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The complexity of graph homomorphisms has been a subject of intense study [11, 12, 4, 42, 21, 17, 6, 20]. The partition function $Z_{\mathbf A}(\cdot)$ of graph homomorphism is defined by a symmetric matrix $\mathbf A$ over $\mathbb C$. We…
In high-dimensional graph learning problems, some topological properties of the graph, such as bounded node degree or tree structure, are typically assumed to hold so that the sample complexity of recovering the graph structure can be…
Dictionary learning is a key tool for representation learning, that explains the data as linear combination of few basic elements. Yet, this analysis is not amenable in the context of graph learning, as graphs usually belong to different…
Graph parameters such as the clique number, the chromatic number, and the independence number are central in many areas, ranging from computer networks to linguistics to computational neuroscience to social networks. In particular, the…
The bandwidth of a graph G is the minimum of the maximum difference between adjacent labels when the vertices have distinct integer labels. We provide a polynomial algorithm to produce an optimal bandwidth labeling for graphs in a special…
Graphical models capture relations between entities in a wide range of applications including social networks, biology, and natural language processing, among others. Graph neural networks (GNN) are neural models that operate over graphs,…
We consider graph representation learning in a self-supervised manner. Graph neural networks (GNNs) use neighborhood aggregation as a core component that results in feature smoothing among nodes in proximity. While successful in various…
Many real-world networks are inherently decentralized. For example, in social networks, each user maintains a local view of a social graph, such as a list of friends and her profile. It is typical to collect these local views of social…
Can neural networks learn to compare graphs without feature engineering? In this paper, we show that it is possible to learn representations for graph similarity with neither domain knowledge nor supervision (i.e.\ feature engineering or…
The quadratic embedding constant (QEC) of a graph $G$ is a new numeric invariant, which is defined in terms of the distance matrix and is denoted by $\mathrm{QEC}(G)$. By observing graph structure of the maximal cliques (clique graph), we…
In a seminal paper, Erdos and Renyi identified the threshold for connectivity of the random graph G(n,p). In particular, they showed that if p >> log(n)/n then G(n,p) is almost always connected, and if p << log(n)/n then G(n,p) is almost…
We study unit disk visibility graphs, where the visibility relation between a pair of geometric entities is defined by not only obstacles, but also the distance between them. That is, two entities are not mutually visible if they are too…
Graph Contrastive Learning (GCL) has emerged as a powerful paradigm for training Graph Neural Networks (GNNs) in the absence of task-specific labels. However, its scalability on large-scale graphs is hindered by the intensive message…
Two graphs $G$ and $H$ are homomorphism indistinguishable over a class of graphs $\mathcal{F}$ if for all graphs $F \in \mathcal{F}$ the number of homomorphisms from $F$ to $G$ is equal to the number of homomorphisms from $F$ to $H$. Many…
A graph H is a vertex-minor of a graph G if it can be reached from G by the successive application of local complementations and vertex deletions. Vertex-minors have been the subject of intense study in graph theory over the last decades…
We investigate graph based secret sharing schemes and its information ratio, also called complexity, measuring the maximal amount of information the vertices has to store. It was conjectured that in large girth graphs, where the interaction…
We introduce and investigate the solvable graph $\Gamma_\mathfrak{S}(L)$ of a finite-dimensional Lie algebra $L$ over a field $F$. The vertices are the elements outside the solvabilizer $\sol(L)$, and two vertices are adjacent whenever they…
Many scientific datasets are of high dimension, and the analysis usually requires visual manipulation by retaining the most important structures of data. Principal curve is a widely used approach for this purpose. However, many existing…
We study neural architectures in which each hidden layer is defined by the stationary state of a dissipative Schr\"odinger-type dynamics on a learned latent graph. On stable branches, the local stationary problem defines a differentiable…
In this paper we introduce the graph $\Gamma_{sc}(G)$ associated with a group $G$, called the solvable conjugacy class graph (abbreviated as SCC-graph), whose vertices are the nontrivial conjugacy classes of $G$ and two distinct conjugacy…