A classification for bicyclic and tricyclic graphs with respect to Tutte polynomial

Year 2020, Volume: 49 Issue: 5, 1649 - 1659, 06.10.2020

Zahra Yarahmadi , Shiva Mir

https://doi.org/10.15672/hujms.537824

Abstract

The Tutte polynomial of a graph is a polynomial in two variables defined for every simple graph contains information about how the graph is connected. We prove some formulas for computing Tutte polynomial of bicyclic and tricyclic graph and finally classify tricyclic graph with respect to Tutte polynomial.

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Keywords

Tutte polynomial, bicyclic, tricyclic graph

References

• [1] J.M. Burgos, Singularities in Negamis splitting formula for the Tutte polynomial, Discrete Appl. Math. 237, 65-74, 2018.
• [2] H.H. Crapo, The Tutte polynomial, Aequationes Math. 3 (3), 211-229, 1969.
• [3] H. Deng, S. Chen and J. Zhang, The Merrifield-Simmons index in(n, n+1)-graphs, J. Math. Chem. 43, 75-91, 2008.
• [4] A. Dolati, M. Haghighat, S. Golalizadeh and M. Safari, The smallest Hosoya index of con-nected tricyclic graphs, MATCH Commun. Math. Comput. Chem. 65, 57-70, 2011.
• [5] S. Li and Z. Zhu, Sharp lower bound for total number of matching of tricyclic graphs, Electron. J. Comb. 17, 15 pages, 2010.
• [6] S. Ok and T.J. Perrett, Density of real zeros of the Tutte polynomial, Electron. J. Discrete Math. 61, 941-946, 2017.
• [7] Y.M. Tong, J.B. Liu, Z.Z. Jiang and N.N. Lv, Extreme values of the first general Zagreb index in tricyclic graphs, J. Hefei Univ. Nat. Sci. 1, 4-7, 2010.
• [8] W.T. Tutte, A contribution to the theory chromatic polynomials, Canad. J .Math. 6, 80-91, 1953.
• [9] W.T. Tutte, On dichromatic polynomials, J. Combin. Theory 2, 301-320, 1967.
• [10] W.T. Tutte, On chromatic polynomials and the golden ratio, J. Combin. Theory 9, 289-296, 1970.
• [11] W.T. Tutte, Graph-polynomials, Adv. Appl. Math. 32, 5-9, 2004.