定向带状图（dessins d 'enfant）是嵌入在定向曲面中的图。Bollobás-Riordan-Tutte多项式是一个三变量多项式，它将Tutte多项式扩展到定向带状图。带状图的拟树是一个只有一个面的生成子图，它可用有序弦图来描述。我们将Tutte多项式的生成树展开推广到Bollobás-Riordan-Tutte多项式的拟树展开。
英文摘要：
Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented surfaces. The Bollobás-Riordan-Tutte polynomial is a three-variable polynomial that extends the Tutte polynomial to oriented ribbon graphs. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. We generalize the spanning tree expansion of the Tutte polynomial to a quasi-tree expansion of the Bollobás-Riordan-Tutte polynomial.
The Bollobás-Riordan polynomial [Math. Ann. 323, 81 (2002)] is a universal polynomial invariant for ribbon graphs. We find an extension of this polynomial for a particular family of combinatorial objects, called rank 3 weakly-colored stranded graphs. Stranded graphs arise in the study of tensor models for quantum gravity in physics, and generalize graphs and ribbon graphs. We present a seven-variable polynomial invariant of these graphs, which obeys a contraction/deletion recursion relation similar to that of the Tutte and Bollobás-Riordan polynomials. However, it is defined on a much broader …
Bollobás and Riordan, in their paper "Metrics for sparse graphs," proposed a number of provocative conjectures extending central results of quasirandom graphs and graph limits to sparse graphs. We refute these conjectures by exhibiting a sequence of graphs with convergent normalized subgraph densities (and pseudorandom $C_4$-counts), but with no limit expressible as a kernel.
A recipe theorem for the topological Tutte polynomial of Bollobas and Riordan
在[A polynomial invariant of graphs on可定向曲面，Proc. Lond.数学会，III Ser.83，No.3，513-531（2001）]和[A polynomial of graphs on surfaces，Math.Ann.323，81-96（2002）]中，Bollobas和Riordan将经典的Tutte多项式推广到嵌入在表面中的细胞图，即带状图，从而编码经典的Tutte多项式未捕获的拓扑信息。我们为他们的新拓扑Tutte多项式R（G）提供了一个“配方定理”。然后，我们通过一个中间图的结构，R（G）的广义转移多项式Q（G），从而扩展经典Tutte多项式和马丁，或电路划分，带状图多项式之间的关系。我们利用这个关系证明了R（G）的一个对偶性质，它对有向和无向带状图都成立。我们通过将Chumutov和Pak [The Kauffman bracket and the Bollobas-Riordan polynomial of ribbon graphs，Moscow Mathematical Journal 7（3）（2007）409-418]的结果用于R（G）和Q（R）之间关系的上下文中的虚拟链接来得出结论。
We take an elementary and systematic approach to the problem of extending the Tutte polynomial to the setting of embedded graphs. Four notions of embedded graphs arise naturally when considering deletion and contraction operations on graphs on surfaces. We give a description of each class in terms of coloured ribbon graphs. We then identify a universal deletion-contraction invariant (i.e., a `Tutte polynomial') for each class. We relate these to graph polynomials in the literature, including the Bollobás--Riordan, Krushkal, and Las Vergnas polynomials, and give state-sum formulations, duality …
Las Vergnas多项式是Tutte多项式到胞腔嵌入图的扩展。它是由Michel Las Vergnas在1978年引入的，作为拟阵态射的Tutte多项式的特例。虽然一般的Tutte多项式的态射拟阵有一套完整的删除-收缩关系，其专门细胞嵌入图没有。在这里，我们扩展Las Vergnas多项式的伪曲面图。我们表明，在这种设置中，我们可以定义删除和收缩嵌入图一致的删除和收缩的基本拟阵的角度来看，从而产生一个版本的拉斯维加斯Vergnas多项式完整的递归定义。这也使我们能够更深入地了解Las Vergnas多项式，Bollobas-Riordan多项式和Krushkal多项式之间的关系。我们也借此机会扩展一些拉斯维加斯Vergnas的结果欧拉电路从图表面的低亏格表面的任意亏格。
Graph-bootstrap percolation, also known as weak saturation, was introduced by Bollobás in 1968. In this process, we start with initial "infected" set of edges $E_0$, and we infect new edges according to a predetermined rule. Given a graph $H$ and a set of previously infected edges $E_t\subseteq E(K_n)$, we infect a non-infected edge $e$ if it completes a new copy of $H$ in $G=([n],E_t\cup e)$. A question raised by Bollobás asks for the maximum time the process can run before it stabilizes. Bollobás, Przykucki, Riordan, and Sahasrabudhe considered this problem for the most natural case where $H…
In this paper, we analyze the Bollobás and Riordan polynomial $\mathcal{R}$ for ribbon graphs with half-ribbons introduced in [Combinatorics, Probability and Computing 31, 507-549, 2022]. We prove the universality property of a multivariate version of $\mathcal{R}$ whereas $\mathcal{R}$ itself turns out to be universal for a subclass of ribbon graphs with half-ribbons. We also show that $\mathcal{R}$ can be defined on some equivalence classes of ribbon graphs involving half-ribbons moves and that the new polynomial is universal on these classes.
Motivated by applications, the last few years have witnessed tremendous interest in understanding the structure as well as the behavior of dynamics for inhomogeneous random graph models. In this study we analyze the maximal components at criticality of one famous class of such models, the rank-one inhomogeneous random graph model. Viewing these components as measured random metric spaces, under finite moment assumptions for the weight distribution, we show that the components in the critical scaling window with distances scaled by $n^{-1/3}$ converge in the Gromov-Haussdorf-Prokhorov metric to…
