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DAG models are statistical models satisfying a collection of conditional independence relations encoded by the nonedges of a directed acyclic graph (DAG) $\mathcal{G}$. Such models are used to model complex cause-effect systems across a…

Let $I(G;x)$ denote the independence polynomial of a graph $G$. In this paper we study the unimodality properties of $I(G;x)$ for some composite graphs $G$. Given two graphs $G_1$ and $G_2$, let $G_1[G_2]$ denote the lexicographic product…

The pivotal quality of proximity graphs is connectivity, i.e. all nodes in the graph are connected to one another either directly or via intermediate nodes. These types of graphs are robust, i.e., they are able to function well even if they…

These notes concern aspects of various graphs whose vertex set is a group $G$ and whose edges reflect group structure in some way (so that they are invariant under the action of the automorphism group of $G$). The graphs I will discuss are…

We study the zero sets of the independence polynomial on recursive sequences of graphs. We prove that for a maximally independent starting graph and a stable and expanding recursion algorithm, the zeros of the independence polynomial are…

Graphical network inference is used in many fields such as genomics or ecology to infer the conditional independence structure between variables, from measurements of gene expression or species abundances for instance. In many practical…

We introduce a novel class of labeled directed acyclic graph (LDAG) models for finite sets of discrete variables. LDAGs generalize earlier proposals for allowing local structures in the conditional probability distribution of a node, such…

We present a detailed analytical study of a paradigmatic scale-free network model, the Static Model. Analytical expressions for its main properties are derived by using the hidden variables formalism. We map the model into a canonic hidden…

Large-scale analysis of the distributions of the network graphs observed in naturally-occurring phenomena has revealed that the degrees of such graphs follow a power-law or lognormal distribution. Seshadhri, Pinar, and Kolda (J. ACM, 2013)…

Petersen's seminal work in 1891 asserts that the edge-set of a cubic graph can be covered by distinct perfect matchings if and only if it is bridgeless. Actually, it is known that for a very large fraction of bridgeless cubic graphs, every…

The independence gap of a graph was introduced by Ekim et al. (2018) as a measure of how far a graph is from being well-covered. It is defined as the difference between the maximum and minimum size of a maximal independent set. We…

Why do many modern neural-network-based graph generative models fail to reproduce typical real-world network characteristics, such as high triangle density? In this work we study the limitations of edge independent random graph models, in…

Learning graphs from sets of nodal observations represents a prominent problem formally known as graph topology inference. However, current approaches are limited by typically focusing on inferring single networks, and they assume that…

We consider distributed model-checking of Monadic Second-Order logic (MSO) on graphs which constitute the topology of communication networks. The graph is thus both the structure being checked and the system on which the distributed…

Random K-out graphs are garnering interest in designing distributed systems including secure sensor networks, anonymous crypto-currency networks, and differentially-private decentralized learning. In these security-critical applications, it…

Real-world social and economic networks typically display a number of particular topological properties, such as a giant connected component, a broad degree distribution, the small-world property and the presence of communities of densely…

Each graphon $W:\Omega^2\rightarrow[0,1]$ yields an inhomogeneous random graph model $G(n,W)$. We show that $G(n,W)$ is asymptotically almost surely connected if and only if (i) $W$ is a connected graphon and (ii) the measure of elements of…

Let $H$ be a fixed graph. What can be said about graphs $G$ that have no subgraph isomorphic to a subdivision of $H$? Grohe and Marx proved that such graphs $G$ satisfy a certain structure theorem that is not satisfied by graphs that…

An isolating set in a graph $G$ is a set $S$ of vertices such that removing $S$ and its neighborhood leaves no edge. The isolation number $\iota(G)$ of $G$ (also known as the vertex-edge domination number) is the minimum size among all…

While Graph Neural Networks (GNNs) have recently become the de facto standard for modeling relational data, they impose a strong assumption on the availability of the node or edge features of the graph. In many real-world applications,…