# A Polynomial Invariant of Graphs On Orientable Surfaces

@article{Bollobs2001API,
title={A Polynomial Invariant of Graphs On Orientable Surfaces},
author={B. Bollob{\'a}s and O. Riordan},
journal={Proceedings of The London Mathematical Society},
year={2001},
volume={83},
pages={513-531}
}
• Published 2001
• Mathematics
• Proceedings of The London Mathematical Society
Our aim in this paper is to construct a polynomial invariant of cyclic graphs, that is, graphs with cyclic orders at the vertices, or, equivalently, of 2-cell embeddings of graphs into closed orientable surfaces. We shall call this invariant the cyclic graph polynomial, and denote it by the letter C . The cyclic graph polynomial is a three-variable polynomial which generalizes the Tutte polynomial in an essential way. In the next section we de®ne cyclic graphs from two different viewpoints, and… Expand
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#### References

SHOWING 1-10 OF 36 REFERENCES
A Tutte polynomial for signed graphs
• L. Kauffman
• Computer Science, Mathematics
• Discret. Appl. Math.
• 1989
A generalization of the Tutte polynomial that is defined for signed graphs to provide a link between knot theory and graph theory, and to explore a context embracing both subjects. Expand
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 essentiallyExpand
A contribution to the theory of chromatic polynomials
Two polynomials 6(G, n) and <f>(G, n) connected with the colourings of a graph G or of associated maps are discussed. A result believed to be new is proved for the lesser-known polynomial <f>(G, n).Expand
Vassiliev knot invariants. III: Forest algebra and weighted graphs
• Mathematics
• 1994
The main tool used in the investigation of Vassiliev knot invariants is the Hopf algebra of chord diagrams CDL1]. This algebra, simple as it seems at rst sight, upon a closer examination proves to beExpand
On a universal perturbative invariant of 3-manifolds
• Mathematics
• 1998
Using finite type invariants (or Vassiliev invariants) of framed links and the Kirby calculus we construct an invariant of closed oriented three-dimensional manifolds with values in a graded HopfExpand
Vassiliev Knot invariants. II: Intersection graph conjecture for trees
• Mathematics
• 1994
The space of Vassiliev knot invariants has a natural ltration by order V]. Kontsevich's theorem K] gives a purely combinatorial description of the corresponding graded space as a space of functions vExpand
On finite type 3-manifold invariants III: Manifold weight systems
• Mathematics
• 1998
Abstract The present paper is a continuation of [11,6] devoted to the study of finite type invariants of integral homology 3-spheres. We introduce the notion of manifold weight systems , and showExpand
On the Vassiliev knot invariants
The theory of knot invariants of finite type (Vassiliev invariants) is described. These invariants turn out to be at least as powerful as the Jones polynomial and its numerous generalizations comingExpand
The Coloring of Graphs.
• H. Whitney
• Mathematics, Medicine
• Proceedings of the National Academy of Sciences of the United States of America
• 1931
The author has given a proof of a formula for M(λ), the number of ways of coloring a graph in λ colors, due to Birkhoff, which is here studied in detail. Expand