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Graph machine learning architectures are typically tailored to specific tasks on specific datasets, which hinders their broader applicability. This has led to a new quest in graph machine learning: how to build graph foundation models…
In this paper we resolve the complexity of the isomorphism problem on all but finitely many of the graph classes characterized by two forbidden induced subgraphs. To this end we develop new techniques applicable for the structural and…
The computational complexity of the graph isomorphism problem is considered to be a major open problem in theoretical computer science. It is known that testing isomorphism of chordal graphs is polynomial-time equivalent to the general…
In a Euclidean Jordan algebra V of rank n which carries the trace inner product, to each element x we associate the eigenvalue vector whose components are the eigenvalues of x written in the decreasing order. For any number p between (and…
Graph transformers typically embed every node in a single Euclidean space, blurring heterogeneous topologies. We prepend a lightweight Riemannian mixture-of-experts layer that routes each node to various kinds of manifold, mixture of…
The graph fractional Fourier transform (GFRFT) for unitary graph Fourier transform (GFT) matrices can be interpreted through the scalar function $e^{j\alpha\theta}$ on the unit circle. Under the principal branch, its Fourier-series…
Let $(X,E_X)$ and $(V,E_V)$ be finite connected graphs without loops. We assume that $V$ has two distinguished vertices $a,b$ and an automorphism $\gamma$ which exchanges $a$ and~$b$. The $V$-edge substitution of $X$ is the graph $X[V]$…
Deep learning models have achieved huge success in numerous fields, such as computer vision and natural language processing. However, unlike such fields, it is hard to apply traditional deep learning models on the graph data due to the…
Two graphs are homomorphism indistinguishable over a graph class $\mathcal{F}$, denoted by $G \equiv_{\mathcal{F}} H$, if $\operatorname{hom}(F,G) = \operatorname{hom}(F,H)$ for all $F \in \mathcal{F}$ where $\operatorname{hom}(F,G)$…
We introduce a hypergraph matrix, named the unified matrix, and use it to represent the hypergraph as a graph. We show that the unified matrix of a hypergraph is identical to the adjacency matrix of the associated graph. This enables us to…
We examine the capacity of the complementarity spectrum to distinguish non-isomorphic digraphs. We focus on the seven families with exactly three complementarity eigenvalues. Our findings reveal that in some, but not all families, any two…
We compare the spectral properties of two kinds of linear operators characterizing the (classical) geodesic flow and its quantization on connected locally finite graphs without dead ends. The first kind are transfer operators acting on…
We solve the subgraph isomorphism problem in planar graphs in linear time, for any pattern of constant size. Our results are based on a technique of partitioning the planar graph into pieces of small tree-width, and applying dynamic…
Consider the collection of edge bicolorings of a graph that is cellularly embedded on an orientable surface. In this work, we count the number of equivalence classes of such colorings under two relations: reversing colors around a face and…
A function $f$ of a graph is called a complete graph invariant if the isomorphism of graphs $G$ and $H$ is equivalent to the equality $f(G)=f(H)$. If, in addition, $f(G)$ is a graph isomorphic to $G$, then $f$ is called a canonical form for…
Let E be a rank two vector bundle on a scheme X. The following three structures are shown to be equivalent : a) A primitive quadratic map q: E --> L, with values in an invertible module L. b) A double covering f: Y --> X endowed with an…
This article studies the set of R-equivalence classes of the group of proper projective similitudes of an algebra with involution of the first kind. The main results concern base fields of characteristic different from 2 over which every…
We present several new examples of reflection principles which apply to both class groups of number fields and picard groups of of curves over $\mathbb{P}^{1}/\mathbb{F}_{p}$. This proves a conjecture of Lemmermeyer about equality of 2-rank…
In this draft paper, we introduce a novel architecture for graph networks which is equivariant to the Euclidean group in $n$-dimensions. The model is designed to work with graph networks in their general form and can be shown to include…
The overlap graphs of subtrees of a tree are equivalent to subtree filament graphs, the overlap graphs of subtrees of a star are cocomparability graphs, and the overlap graphs of subtrees of a caterpillar are interval filament graphs. In…