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We determine all graphs whose adjacency matrix has at most two eigenvalues (multiplicities included) different from $\pm 1$ and decide which of these graphs are determined by their spectrum. This includes the so-called friendship graphs,…
We introduce a new class of countably infinite random geometric graphs, whose vertices are points in a metric space, and vertices are adjacent independently with probability p if the metric distance between the vertices is below a given…
A theory of orientation on gain graphs (voltage graphs) is developed to generalize the notion of orientation on graphs and signed graphs. Using this orientation scheme, the line graph of a gain graph is studied. For a particular family of…
Quantum graphs have attracted attention from mathematicians for some time. A quantum graph is defined by having a Laplacian on each edge of a metric graph and imposing boundary conditions at the vertices to get an eigenvalue problem. A…
This paper consists of a few results, discovered and proved during the 2012-2013 research group at Eastern Oregon University. Inertia tables are a visual representation of the possible inertias of a given graph. The inertia of a graph…
The Lipschitz space of an infinite (locally-finite) graph is defined as the set of functions on the vertices of the graph such that the differences of the values between adjacent vertices remain bounded. In this paper we prove that this set…
In this article we give an algorithm for determining the generators and relations for the rings of semi-invariant functions on irreducible components of representation spaces for gentle string algebras. These rings of semi-invariants turn…
Weight-equitable partitions of graphs, which are a natural extension of the well-known equitable partitions, have been shown to be a powerful tool to weaken the regularity assumption in several well-known eigenvalue bounds. In this work we…
Data analysts commonly utilize statistics to summarize large datasets. While it is often sufficient to explore only the summary statistics of a dataset (e.g., min/mean/max), Anscombe's Quartet demonstrates how such statistics can be…
In this paper we establish spectral comparison results for Schr\"odinger operators on a certain class of infinite quantum graphs, using recent results obtained in the finite setting. We also show that new features do appear on infinite…
Random graphs are more and more used for modeling real world networks such as evolutionary networks of proteins. For this purpose we look at two different models and analyze how properties like connectedness and degree distributions are…
Recently, Bollob\'as, Janson and Riordan introduced a family of random graph models producing inhomogeneous graphs with $n$ vertices and $\Theta(n)$ edges whose distribution is characterized by a kernel, i.e., a symmetric measurable…
Let $\XX$ be a compact, smooth, connected, Riemannian manifold without boundary, $G:\XX\times\XX\to \RR$ be a kernel. Analogous to a radial basis function network, an eignet is an expression of the form $\sum_{j=1}^M a_jG(\circ,y_j)$, where…
We consider matrices on infinite trees which are universal covers of Jacobi matrices on finite graphs. We are interested in the question of the existence of sequences of finite covers whose normalized eigenvalue counting measures converge…
Graphons are symmetric measurable functions that arise from a sequence of graphs. A graphon variety is the a set of all graphons defined by a condition of the form $t(g, W) = 0$ for a fixed quantum graph $g$, where $t(.,.)$ is the…
Constructing the adjacency graph is fundamental to graph-based clustering. Graph learning in kernel space has shown impressive performance on a number of benchmark data sets. However, its performance is largely determined by the chosen…
We present sharp inequalities relating the number of vertices, edges, and triangles of a graph to the smallest eigenvalue of its adjacency matrix and the largest eigenvalue of its Laplacian.
Infinite graphs are finitary in the sense that their points are connected via finite paths. So what would an infinitary generalization of finite graphs look like? Usually this question is answered with the aid of topology, e.g. in the case…
Consider a random regular graph of fixed degree $d$ with $n$ vertices. We study spectral properties of the adjacency matrix and of random Schr\"odinger operators on such a graph as $n$ tends to infinity. We prove that the integrated density…
We introduce a family of multilayer graph kernels and establish new links between graph convolutional neural networks and kernel methods. Our approach generalizes convolutional kernel networks to graph-structured data, by representing…