Relative Tutte Polynomials of Tensor Products of Coloured Graphs

Tutte Polynomials of Flower Graphs

Tutte Polynomials of Tensor Products of Signed Graphs and Their Applications in Knot Theory

It is well-known that the Jones polynomial of an alternating knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. Relying on the results of Bollobás and Riordan, we introduce a generalization of Kauffman’s Tutte polynomial of signed graphs for which describing the effect of taking a signed tensor product of signed graphs is very simp...

Invariants of Composite Networks Arising as a Tensor Product

We show how the results of Brylawski and Oxley [4] on the Tutte polynomial of a tensor product of graphs may be generalized to colored graphs and the Tutte polynomials introduced by Bollobás and Riordan [1]. This is a generalization of our earlier work on signed graphs with applications to knot theory. Our result makes the calculation of certain invariants of many composite networks easier, pro...

Relative Tutte Polynomials for Coloured Graphs and Virtual Knot Theory

We introduce the concept of a relative Tutte polynomial. We show that the relative Tutte polynomial can be computed in a way similar to the classical spanning tree expansion used by Tutte in his original paper on this subject. We then apply the relative Tutte polynomial to virtual knot theory. More specifically, we show that the Kauffman bracket polynomial (hence the Jones polynomial) of a virt...

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 Bollobás and Riordan. We give a generalized Zaslavsky– Bollobás–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 gener...