-分拆系统研究自然标号锯齿形偏序集的 Eulerian 多项式，统一既有结果、建立精细化递推，并解释其与交错排列大回报分布的联系。

Sequences involving square zig-zag shapes

We define a so-called square $k$-zig-zag shape as a part of the regular square grid. Considering the shape as a $k$-zig-zag digraph, we give values of its vertices according to the number of the shortest paths from a base vertex. It provides several integer sequences, whose higher-order homogeneous recurrences are determined by the help of a special matrix recurrence.

The $h^*$-polynomial of the order polytope of the zig-zag poset

We describe a family of shellings for the canonical triangulation of the order polytope of the zig-zag poset. This gives a new combinatorial interpretation for the coefficients in the numerator of the Ehrhart series of this order polytopein terms of the swap statistic on alternating permutations.

Containment game played on random graphs: another zig-zag theorem

We consider a variant of the game of Cops and Robbers, called Containment, in which cops move from edge to adjacent edge, the robber moves from vertex to adjacent vertex (but cannot move along an edge occupied by a cop). The cops win by "containing" the robber, that is, by occupying all edges incident with a vertex occupied by the robber. The minimum number of cops, $ξ(G)$, required to contain a robber played on a graph $G$ is called the containability number, a natural counterpart of the well-known cop number $c(G)$. This variant of the game was recently introduced by Komarov and Mackey, who …

Primality of closed path polyominoes

In this paper we introduce a new class of polyominoes, called closed paths, and we study the primality of their associated ideal. Inspired by an existing conjecture that characterizes the primality of a polyomino ideal by nonexistence of zig-zag walks, we classify all closed paths which do not contain zig-zag walks, and we give opportune toric representations of the associated ideals. To support the conjecture we prove that having no zig-zag walks is a necessary and sufficient condition for the primality of the associated ideal of a closed path. Finally, we present some classes of prime polyom…

On extremal multiplicative Zagreb indices of trees with given domination number

For a graph $G$, the first multiplicative Zagreb index $\prod_1$ is equal to the product of squares of the vertex degrees, and the second multiplicative Zagreb index $\prod_2$ is equal to the product of the products of degrees of pairs of adjacent vertices. The (mutiplicative) Zagreb indices have been the focus of considerable research in computational chemistry dating back to Gutman and Trinajstić in 1972. In this paper, we explore the mutiplicative Zagreb indices in terms of arbitrary domination number. The sharp upper and lower bounds of $\prod_1(G)$ and $\prod_2(G)$ are given. In addition,…

On the sharp upper and lower bounds of multiplicative Zagreb indices of graphs with connectivity at most k

For a (molecular) graph, the first multiplicative Zagreb index $\prod_1(G) $ is the product of the square of every vertex degree, and the second multiplicative Zagreb index $\prod_2(G) $ is the product of the products of degrees of pairs of adjacent vertices. In this paper, we explore graphs in terms of (edge) connectivity. The maximum and minimum values of $\prod_1(G) $ and $\prod_2(G) $ of graphs with connectivity at most $k$ are provided. In addition, the corresponding extremal graphs are characterized, and our results extend and enrich some known conclusions.

On Neighbourhood Zagreb index of product graphs

There is powerful relation between the chemical behaviour of chemical compounds and their molecular structures. Topological indices defined on these chemical molecular structures are capable to predict physical properties, chemical reactivity and biological activity. In this article, a new topological index named as Neighbourhood Zagreb index $(M_N)$ is presented. Here the chemical importance of this newly introduced index is studied and some explicit results for this index of different product graphs such as Cartesian, Tensor and Wreath product is derived. Some of these results are applied to…