Abstract
Let \(t_{i,j}\) be the coefficient of \(x^iy^j\) in the Tutte polynomial T(G; x, y) of a connected bridgeless and loopless graph G with order v and size e. It is trivial that \(t_{0,e-v+1}=1\) and \(t_{v-1,0}=1\). In this paper, we obtain expressions for another six extreme coefficients \(t_{i,j}\)’s with \((i,j)=(0,e-v)\),\((0,e-v-1)\),\((v-2,0)\),\((v-3,0)\),\((1,e-v)\) and \((v-2,1)\) in terms of small substructures of G. We also discuss their duality properties and their specializations to extreme coefficients of the Jones polynomial.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Biggs, N.: Algebraic Graph Theory. Cambridge University Press, Cambridge (1993)
Dasbach, O., Lin, X.-S.: A volumish theorem for the Jones polynomial of alternating knots. Pac. J. Math. 231, 279–291 (2007)
Dong, F.M., Jin, X.: Zeros of Jones polynomials of graphs. Eletron. J. Comb. 22 3:#P3.23 (2015)
Jones, V.F.R.: A polynomial invariant for knots via Von Neumann algebras. Bull. Am. Math. Soc. 12, 103–111 (1985)
Kauffman, L.H.: A Tutte polynomial for signed graphs. Discrete Appl. Math. 25, 105–127 (1989)
Kook, W., Reiner, V., Stanon, D.: A convolution formula for the Tutte polynomial. J. Combin. Theory Ser. B 76, 297–300 (1999)
Meredith, G.H.J.: Coefficients of chromatic polynomials. J. Combin. Theory Ser. B 13, 14–17 (1972)
Read, R.C.: An introduction to chromatic polynomial. J. Combin. Theory Ser. B 4, 52–71 (1968)
Rota, G.C.: On the foundation of combinatorial theory I. theory of Möbius funcition. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2, 340–368 (1964)
Tutte, W.T.: A contribution to the theory of chromatic polynomials. Can. J. Math. 6, 80–91 (1954)
Acknowledgements
This work is supported by NSFC (No. 11671336) and Fundamental Research Funds for the Central Universities (No. 20720190062). We thank the anonymous referees and A/P Fengming Dong for some helpful comments: our results could be extended to matroids; by using the duality, one could reduce half of the theorems and the proofs; results on the coefficients \(t_{1,e-v}\) and \(t_{v-2,1}\) could be proved by induction on the number of edges and using deletion and contraction, and so on.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Gong, H., Jin, X. & Li, M. Several Extreme Coefficients of the Tutte Polynomial of Graphs. Graphs and Combinatorics 36, 445–457 (2020). https://doi.org/10.1007/s00373-019-02126-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-019-02126-y