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We study the independence structure of finitely exchangeable distributions over random vectors and random networks. In particular, we provide necessary and sufficient conditions for an exchangeable vector so that its elements are completely…
We consider the problem of devising algorithms to count exactly the number of independent sets of a graph G . We show that there is a polynomial time algorithm for this problem when G is restricted to the class of strongly orderable graphs,…
We prove the unexpected result that almost uniform sampling of independent sets in graphs is possible via a probabilistic polynomial time algorithm. Note that our sampling algorithm (if correct) has extremely surprising consequences; the…
The notion of a Riordan graph was introduced recently, and it is a far-reaching generalization of the well-known Pascal graphs and Toeplitz graphs. However, apart from a certain subclass of Toeplitz graphs, nothing was known on independent…
We generalize two main theorems of matching polynomials of undirected simple graphs, namely, real-rootedness and the Heilmann-Lieb root bound. Viewing the matching polynomial of a graph $G$ as the independence polynomial of the line graph…
The graph theoretic concept of maximal independent set arises in several practical problems in computer science as well as in game theory. A maximal independent set is defined by the set of occupied nodes that satisfy some packing and…
We describe algorithms, based on Avis and Fukuda's reverse search paradigm, for listing all maximal independent sets in a sparse graph in polynomial time and delay per output. For bounded degree graphs, our algorithms take constant time per…
This work presents a graph neural network (GNN) framework for solving the maximum independent set (MIS) problem, inspired by dynamic programming (DP). Specifically, given a graph, we propose a DP-like recursive algorithm based on GNNs that…
We prove the existence of an effective universal upper bound for the order of any integral periodic orbit of any integral algebraic dynamical system in a fixed ambient space. Using this, we demonstrate the decidability of periodicity in…
We use methods from algebra and discrete geometry to study the irreducibility of the dispersion polynomial of a discrete periodic operator associated to a periodic graph after changing the period lattice. We provide numerous applications of…
This paper studies a class of binomial ideals associated to graphs with finite vertex sets. They generalize the binomial edge ideals, and they arise in the study of conditional independence ideals. A Gr\"obner basis can be computed by…
In this paper we generalize the involutive methods and algorithms devised for polynomial ideals to differential ones generated by a finite set of linear differential polynomials in the differential polynomial ring over a zero characteristic…
In [1] Peters and Regts confirmed a conjecture by Sokal by showing that for every $\Delta \in \mathbb{Z}_{\geq 3}$ there exists a complex neighborhood of the interval $\left[0, \frac{\left(\Delta - 1\right)^{\Delta -…
We consider graphs on monomials in $n$ variables of a fixed degree $d$ where two monomials are adjacent if and only if their least common multiple has degree $d+1$. We prove that when $n = 3$ and $d$ is divisible by $3$ as well as when…
The Havel-Hakimi algorithm iteratively reduces the degree sequence of a graph to a list of zeroes. As shown by Favaron, Mah\'eo, and Sacl\'e, the number of zeroes produced, known as the residue, is a lower bound on the independence number…
Generalising the Heilman-Lieb Theorem from statistical physics, Chudnovsky and Seymour [J. Combin. Theory Ser. B, 97(3):350--357] showed that the univariate independence polynomial of any claw-free graph has all of its zeros on the negative…
We construct several pairwise-incomparable bounds on the projective dimensions of edge ideals. Our bounds use combinatorial properties of the associated graphs; in particular we draw heavily from the topic of dominating sets. Through…
A $k$-independent set in a connected graph is a set of vertices such that any two vertices in the set are at distance greater than $k$ in the graph. The $k$-independence number of a graph, denoted $\alpha_k$, is the size of a largest…
We study an infinite class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. This generalizes a sequence of sparse polynomials which arises in a natural way as graph theoretic…
This paper studies infinite graphs produced from a natural unfolding operation applied to finite graphs. Graphs produced via such operations are of finite degree and automatic over the unary alphabet (that is, they can be described by…