The algebra of flows in graphs

We define a contravariant functor K from the category of finite graphs and graph morphisms to the category of finitely generated graded abelian groups and homomorphisms. For a graph X, an abelian group B, and a nonnegative integer j, an element of Hom(K^j(X),B) is a coherent family of B-valued flows on the set of all graphs obtained by contracting some (j-1)-set of edges of X; in particular, Hom(K^1(X),R) is the familiar (real) ``cycle-space'' of X. We show that K(X) is torsion-free and that its Poincare polynomial is the specialization t^{n-k}T_X(1/t,1+t) of the Tutte polynomial of X (here X has n vertices and k components). Functoriality of K induces a functorial coalgebra structure on K(X); dualizing, for any ring B we obtain a functorial B-algebra structure on Hom(K(X),B). When B is commutative we present this algebra as a quotient of a divided power algebra, leading to some interesting inequalities on the coefficients of the above Poincare polynomial. We also provide a formula for the theta function of the lattice of integer-valued flows in X, and conclude with ten open problems.Comment: 31 pages, 1 figur

Relationship-based access control: its expression and enforcement through hybrid logic

Access control policy is typically de ned in terms of attributes, but in many applications it is more natural to de- ne permissions in terms of relationships that resources, systems, and contexts may enjoy. The paradigm of relationshipbased access control has been proposed to address this issue, and modal logic has been used as a technical foundation. We argue here that hybrid logic { a natural and wellestablished extension of modal logic { addresses limitations in the ability of modal logic to express certain relationships. Also, hybrid logic has advantages in the ability to e ciently compute policy decisions relative to a relationship graph. We identify a fragment of hybrid logic to be used for expressing relationship-based access-control policies, show that this fragment supports important policy idioms, and study its expressiveness. We also capture the previously studied notion of relational policies in a static type system. Finally, we point out that use of our hybrid logic removes an exponential penalty in existing attempts of specifying complex relationships such as \at least three friends"

An Algebra of Hierarchical Graphs and its Application to Structural Encoding

We define an algebraic theory of hierarchical graphs, whose axioms characterise graph isomorphism: two terms are equated exactly when they represent the same graph. Our algebra can be understood as a high-level language for describing graphs with a node-sharing, embedding structure, and it is then well suited for defining graphical representations of software models where nesting and linking are key aspects. In particular, we propose the use of our graph formalism as a convenient way to describe configurations in process calculi equipped with inherently hierarchical features such as sessions, locations, transactions, membranes or ambients. The graph syntax can be seen as an intermediate representation language, that facilitates the encodings of algebraic specifications, since it provides primitives for nesting, name restriction and parallel composition. In addition, proving soundness and correctness of an encoding (i.e. proving that structurally equivalent processes are mapped to isomorphic graphs) becomes easier as it can be done by induction over the graph syntax

Forbidden Subgraphs in Connected Graphs

Given a set $\xi=\{H_1,H_2,...\}$ of connected non acyclic graphs, a $\xi$-free graph is one which does not contain any member of $$ as copy. Define the excess of a graph as the difference between its number of edges and its number of vertices. Let {\gr{W}}_{k,\xi} be theexponential generating function (EGF for brief) of connected $\xi$-free graphs of excess equal to $k$ ($k \geq 1$). For each fixed $\xi$, a fundamental differential recurrence satisfied by the EGFs {\gr{W}}_{k,\xi} is derived. We give methods on how to solve this nonlinear recurrence for the first few values of $k$ by means of graph surgery. We also show that for any finite collection $\xi$ of non-acyclic graphs, the EGFs {\gr{W}}_{k,\xi} are always rational functions of the generating function, $T$, of Cayley's rooted (non-planar) labelled trees. From this, we prove that almost all connected graphs with $n$ nodes and $n+k$ edges are $\xi$-free, whenever $k=o(n^{1/3})$ and $|\xi| < \infty$ by means of Wright's inequalities and saddle point method. Limiting distributions are derived for sparse connected $\xi$-free components that are present when a random graph on $n$ nodes has approximately $\frac{n}{2}$ edges. In particular, the probability distribution that it consists of trees, unicyclic components, $...$, $(q+1)$-cyclic components all $\xi$-free is derived. Similar results are also obtained for multigraphs, which are graphs where self-loops and multiple-edges are allowed

Finding branch-decompositions of matroids, hypergraphs, and more

Given $n$ subspaces of a finite-dimensional vector space over a fixed finite field $\mathcal F$, we wish to find a "branch-decomposition" of these subspaces of width at most $k$, that is a subcubic tree $T$ with $n$ leaves mapped bijectively to the subspaces such that for every edge $e$ of $T$, the sum of subspaces associated with leaves in one component of $T-e$ and the sum of subspaces associated with leaves in the other component have the intersection of dimension at most $k$. This problem includes the problems of computing branch-width of $\mathcal F$-represented matroids, rank-width of graphs, branch-width of hypergraphs, and carving-width of graphs. We present a fixed-parameter algorithm to construct such a branch-decomposition of width at most $k$, if it exists, for input subspaces of a finite-dimensional vector space over $\mathcal F$. Our algorithm is analogous to the algorithm of Bodlaender and Kloks (1996) on tree-width of graphs. To extend their framework to branch-decompositions of vector spaces, we developed highly generic tools for branch-decompositions on vector spaces. The only known previous fixed-parameter algorithm for branch-width of $\mathcal F$-represented matroids was due to Hlin\v{e}n\'y and Oum (2008) that runs in time $O(n^3)$ where $n$ is the number of elements of the input $\mathcal F$-represented matroid. But their method is highly indirect. Their algorithm uses the non-trivial fact by Geelen et al. (2003) that the number of forbidden minors is finite and uses the algorithm of Hlin\v{e}n\'y (2005) on checking monadic second-order formulas on $\mathcal F$-represented matroids of small branch-width. Our result does not depend on such a fact and is completely self-contained, and yet matches their asymptotic running time for each fixed $k$.Comment: 73 pages, 10 figure

The Noncommutative Geometry of k-graph C*-Algebras

This paper is comprised of two related parts. First we discuss which k-graph algebras have faithful gauge invariant traces, where the gauge action of \T^k is the canonical one. We give a sufficient condition for the existence of such a trace, identify the C*-algebras of k-graphs satisfying this condition up to Morita equivalence, and compute their K-theory. For k-graphs with faithful gauge invariant trace, we construct a smooth $(k,\infty)$-summable semifinite spectral triple. We use the semifinite local index theorem to compute the pairing with K-theory. This numerical pairing can be obtained by applying the trace to a KK-pairing with values in the K-theory of the fixed point algebra of the \T^k action. As with graph algebras, the index pairing is an invariant for a finer structure than the isomorphism class of the algebra.Comment: 38 pages, some pictures drawn in picTeX Some minor technical revisions. Material has been reorganised with detailed discussion of k-graphs admitting graph traces shortened and moved to an appendix. This version to appear in K-theor