Related papers: Regularity in Weighted Graphs: A Symmetric Functio…

Motivated by the question of how macromolecules assemble, the notion of an {\it assembly tree} of a graph is introduced. Given a graph $G$, the paper is concerned with enumerating the number of assembly trees of $G$, a problem that applies…

A realization of a graph $G=(V,E)$ is a map $v\colon V\to\Bbb R^d$ that assigns to each vertex a point in $d$-dimensional Euclidean space. We study graph realizations from the perspective of representation theory (expressing certain…

In a previous work we introduced an elementary method to analyze the periodicity of a generating function defined by a single equation y=G(x,y). This was based on deriving a single set-equation Y = Gammma(Y) defining the spectrum of the…

A key feature of neural network architectures is their ability to support the simultaneous interaction among large numbers of units in the learning and processing of representations. However, how the richness of such interactions trades off…

Any homogeneous polynomial $P(x, y, z)$ of degree $d$, being restricted to a unit sphere $S^2$, admits essentially a unique representation of the form $\lambda + \sum_{k = 1}^d [\prod_{j = 1}^k L_{kj}]$, where $L_{kj}$'s are linear forms in…

We introduce and study a mathematical framework for a broad class of regularization functionals for ill-posed inverse problems: Regularization Graphs. Regularization graphs allow to construct functionals using as building blocks linear…

A graph is $k$-planar if it can be drawn in the plane such that no edge is crossed more than $k$ times. While for $k=1$, optimal $1$-planar graphs, i.e., those with $n$ vertices and exactly $4n-8$ edges, have been completely characterized,…

Hypergraphs are structures that can be decomposed or described; in other words they are recursively countable. Here, we get exact and asymptotic enumeration results on hypergraphs by means of exponential generating functions. The number of…

We extend the notion of nonbacktracking walks from unweighted graphs to graphs whose edges have a nonnegative weight. Here the weight associated with a walk is taken to be the product over the weights along the individual edges. We give two…

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…

Using the technique of the metrization theorem of uniformities with countable bases, in this note we provide, test and compare an explicit algorithm to produce a metric $d(x,y)$ between the vertices $x$ and $y$ of an affinity weighted…

The generation of random graphs using edge swaps provides a reliable method to draw uniformly random samples of sets of graphs respecting some simple constraints, e.g. degree distributions. However, in general, it is not necessarily…

There exist very lucid explanations of the combinatorial origins of rational and algebraic functions, in particular with respect to regular and context free languages. In the search to understand how to extend these natural correspondences,…

We consider a Random Graph Model on $\mathbb{Z}^{d}$ that incorporates the interplay between the statistics of the graph and the underlying space where the vertices are located. Based on a graphical construction of the model as the…

For an edge-ordered graph $G$, we say that an $n$-vertex edge-ordered graph $H$ is $G$-saturated if it is $G$-free and adding any new edge with any new label to $H$ introduces a copy of $G$. The saturation function describes the minimum…

Following Penrose, we introduce a family of graph functions defined in terms of contractions of certain products of symmetric tensors along the edges of a graph. Special cases of these functions enumerate edge colorings and cycles of…

Consider the normalized adjacency matrices of random $d$-regular graphs on $N$ vertices with fixed degree $d\geq3$. We prove that, with probability $1-N^{-1+{\varepsilon}}$ for any ${\varepsilon} >0$, the following two properties hold as $N…

The $k$-th symmetric product of a graph $G$ with vertex set $V$ with edge set $E$ is a graph with vertices as $k$-sets of $V$, where two $k$-sets are connected by an edge if and only if their symmetric difference is an edge in $E$. Using…

Graph product structure theory expresses certain graphs as subgraphs of the strong product of much simpler graphs. In particular, an elegant formulation for the corresponding structural theorems involves the strong product of a path and of…

Graph editing problems offer an interesting perspective on sub- and supergraph identification problems for a large variety of target properties. They have also attracted significant attention in recent years, particularly in the area of…