For a graph G embedded in an orientable surface Σ, we consider associated links L(G) in the thickened surface Σ\times I. We relate the HOMFLY polynomial of L(G) to the recently defined Bollobas-Riordan polynomial of a ribbon graph. This generalizes celebrated results of Jaeger and Traldi. We use knot theory to prove results about graph polynomials and, after discussing questions of equivalence of the polynomials, we go on to use our formulae to prove a duality relation for the Bollobas-Riordan polynomial. We then consider the specialization to the Jones polynomial and recent results of Chmutov and Pak to relate the Bollobas-Riordan polynomials of an embedded graph and its tensor product with a cycle.
Quasi-tree expansion for the Bollobás-Riordan-Tutte polynomial
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。
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。
Bollobás-Riordan random pairing model of a preferential attachment graph $G_m^n$ is studied. Let $\{W_j\}_{j\le mn+1}$ be the process of sums of independent exponentials with mean $1$. We prove that the degrees of the first $ν_m^n:=n^{\frac{m}{m+2}-ε}$ vertices are jointly, and uniformly, asymptotic to $\{2(mn)^{1/2}\bigl(W^{1/2}_{mj}-W^{1/2}_{m(j-1)}\bigr)\}_{j\in [t]}$, and that with high probability (whp) the smallest of these degrees is $n^{\frac{ε(m+2)}{2m}}$, at least. In contrast, the degrees of vertices below the top by any fraction of $n$ are whp of $O(\log n)$ order. Next we bound th…
We generalize theories of graph, matroid, and ribbon-graph activities to delta-matroids. As a result, we obtain an activities based feasible-set expansion for a transition polynomial of delta-matroids defined by Brijder and Hoogeboom. This result yields feasible-set expansions for the two-variable Bollobás-Riordan and interlace polynomials of a delta-matroid. In the former case, the expansion obtained directly generalizes the activities expansions of the Tutte polynomial of graphs and matroids.
We construct a new polynomial invariant of maps (graphs embedded in a compact surface, orientable or non-orientable), which contains as specializations the Krushkal polynomial, the Bollobás--Riordan polynomial, the Las Vergnas polynomial, and their extensions to non-orientable surfaces, and hence in particular the Tutte polynomial. Other evaluations include the number of local flows and local tensions taking non-identity values in a given finite group.
The $2$-decomposition for ribbon graphs was introduced in [Annals of Combinatorics 15 (2011), pp 675-706]. We extend this result to half-edged ribbon graphs and to rank $D$-weakly colored graphs [SIGMA 12 (2016), 030], generalizing therefore the $2$-sums and tensor products of these graphs. Using this extension for the $2$-decompositions, we provide new expansion formulas for the Bollobás Riordan polynomial for half-edged ribbon graphs and also for the polynomial invariant for weakly colored stranded graphs.
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.
A k-valuation is a special type of edge k-colouring of a medial graph. Various graph polynomials, such as the Tutte, Penrose, Bollobás-Riordan, and transition polynomials, admit combinatorial interpretations and evaluations as weighted counts of k-valuations. In this paper, we consider a multivariate generating function of k-valuations. We show that this is a polynomial in k and hence defines a graph polynomial. We then show that the resulting polynomial has several desirable properties, including a recursive deletion-contraction-type definition, and specialises to the graph polynomials mentio…
We follow the example of Tutte in his construction of the dichromate of a graph (that is, the Tutte polynomial) as a unification of the chromatic polynomial and the flow polynomial in order to construct a new polynomial invariant of maps (graphs embedded in orientable surfaces). We call this the surface Tutte polynomial. The surface Tutte polynomial of a map contains the Las Vergnas polynomial, Bollobás-Riordan polynomial and Kruskhal polynomial as specializations. By construction, the surface Tutte polynomial includes among its evaluations the number of local tensions and local flows taking v…
Polynomials on stranded graphs are higher dimensional generalization of Tutte and Bollobás-Riordan polynomials [Math. Ann. 323 (2002), 81-96]. Here, we deepen the analysis of the polynomial invariant defined on rank 3 weakly-colored stranded graphs introduced in arXiv:1301.1987. We successfully find in dimension $D\geq3$ a modified Euler characteristic with $D-2$ parameters. Using this modified invariant, we extend the rank 3 weakly-colored graph polynomial, and its main properties, on rank 4 and then on arbitrary rank $D$ weakly-colored stranded graphs.
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。
By considering Tutte polynomials of Hopf algebras, we show how a Tutte polynomial can be canonically associated with combinatorial objects that have some notions of deletion and contraction. We show that several graph polynomials from the literature arise from this framework. These polynomials include the classical Tutte polynomial of graphs and matroids, Las Vergnas' Tutte polynomial of the morphism of matroids and his Tutte polynomial for embedded graphs, Bollobas and Riordan's ribbon graph polynomial, the Krushkal polynomial, and the Penrose polynomial. We show that our Tutte polynomials of…
