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We consider a class of linear programs on graphs with total variation regularization and a budgetary constraint. For these programs, we give a characterization of basic solutions in terms of rooted spanning forests with orientation on the…
Embedding the vertices of arbitrary graphs into trees while minimizing some measure of overlap is an important problem with applications in computer science and physics. In this work, we consider the problem of bijectively embedding the…
It is conjectured that the Laplacian spectrum of a graph is majorized by its conjugate degree sequence. In this paper, we prove that this majorization holds for a class of graphs including trees. We also show that a generalization of this…
(DRAFT VERSION) In this article we present a proof of the famous Kirchoff's Matrix-Tree theorem, which relates the number of spanning trees in a connected graph with the cofactors (and eigenvalues) of its combinatorial Laplacian matrix.…
Sparse structures are frequently sought when pursuing tractability in optimization problems. They are exploited from both theoretical and computational perspectives to handle complex problems that become manageable when sparsity is present.…
Graph sparsification aims to reduce the number of edges of a graph while maintaining its structural properties. In this paper, we propose the first general and effective information-theoretic formulation of graph sparsification, by taking…
We present a simple spectral approach to the well-studied constrained clustering problem. It captures constrained clustering as a generalized eigenvalue problem with graph Laplacians. The algorithm works in nearly-linear time and provides…
We study the Laplacian on family preserving metric graphs. These are graphs that have a certain symmetry that, as we show, allows for a decomposition into a direct sum of one-dimensional operators whose properties are explicitly related to…
We study the crossing-minimization problem in a layered graph drawing of planar-embedded rooted trees whose leaves have a given total order on the first layer, which adheres to the embedding of each individual tree. The task is then to…
We introduce a family of post-critically finite fractal trees indexed by the number of branches they possess. Then we produce a Laplacian operator on graph approximations to these fractals and use spectral decimation to describe the…
We present an algorithm that enumerates all the minimal triangulations of a graph in incremental polynomial time. Consequently, we get an algorithm for enumerating all the proper tree decompositions, in incremental polynomial time, where…
Contour trees offer an abstract representation of the level set topology in scalar fields and are widely used in topological data analysis and visualization. However, applying contour trees to large-scale scientific datasets remains…
The spectrum of the normalized graph Laplacian yields a very comprehensive set of invariants of a graph. In order to understand the information contained in those invariants better, we systematically investigate the behavior of this…
Treewidth is a graph parameter that plays a fundamental role in several structural and algorithmic results. We study the problem of decomposing a given graph $G$ into node-disjoint subgraphs, where each subgraph has sufficiently large…
We consider the class of simple graphs with large algebraic connectivity (the second-smallest eigenvalue of the Laplacian matrix). For this class of graphs we determine the asymptotic behavior of the number of Eulerian orientations. In…
We study the evolution of graphs densifying by adding edges: Two vertices are chosen randomly, and an edge is (i) established if each vertex belongs to a tree; (ii) established with probability $p$ if only one vertex belongs to a tree;…
Random spanning trees of a graph $G$ are governed by a corresponding probability mass distribution (or "law"), $\mu$, defined on the set of all spanning trees of $G$. This paper addresses the problem of choosing $\mu$ in order to utilize…
High-dimensional graphical models are often estimated using regularization that is aimed at reducing the number of edges in a network. In this work, we show how even simpler networks can be produced by aggregating the nodes of the graphical…
We use two variational techniques to prove upper bounds for sums of the lowest several eigenvalues of matrices associated with finite, simple, combinatorial graphs. These include estimates for the adjacency matrix of a graph and for both…
The evolution of aligned DNA sequence sites is generally modeled by a Markov process operating along the edges of a phylogenetic tree. It is well known that the probability distribution on the site patterns at the tips of the tree…