Lorenzo Traldi

A dichromatic polynomial for weighted graphs and link polynomials

A dichromatic polynomial for weighted graphs is presented. The Kauffman bracket of a signed graph, an invariant inspired by the Jones poly- nomial of a link in three-space, is shown to be essentially an evaluation of this dichromatic polynomial, as are the homfly polynomials of certain particular types of links. 1. THE DICHROMATIC POLYNOMIAL OF A WEIGHTED GRAPH In this note we use the term graph to refer to a finite multigraph, i.e. a graph G = (V(G),E(G)) which has a finite number zz > 0 of vertices and a finite number zzz > 0 of edges, some of which may be multiple edges or loops. The zeroth Betti number p0(G) of such a graph is the number of its connected components, and the first Betti number px(G) is px(G) = m - n + p0(G). A weighted graph is a graph G together with a function w mapping E(G) into some commutative ring with unity R ; if e is an edge of G then w(e) is the weight or conductance of e . If S is any set of edges of a graph G then we denote by G : S the subgraph of G which includes all the vertices of G but only the edges in S. Using this notation, we define the dichromatic polynomial Q(G ;t,z) of a weighted graph

Milnor's invariants and the completions of link modules

Let L be a tame link of k > 2 components in S3, H the abelianization of its group 7TJ(S3 L), and IH the augmentation ideal of the integral group ring ZH. The IH-adic completions of the Alexander module and Alexander invariant of L are shown to possess presentation matrices whose entries are given in terms of certain integers t4(i1,...,iq) introduced by J. Milnor. Various applications to the theory of the elementary ideals of these modules are given, including a condition on the Alexander polynomial necessary for the linking numbers of the components of L with each other to all be zero. In the special case /. = 2, it is shown that the various Milnor invariants f([r + 1, s + 1]) are determined (up to sign) by the Alexander polynomial of L, and that this Alexander polynomial is 0 iff i([r + 1, s + 1]) = 0 for all r, s >0 with r + s even; also, the Chen groups of L are determined (up to isomorphism) by those nonzero i([r + 1, s + 1]) with r + s minimal. In contrast, it is shown by example that for k > 3 the Alexander polynomials of a link and its sublinks do not determine its Chen groups.

Preprocessing minpaths for sum of disjoint products

Network reliability algorithms which produce sums of disjoint products (SDP) are sensitive to the order in which the minimal pathsets are analyzed. The minpaths are preprocessed by choosing this order in the hope that an SDP algorithm will then provide a relatively efficient analysis. The most commonly used preprocessing strategy is to list the minpaths in order of increasing size. This paper gives examples for which this strategy is not optimal. A new preprocessing strategy which works well for SDP algorithms with single-variable inversion (SVI) is introduced. It is also observed that optimal preprocessing for SVI-SDP can be different from optimal preprocessing for SDP algorithms which use multiple-variable inversion; one reason for this is that MVI-SDP algorithms handle disjoint minpaths much more effectively than SVI-SDP algorithms do. Both kinds of SDP algorithms profit from prior reduction of elements and of subsystems which are in parallel or in series.

A bracket polynomial for graphs, I

A knot diagram has an associated looped interlacement graph, obtained from the intersection graph of the Gauss diagram by attaching loops to the vertices that correspond to negative crossings. This construction suggests an extension of the Kauffman bracket to an invariant of looped graphs, and an extension of Reidemeister equivalence to an equivalence relation on looped graphs. The graph bracket polynomial can be defined recursively using the same pivot and local complementation operations used to define the interlace polynomial, and it gives rise to a graph Jones polynomial that is invariant under the graph Reidemeister moves.

Parametrized Tutte Polynomials of Graphs and Matroids

We generalize and unify results on parametrized and coloured Tutte polynomials of graphs and matroids due to Zaslavsky, and Bollobas and Riordan. We give a generalized Zaslavsky–Bollobas–Riordan theorem that characterizes parametrized contraction–deletion functions on minor-closed classes of matroids, as well as the modifications necessary to apply the discussion to classes of graphs. In general, these parametrized Tutte polynomials do not satisfy analogues of all the familiar properties of the classical Tutte polynomial. We give conditions under which they do satisfy corank-nullity formulas, and also conditions under which they reflect the structure of series-parallel connections.

A bracket polynomial for graphs, II. Links, Euler circuits and marked graphs.

Let D be an oriented classical or virtual link diagram with directed universe ~ U. Let C denote a set of directed Euler circuits, one in each connected component of U. There is then an associated looped interlacement graph L(D, C) whose construction involves very little geometric information about the way D is drawn in the plane; consequently L(D, C) is different from other combinatorial structures associated with classical link diagrams, like the checkerboard graph, which can be difficultto extend to arbitrary virtual links. L(D, C) is determined by three things: the structure of ~ U as a 2-in, 2-out digraph, the distinction between crossings that make a positive contribution to the writhe and those that make a negative contribution, and the relationship between C and the directed circuits in ~ U arising from the link components; this relationship is indicated by marking the vertices where C does not follow the incident link component(s). We introduce a bracket polynomial for arbitrary marked graphs, defined using either a formula involving matrix nullities or a recursion involving the local complement and pivot operations; the marked-graph bracket of L(D, C) is the same as the Kauffman bracket of D. This provides a unified combinatorial description of the Jones polynomial that applies seamlessly to both classical and non-classical virtual links.

Chain polynomials and Tutte polynomials

The recently introduced chain and sheaf polynomials of a graph are shown to be essentially equivalent to a weighted version of the Tutte polynomial.

On the colored Tutte polynomial of a graph of bounded treewidth

We observe that a formula given by Negami [Polynomial invariants of graphs, Trans. Amer. Math. Soc. 299 (1987) 601-622] for the Tutte polynomial of a k-sum of two graphs generalizes to a colored Tutte polynomial. Consequently, an algorithm of Andrzejak [An algorithm for the Tutte polynomials of graphs of bounded treewidth, Discrete Math. 190 (1998) 39-54] may be directly adapted to compute the colored Tutte polynomial of a graph of bounded treewidth in polynomial time. This result has also been proven by Makowsky [Colored Tutte polynomials and Kauffman brackets for graphs of bounded tree width, Discrete Appl. Math. 145 (2005) 276-290], using a different algorithm based on logical techniques.

The transition matroid of a 4-regular graph

Given a 4-regular graph F , we introduce a binary matroid M ? ( F ) on the set of transitions of F . Parametrized versions of the Tutte polynomial of M ? ( F ) yield several well-known graph and knot polynomials, including the Martin polynomial, the homflypt polynomial, the Kauffman polynomial and the Bollobas-Riordan polynomial.

Weighted interlace polynomials

The interlace polynomials introduced by Arratia, Bollobas and Sorkin extend to invariants of graphs with vertex weights, and these weighted interlace polynomials have several novel properties. One novel property is a version of the fundamental three-term formula