This paper develops an algebraic multigrid preconditioner for the graph Laplacian. The proposed approach uses aggressive coarsening based on the aggregation framework in the setup phase and a polynomial smoother with sufficiently large degree within a (nonlinear) Algebraic Multilevel Iteration as a preconditioner to the flexible Conjugate Gradient iteration in the solve phase. We show that by c...

We present an extremely fast graph drawing algorithm for very large graphs, which we term ACE (for Algebraic multigrid Computation of Eigenvectors). ACE exhibits a vast improvement over the fastest algorithms we are currently aware of; it draws graphs of millions of nodes in less than a minute. ACE finds an optimal drawing by minimizing a quadratic energy function. The minimization problem is e...

Many well-known graph drawing techniques, including force directed drawings, spectral graph layouts, multidimensional scaling, and circle packings, have algebraic formulations. However, practical methods for producing such drawings ubiquitously use iterative numerical approximations rather than constructing and then solving algebraic expressions representing their exact solutions. To explain th...

Let G be a connected graph of order n. The algebraic connectivity of G is the second smallest eigenvalue of the Laplacian matrix of G. A dominating set in G is a vertex subset S such that each vertex of G that is not in S is adjacent to a vertex in S. The least cardinality of a dominating set is the domination number. In this paper, we prove a sharp upper bound on the algebraic connectivity of ...

Let G be a weighted graph. Let v be a vertex of G and let Gω denote the graph obtained by adding a vertex u and an edge {v, u} with weight ω to G. Then the algebraic connectivity μ(Gω) of G v ω is a nondecreasing function of ω and is bounded by the algebraic connectivity μ(G) of G. The question of when lim ω→∞ v ω) is equal to μ(G) is considered and answered in the case that G is a tree.

The study of graph vertex colorability from an algebraic perspective has introduced novel techniques and algorithms into the field. For instance, it is known that k-colorability of a graph G is equivalent to the condition 1 ∈ IG,k for a certain ideal IG,k ⊆ k[x1, . . . , xn]. In this paper, we extend this result by proving a general decomposition theorem for IG,k . This theorem allows us to giv...

We previously defined collagories essentially as “distributive allegories without zero morphisms”. Collagories are sufficient for accommodating the relation-algebraic approach to graph transformation, and closely correspond to the adhesive categories important for the categorical DPO approach to graph transformation. Heindel and Sobociński have recently characterised the Van-Kampen colimits use...

Several cryptographic methods have been developed based on the difficulty to determine the set of solutions of a polynomial system over a given field. We build a polynomial ideal whose algebraic set is related to the set of isomorphisms between two graphs. The problem isomorphism, posed in the context of Graph Theory, has been extensively used in zero knowledge authentication protocols. Thus, a...

The paper investigates relationship between algebraic expressions and Fibonacci graphs (which give a generic example of non-series-parallel graphs). We propose the uniform generalized decomposition method for constructing Fibonacci graph expressions. On every step this method divides the graph of size n into k parts of the same size. We prove that to reach the smallest possible length of the co...

We present an extremely fast graph drawing algorithm for very large graphs, which we term ACE (for Algebraic multigrid Computation of Eigenvectors). ACE exhibits a vast improvement over the fastest algorithms we are currently aware of; using a serial PC, it draws graphs of millions of nodes in less than a minute. ACE finds an optimal drawing by minimizing a quadratic energy function. The minimi...