Related papers: Identities between Modular Graph Forms
In this paper we explore the connection between the ranks of the magnitude homology groups of a graph and the structure of its subgraphs. To this end, we introduce variants of magnitude homology called eulerian magnitude homology and…
Pairs of graded graphs, together with the Fomin property of graded graph duality, are rich combinatorial structures providing among other a framework for enumeration. The prototypical example is the one of the Young graded graph of integer…
We extend the notion of graph homomorphism to cellularly embedded graphs (maps) by designing operations on vertices and edges that respect the surface topology; we thus obtain the first definition of map homomorphism that preserves both the…
Clustering heterogeneous relational data remains a central challenge in graph learning, particularly when interactions involve more than two types of entities. While differentiable modularity objectives such as DMoN have enabled end-to-end…
This paper studies modular forms of rank four and level one. There are two possiblities for the isomorphism type of the space of modular forms that can arise from an irreducible representation of the modular group of rank four, and we…
There are good arguments to support the claim that deep neural networks (DNNs) capture better feature representations than the previous hand-crafted feature engineering, which leads to a significant performance improvement. In this paper,…
An arbitrary Feynman graph for string field theory interactions is analysed and the homeomorphism type of the corresponding world sheet surface is completely determined even in the non-orientable cases. Algorithms are found to mechanically…
An earlier work of the author's showed that it was possible to adapt the Alekseev-Meinrenken Chern-Weil proof of the Duflo isomorphism to obtain a completely combinatorial proof of the Wheeling isomorphism. That work depended on a certain…
This article approaches the counting of subgraphs, in terms of signature-type functionals defined over combinatorial Hopf algebras of graphs. Well-known algebraic identities that arise in the context of counting subgraphs are then captured…
We consider constrained variants of graph homomorphisms such as embeddings, monomorphisms, full homomorphisms, surjective homomorpshims, and locally constrained homomorphisms. We also introduce a new variation on this theme which derives…
We continue the study of non-commutative operator graphs generated by resolutions of identity covariant with respect to unitary actions of the circle group and the Heisenber-Weyl group as well. It is shown that the graphs generated by the…
Let ${\mathbb{D}}^{m\times n}$ be the set of $m\times n$ matrices over a division ring $\mathbb{D}$. Two matrices $A,B\in {\mathbb{D}}^{m\times n}$ are adjacent if ${\rm rank}(A-B)=1$. By the adjacency, ${\mathbb{D}}^{m\times n}$ is a…
The problem of finding graph structure of functions commuting with a given function in terms of their functional graphs is considered. Structure of functional graphs of commuting functions is described. The problem is reduced to describing…
Generalized Halphen systems are solved in terms of functions that uniformize genus zero Riemann surfaces, with automorphism groups that are commensurable with the modular group. Rational maps relating these functions imply subgroup…
It is by now well established that, by means of the integration by part identities, all the integrals occurring in the evaluation of a Feynman graph of given topology can be expressed in terms of a few independent master integrals. It is…
Molecular graphs generally contain subgraphs (known as groups) that are identifiable and significant in composition, functionality, geometry, etc. Flat latent representations (node embeddings or graph embeddings) fail to represent, and…
The Feynman rules assign to every graph an integral which can be written as a function of a scaling parameter L. Assuming L for the process under consideration is very small, so that contributions to the renormalizaton group are small, we…
Context: Edge graphs are graphs whose edges are labelled with identifiers, and nodes can have multiple edges between them. They are used to model a wide range of systems, including networks with distances or degrees of connection and…
Motivated by discrete Laplacian differential operators with various accuracy orders in numerical analysis, we introduce new matrices attached to a simple graph that can be considered graph Laplacians with higher accuracy. In particular, we…
The aim of this article is to give a concise algebraic treatment of the modular symbols formalism, generalised from modular curves to Hecke triangle surfaces. A sketch is included of how the modular symbols formalism gives rise to the…