In directed network, algebraic connectivity is defined as the second smallest eigenvalue of graph Laplacian, and it captures the most conservative estimate of convergence rate and synchronicity of networked systems. In this paper, distributed estimation of algebraic connectivity of directed and connected graphs is studied using a decentralized power iteration scheme. Specifically, the proposed ...

An algebraic approach to finding all edge-weighted-colored paths within a weighted colored multidigraph is developed. Generally, the adjacency matrix represents a simple digraph and determines all paths between any two vertices, and is not readily extendable to colored multidigraphs. To bridge the gap, a conversion function is proposed to transform the original problem of searching edge-colored...

We investigate the computation of Hessian matrices via Automatic Differentiation, using a graph model and an algebraic model. The graph model reveals the inherent symmetries involved in calculating the Hessian. The algebraic model, based on Griewank and Walther’s state transformations [7], synthesizes the calculation of the Hessian as a formula. These dual points of view, graphical and algebrai...

We present fast and simple algebraic algorithms for the linear matroid parity problem and its applications. For the linear matroid parity problem, we obtain a simple randomized algorithm with running time O(mrω−1) where m and r are the number of columns and the number of rows and ω ≈ 2.3727 is the matrix multiplication exponent. This improves the O(mrω)-time algorithm by Gabow and Stallmann, an...

A set of edges of a hypergraph H is an algebraic set if its characteristic vector can be expressed as a linear combination of rows of the (node-edge) incidence matrix of H. Recently it was proven that deciding whether or not a given edge-set of H contains a non-empty algebraic set is an NP complete problem. In this paper we give a linear time algorithm to decide if a given edge-set contains a n...

Zilber's Exponential Algebraic Closedness conjecture (also known as Nullstellensatz) gives conditions under which a complex algebraic variety should intersect the graph of exponential map semiabelian variety. We prove special case where has dominant projection to domain map, for abelian varieties and tori. Furthermore, in situation intersection is 0-dimensional, we exhibit structure by parametr...

We propose a class of hierarchical graphs equipped with a simple algebraic syntax as a convenient way to describe the configurations of languages equipped with inherently hierarchical features such as sessions, fault-handling scopes, transactions or locations. The graph syntax can be seen as an intermediate representation language, that facilitates the encoding of algebraic specifications and, ...