A Note on Hodge Structures Associated to Graphs
نویسنده
چکیده
Feynman amplitudes, which play a central role in perturbative quantum field theory, are algebro-geometric periods associated to graphs. These periods have been investigated by Broadhurst and Kreimer (see [4] and the references cited there) and shown for many special graphs to be sums of multiple zeta values. On the other hand, Belkale and Brosnan have shown [2] that the related graph motives are not in general mixed Tate. The motive associated to a graph Γ is the motive of the graph hypersurface XΓ [3]. If the cohomology of XΓ were mixed Tate, the function p 7→ #X(Fp) would be a polynomial in p. In [2] it is shown that this function can be quite general. In particular it is not always a polynomial in p (not even if one omits a finite set of p). The purpose of this note is to consider the Hodge structure associated to the Betti cohomology of XΓ. Our main tool is another variety ΛΓ which sits as a birational cover f : ΛΓ → XΓ. The variety ΛΓ has mixed Tate cohomology. As a consequence we show
منابع مشابه
A note on vague graphs
In this paper, we introduce the notions of product vague graph, balanced product vague graph, irregularity and total irregularity of any irregular vague graphs and some results are presented. Also, density and balanced irregular vague graphs are discussed and some of their properties are established. Finally we give an application of vague digraphs.
متن کاملA Note on a graph associated to a commutative ring
The rings considered in this article are commutative with identity. This article is motivated by the work on comaximal graphs of rings. In this article, with any ring $R$, we associate an undirected graph denoted by $G(R)$, whose vertex set is the set of all elements of $R$ and distinct vertices $x,y$ are joined by an edge in $G(R)$ if and only if $Rxcap Ry = Rxy$. In Section 2 of this articl...
متن کاملA Note on Tensor Product of Graphs
Let $G$ and $H$ be graphs. The tensor product $Gotimes H$ of $G$ and $H$ has vertex set $V(Gotimes H)=V(G)times V(H)$ and edge set $E(Gotimes H)={(a,b)(c,d)| acin E(G):: and:: bdin E(H)}$. In this paper, some results on this product are obtained by which it is possible to compute the Wiener and Hyper Wiener indices of $K_n otimes G$.
متن کاملINDEPENDENT SETS OF SOME GRAPHS ASSOCIATED TO COMMUTATIVE RINGS
Let $G=(V,E)$ be a simple graph. A set $Ssubseteq V$ isindependent set of $G$, if no two vertices of $S$ are adjacent.The independence number $alpha(G)$ is the size of a maximumindependent set in the graph. In this paper we study and characterize the independent sets ofthe zero-divisor graph $Gamma(R)$ and ideal-based zero-divisor graph $Gamma_I(R)$of a commutative ring $R$.
متن کاملA NOTE ON THE COMMUTING GRAPHS OF A CONJUGACY CLASS IN SYMMETRIC GROUPS
The commuting graph of a group is a graph with vertexes set of a subset of a group and two element are adjacent if they commute. The aim of this paper is to obtain the automorphism group of the commuting graph of a conjugacy class in the symmetric groups. The clique number, coloring number, independent number, and diameter of these graphs are also computed.
متن کامل