Abstract
Benzenoid systems are natural graph representation of benzenoid hydrocarbons. Many chemically and combinatorially interesting indices and polynomials for bezenoid systems have been widely researched by both chemists and graph theorists. The Tutte polynomial of benzenoid chains without branched hexagons has already been computed by the recursive method. In this paper, by multiple recursion schema, an explicit expression for the Tutte polynomial of benzenoid systems with exactly one branched hexagon is obtained in terms of the number of hexagons on three linear or kinked chains. As a by-product, the number of spanning trees for these kind of benzenoid systems is determined.
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Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant No. 11271307.
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Appendices
Appendix 1: Tutte polynomials of \(H_{l,m,n}\)
\(T (H_{1,1,1}; x, y)\) | |
\(x^{17} +\) \( 4x^{16} +\) \( 10x^{15} +\) \( 20x^{14} +\) \( 4yx^{12} +\) \( 35x^{13} +\) \( 12yx^{11} +\) \( 52x^{12} +\) \( 24yx^{10} +\) \( 68x^{11} +\) \( 3y^2x^8 +\) \( 40yx^9 +\) \( 80x^{10} +\) \( 9y^2x^7 +\) \( 57yx^8 +\) \( 85x^9 +\) \( 15y^2x^6 +\) \( 66yx^7 +\) \( 80x^8 +\) \( 3y^3x^4 +\) \( 21y^2x^5 +\) \( 67yx^6 +\) \( 68x^7 +\) \( 3y^3x^3 +\) \( 24y^2x^4 +\) \( 60yx^5 +\) \( 52x^6 +\) \( 4y^3x^2 +\) \( 21y^2x^3 +\) \( 45yx^4 +\) \( 35x^5 +\) \( y^4 +\) \( 4y^3x +\) \( 15y^2x^2 +\) \( 28yx^3 +\) \( 20x^4 +\) \( 3y^3 +\) \( 9y^2x +\) \( 15yx^2 +\) \( 10x^3 +\) \( 3y^2 +\) \( 6yx +\) \( 4x^2 +\) \( y +\) x | |
\(T (H_{1,1,2}; x, y)\) | |
\( x^{21} +\) \( 5x^{20} +\) \( 15x^{19} +\) \( 35x^{18} +\) \( 5yx^{16} +\) \( 70x^{17} +\) \( 20yx^{15} +\) \( 121x^{16} +\) \( 50yx^{14} +\) \( 185x^{15} +\) \( 4y^2x^{12} +\) \( 100yx^{13} +\) \( 255x^{14} +\) \( 18y^2x^{11} +\) \( 171yx^{12} +\) \( 320x^{13} +\) \( 42y^2x^{10} +\) \( 248yx^{11} +\) \( 365x^{12} +\) \( 4y^3x^8 +\) \( 76y^2x^9 +\) \( 316yx^{10} +\) \( 381x^{11} +\) \( 12y^3x^7 +\) \( 116y^2x^8 +\) \( 360yx^9 +\) \( 365x^{10} +\) \( 22y^3x^6 +\) \( 146y^2x^7 +\) \( 365yx^8 +\) \( 320x^9 +\) \( 3y^4x^4 +\) \( 32y^3x^5 +\) \( 156y^2x^6 +\) \( 328yx^7 +\) \( 255x^8 +\) \( 4y^4x^3 +\) \( 39y^3x^4 +\) \( 146y^2x^5 +\) \( 264yx^6 +\) \( 185x^7 +\) \( 5y^4x^2 +\) \( 35y^3x^3 +\) \( 116y^2x^4 +\) \( 188yx^5 +\) \( 121x^6 +\) \( y^5 +\) \( 5y^4x +\) \( 26y^3x^2 +\) \( 76y^2x^3 +\) \( 115yx^4 +\) \( 70x^5 +\) \( 4y^4 +\) \( 16y^3x +\) \( 42y^2x^2 +\) \( 60yx^3 +\) \( 35x^4 +\) \( 6y^3 +\) \( 18y^2x +\) \( 26yx^2 +\) \( 15x^3 +\) \( 4y^2 +\) \( 8yx +\) \( 5x^2 +\) \( y +\) x | |
\(T (H_{1,1,3}; x, y)\) | |
\( x^{25} +\) \( 6x^{24} +\) \( 21x^{23} +\) \( 56x^{22} +\) \( 6yx^{20} +\) \( 126x^{21} +\) \( 30yx^{19} +\) \( 246x^{20} +\) \( 90yx^{18} +\) \( 426x^{19} +\) \( 5y^2x^{16} +\) \( 210yx^{17} +\) \( 666x^{18} +\) \( 30y^2x^{15} +\) \( 415yx^{16} +\) \( 951x^{17} +\) \( 90y^2x^{14} +\) \( 706yx^{15} +\) \( 1246x^{16} +\) \( 5y^3x^{12} +\) \( 200y^2x^{13} +\) \( 1060yx^{14} +\) \( 1506x^{15} +\) \( 25y^3x^{11} +\) \( 370y^2x^{12} +\) \( 1430yx^{13} +\) \( 1686x^{14} +\) \( 65y^3x^{10} +\) \( 580y^2x^{11} +\) \( 1745yx^{12} +\) \( 1751x^{13} +\) \( 4y^4x^8 +\) \( 125y^3x^9 +\) \( 785y^2x^{10} +\) \( 1930yx^{11} +\) \( 1686x^{12} +\) \( 16y^4x^7 +\) \( 201y^3x^8 +\) \( 940y^2x^9 +\) \( 1946yx^{10} +\) \( 1506x^{11} +\) \( 30y^4x^6 +\) \( 265y^3x^7 +\) \( 1000y^2x^8 +\) \( 1790yx^9 +\) \( 1246x^{10} +\) \( 3y^5x^4 +\) \( 45y^4x^5 +\) \( 295y^3x^6 +\) \( 940y^2x^7 +\) \( 1495yx^8 +\) \( 951x^9 +\) \( 5y^5x^3 +\) \( 57y^4x^4 +\) \( 285y^3x^5 +\) \( 785y^2x^6 +\) \( 1130yx^7 +\) \( 666x^8 +\) \( 6y^5x^2 +\) \( 53y^4x^3 +\) \( 235y^3x^4 +\) \( 580y^2x^5 +\) \( 770yx^6 +\) \( 426x^7 +\) \( y^6 +\) \( 6y^5x +\) \( 40y^4x^2 +\) \( 159y^3x^3 +\) \( 370y^2x^4 +\) \( 466yx^5 +\) \( 246x^6 +\) \( 5y^5 +\) \( 25y^4x +\) \( 90y^3x^2 +\) \( 200y^2x^3 +\) \( 245yx^4 +\) \( 126x^5 +\) \( 10y^4 +\) \( 40y^3x +\) \( 90y^2x^2 +\) \( 110yx^3 +\) \( 56x^4 +\) \( 10y^3 +\) \( 30y^2x +\) \( 40yx^2 +\) \( 21x^3 +\) \( 5y^2 +\) \( 10yx +\) \( 6x^2 +\) \( y +\) x | |
\(T (H_{1,1,4}; x, y)\) | |
\( x^{29} +\) \( 7x^{28} +\) \( 28x^{27} +\) \( 84x^{26} +\) \( 7yx^{24} +\) \( 210x^{25} +\) \( 42yx^{23} +\) \( 455x^{24} +\) \( 147yx^{22} +\) \( 875x^{23} +\) \( 6y^2x^{20} +\) \( 392yx^{21} +\) \( 1520x^{22} +\) \( 45y^2x^{19} +\) \( 876yx^{20} +\) \( 2415x^{21} +\) \( 165y^2x^{18} +\) \( 1692yx^{19} +\) \( 3535x^{20} +\) \( 6y^3x^{16} +\) \( 435y^2x^{17} +\) \( 2892yx^{18} +\) \( 4795x^{19} +\) \( 42y^3x^{15} +\) \( 939y^2x^{16} +\) \( 4452yx^{17} +\) \( 6055x^{18} +\) \( 143y^3x^{14} +\) \( 1728y^2x^{15} +\) \( 6237yx^{16} +\) \( 7140x^{17} +\) \( 5y^4x^{12} +\) \( 344y^3x^{13} +\) \( 2769y^2x^{14} +\) \( 7996yx^{15} +\) \( 7875x^{16} +\) \( 33y^4x^{11} +\) \( 675y^3x^{12} +\) \( 3945y^2x^{13} +\) \( 9432yx^{14} +\) \( 8135x^{15} +\) \( 93y^4x^{10} +\) \( 1115y^3x^{11} +\) \( 5055y^2x^{12} +\) \( 10272yx^{13} +\) \( 7875x^{14} +\) \( 4y^5x^8 +\) \( 188y^4x^9 +\) \( 1580y^3x^{10} +\) \( 5847y^2x^{11} +\) \( 10337yx^{12} +\) \( 7140x^{13} +\) \( 20y^5x^7 +\) \( 314y^4x^8 +\) \( 1965y^3x^9 +\) \( 6132y^2x^{10} +\) \( 9612yx^{11} +\) \( 6055x^{12} +\) \( 39y^5x^6 +\) \( 431y^4x^7 +\) \( 2165y^3x^8 +\) \( 5847y^2x^9 +\) \( 8256yx^{10} +\) \( 4795x^{11} +\) \( 3y^6x^4 +\) \( 60y^5x^5 +\) \( 494y^4x^6 +\) \( 2103y^3x^7 +\) \( 5055y^2x^8 +\) \( 6532yx^9 +\) \( 3535x^{10} +\) \( 6y^6x^3 +\) \( 78y^5x^4 +\) \( 489y^4x^5 +\) \( 1807y^3x^6 +\) \( 3945y^2x^7 +\) \( 4737yx^8 +\) \( 2415x^9 +\) \( 7y^6x^2 +\) \( 75y^5x^3 +\) \( 414y^4x^4 +\) \( 1371y^3x^5 +\) \( 2769y^2x^6 +\) \( 3132yx^7 +\) \( 1520x^8 +\) \( y^7 +\) \( 7y^6x +\) \( 57y^5x^2 +\) \( 287y^4x^3 +\) \( 900y^3x^4 +\) \( 1728y^2x^5 +\) \( 1872yx^6 +\) \( 875x^7 +\) \( 6y^6 +\) \( 36y^5x +\) \( 165y^4x^2 +\) \( 499y^3x^3 +\) \( 939y^2x^4 +\) \( 996yx^5 +\) \( 455x^6 +\) \( 15y^5 +\) \( 75y^4x +\) \( 230y^3x^2 +\) \( 435y^2x^3 +\) \( 462yx^4 +\) \( 210x^5 +\) \( 20y^4 +\) \( 80y^3x +\) \( 165y^2x^2 +\) \( 182yx^3 +\) \( 84x^4 +\) \( 15y^3 +\) \( 45y^2x +\) \( 57yx^2 +\) \( 28x^3 +\) \( 6y^2 +\) \( 12yx +\) \( 7x^2 +\) \( y +\) x | |
\(T (H_{1,2,2}; x, y)\) | |
\( x^{25} +\) \( 6x^{24} +\) \( 21x^{23} +\) \( 56x^{22} +\) \( 6yx^{20} +\) \( 126x^{21} +\) \( 30yx^{19} +\) \( 246x^{20} +\) \( 90yx^{18} +\) \( 426x^{19} +\) \( 5y^2x^{16} +\) \( 210yx^{17} +\) \( 666x^{18} +\) \( 30y^2x^{15} +\) \( 415yx^{16} +\) \( 951x^{17} +\) \( 90y^2x^{14} +\) \( 706yx^{15} +\) \( 1246x^{16} +\) \( 5y^3x^{12} +\) \( 200y^2x^{13} +\) \( 1060yx^{14} +\) \( 1506x^{15} +\) \( 25y^3x^{11} +\) \( 370y^2x^{12} +\) \( 1430yx^{13} +\) \( 1686x^{14} +\) \( 65y^3x^{10} +\) \( 580y^2x^{11} +\) \( 1745yx^{12} +\) \( 1751x^{13} +\) \( 5y^4x^8 +\) \( 125y^3x^9 +\) \( 785y^2x^{10} +\) \( 1930yx^{11} +\) \( 1686x^{12} +\) \( 15y^4x^7 +\) \( 200y^3x^8 +\) \( 940y^2x^9 +\) \( 1946yx^{10} +\) \( 1506x^{11} +\) \( 30y^4x^6 +\) \( 265y^3x^7 +\) \( 1000y^2x^8 +\) \( 1790yx^9 +\) \( 1246x^{10} +\) \( 3y^5x^4 +\) \( 45y^4x^5 +\) \( 295y^3x^6 +\) \( 940y^2x^7 +\) \( 1495yx^8 +\) \( 951x^9 +\) \( 5y^5x^3 +\) \( 57y^4x^4 +\) \( 285y^3x^5 +\) \( 785y^2x^6 +\) \( 1130yx^7 +\) \( 666x^8 +\) \( 6y^5x^2 +\) \( 53y^4x^3 +\) \( 235y^3x^4 +\) \( 580y^2x^5 +\) \( 770yx^6 +\) \( 426x^7 +\) \( y^6 +\) \( 6y^5x +\) \( 40y^4x^2 +\) \( 159y^3x^3 +\) \( 370y^2x^4 +\) \( 466yx^5 +\) \( 246x^6 +\) \( 5y^5 +\) \( 25y^4x +\) \( 90y^3x^2 +\) \( 200y^2x^3 +\) \( 245yx^4 +\) \( 126x^5 +\) \( 10y^4 +\) \( 40y^3x +\) \( 90y^2x^2 +\) \( 110yx^3 +\) \( 56x^4 +\) \( 10y^3 +\) \( 30y^2x +\) \( 40yx^2 +\) \( 21x^3 +\) \( 5y^2 +\) \( 10yx +\) \( 6x^2 +\) \( y +\) x | |
\(T (H_{1,2,3}; x, y)\) | |
\( x^{29} +\) \( 7x^{28} +\) \( 28x^{27} +\) \( 84x^{26} +\) \( 7yx^{24} +\) \( 210x^{25} +\) \( 42yx^{23} +\) \( 455x^{24} +\) \( 147yx^{22} +\) \( 875x^{23} +\) \( 6y^2x^{20} +\) \( 392yx^{21} +\) \( 1520x^{22} +\) \( 45y^2x^{19} +\) \( 876yx^{20} +\) \( 2415x^{21} +\) \( 165y^2x^{18} +\) \( 1692yx^{19} +\) \( 3535x^{20} +\) \( 6y^3x^{16} +\) \( 435y^2x^{17} +\) \( 2892yx^{18} +\) \( 4795x^{19} +\) \( 42y^3x^{15} +\) \( 939y^2x^{16} +\) \( 4452yx^{17} +\) \( 6055x^{18} +\) \( 143y^3x^{14} +\) \( 1728y^2x^{15} +\) \( 6237yx^{16} +\) \( 7140x^{17} +\) \( 6y^4x^{12} +\) \( 344y^3x^{13} +\) \( 2769y^2x^{14} +\) \( 7996yx^{15} +\) \( 7875x^{16} +\) \( 33y^4x^{11} +\) \( 674y^3x^{12} +\) \( 3945y^2x^{13} +\) \( 9432yx^{14} +\) \( 8135x^{15} +\) \( 93y^4x^{10} +\) \( 1114y^3x^{11} +\) \( 5055y^2x^{12} +\) \( 10272yx^{13} +\) \( 7875x^{14} +\) \( 5y^5x^8 +\) \( 188y^4x^9 +\) \( 1579y^3x^{10} +\) \( 5847y^2x^{11} +\) \( 10337yx^{12} +\) \( 7140x^{13} +\) \( 19y^5x^7 +\) \( 313y^4x^8 +\) \( 1964y^3x^9 +\) \( 6132y^2x^{10} +\) \( 9612yx^{11} +\) \( 6055x^{12} +\) \( 39y^5x^6 +\) \( 430y^4x^7 +\) \( 2164y^3x^8 +\) \( 5847y^2x^9 +\) \( 8256yx^{10} +\) \( 4795x^{11} +\) \( 3y^6x^4 +\) \( 60y^5x^5 +\) \( 494y^4x^6 +\) \( 2103y^3x^7 +\) \( 5055y^2x^8 +\) \( 6532yx^9 +\) \( 3535x^{10} +\) \( 6y^6x^3 +\) \( 78y^5x^4 +\) \( 489y^4x^5 +\) \( 1807y^3x^6 +\) \( 3945y^2x^7 +\) \( 4737yx^8 +\) \( 2415x^9 +\) \( 7y^6x^2 +\) \( 75y^5x^3 +\) \( 414y^4x^4 +\) \( 1371y^3x^5 +\) \( 2769y^2x^6 +\) \( 3132yx^7 +\) \( 1520x^8 +\) \( y^7 +\) \( 7y^6x +\) \( 57y^5x^2 +\) \( 287y^4x^3 +\) \( 900y^3x^4 +\) \( 1728y^2x^5 +\) \( 1872yx^6 +\) \( 875x^7 +\) \( 6y^6 +\) \( 36y^5x +\) \( 165y^4x^2 +\) \( 499y^3x^3 +\) \( 939y^2x^4 +\) \( 996yx^5 +\) \( 455x^6 +\) \( 15y^5 +\) \( 75y^4x +\) \( 230y^3x^2 +\) \( 435y^2x^3 +\) \( 462yx^4 +\) \( 210x^5 +\) \( 20y^4 +\) \( 80y^3x +\) \( 165y^2x^2 +\) \( 182yx^3 +\) \( 84x^4 +\) \( 15y^3 +\) \( 45y^2x +\) \( 57yx^2 +\) \( 28x^3 +\) \( 6y^2 +\) \( 12yx +\) \( 7x^2 +\) \( y +\) x | |
\(T (H_{2,2,2}; x, y)\) | |
\( x^{29} +\) \( 7x^{28} +\) \( 28x^{27} +\) \( 84x^{26} +\) \( 7yx^{24} +\) \( 210x^{25} +\) \( 42yx^{23} +\) \( 455x^{24} +\) \( 147yx^{22} +\) \( 875x^{23} +\) \( 6y^2x^{20} +\) \( 392yx^{21} +\) \( 1520x^{22} +\) \( 45y^2x^{19} +\) \( 876yx^{20} +\) \( 2415x^{21} +\) \( 165y^2x^{18} +\) \( 1692yx^{19} +\) \( 3535x^{20} +\) \( 6y^3x^{16} +\) \( 435y^2x^{17} +\) \( 2892yx^{18} +\) \( 4795x^{19} +\) \( 42y^3x^{15} +\) \( 939y^2x^{16} +\) \( 4452yx^{17} +\) \( 6055x^{18} +\) \( 143y^3x^{14} +\) \( 1728y^2x^{15} +\) \( 6237yx^{16} +\) \( 7140x^{17} +\) \( 7y^4x^{12} +\) \( 344y^3x^{13} +\) \( 2769y^2x^{14} +\) \( 7996yx^{15} +\) \( 7875x^{16} +\) \( 33y^4x^{11} +\) \( 673y^3x^{12} +\) \( 3945y^2x^{13} +\) \( 9432yx^{14} +\) \( 8135x^{15} +\) \( 93y^4x^{10} +\) \( 1113y^3x^{11} +\) \( 5055y^2x^{12} +\) \( 10272yx^{13} +\) \( 7875x^{14} +\) \( 6y^5x^8 +\) \( 188y^4x^9 +\) \( 1578y^3x^{10} +\) \( 5847y^2x^{11} +\) \( 10337yx^{12} +\) \( 7140x^{13} +\) \( 18y^5x^7 +\) \( 312y^4x^8 +\) \( 1963y^3x^9 +\) \( 6132y^2x^{10} +\) \( 9612yx^{11} +\) \( 6055x^{12} +\) \( 39y^5x^6 +\) \( 429y^4x^7 +\) \( 2163y^3x^8 +\) \( 5847y^2x^9 +\) \( 8256yx^{10} +\) \( 4795x^{11} +\) \( 3y^6x^4 +\) \( 60y^5x^5 +\) \( 494y^4x^6 +\) \( 2103y^3x^7 +\) \( 5055y^2x^8 +\) \( 6532yx^9 +\) \( 3535x^{10} +\) \( 6y^6x^3 +\) \( 78y^5x^4 +\) \( 489y^4x^5 +\) \( 1807y^3x^6 +\) \( 3945y^2x^7 +\) \( 4737yx^8 +\) \( 2415x^9 +\) \( 7y^6x^2 +\) \( 75y^5x^3 +\) \( 414y^4x^4 +\) \( 1371y^3x^5 +\) \( 2769y^2x^6 +\) \( 3132yx^7 +\) \( 1520x^8 +\) \( y^7 +\) \( 7y^6x +\) \( 57y^5x^2 +\) \( 287y^4x^3 +\) \( 900y^3x^4 +\) \( 1728y^2x^5 +\) \( 1872yx^6 +\) \( 875x^7 +\) \( 6y^6 +\) \( 36y^5x +\) \( 165y^4x^2 +\) \( 499y^3x^3 +\) \( 939y^2x^4 +\) \( 996yx^5 +\) \( 455x^6 +\) \( 15y^5 +\) \( 75y^4x +\) \( 230y^3x^2 +\) \( 435y^2x^3 +\) \( 462yx^4 +\) \( 210x^5 +\) \( 20y^4 +\) \( 80y^3x +\) \( 165y^2x^2 +\) \( 182yx^3 +\) \( 84x^4 +\) \( 15y^3 +\) \( 45y^2x +\) \( 57yx^2 +\) \( 28x^3 +\) \( 6y^2 +\) \( 12yx +\) \( 7x^2 +\) \( y +\) x |
Appendix 2: The number of spanning trees of \(H_{l,m,n}\)
(l, m, n) | \(\tau (H_{l,m,n})\) | (l, m, n) | \(\tau (H_{l,m,n})\) | (l, m, n) | \(\tau (H_{l,m,n})\) |
|---|---|---|---|---|---|
(1, 1, 1) | 1188 | (1, 1, 2) | 6924 | (1, 1, 3) | 40356 |
(1, 1, 4) | 235212 | (1, 1, 5) | 1370916 | (1, 1, 6) | 7990284 |
(1, 1, 7) | 46570788 | (1, 1, 8) | 271434444 | (1, 1, 9) | 1582035876 |
(1, 1, 10) | 9220780812 | (1, 1, 11) | 53742648996 | (1, 1, 12) | 313235113164 |
(1, 1, 13) | 1825668029988 | (1, 1, 14) | 10640773066764 | (1, 1, 15) | 62018970370596 |
(1, 1, 16) | 361473049156812 | (1, 1, 17) | 2106819324570276 | (1, 1, 18) | 12279442898264844 |
(1, 2, 2) | 40355 | (1, 2, 3) | 235206 | (1, 2, 4) | 1370881 |
(1, 2, 5) | 7990080 | (1, 2, 6) | 46569599 | (1, 2, 7) | 271427514 |
(1, 2, 8) | 1581995485 | (1, 2, 9) | 9220545396 | (1, 2, 10) | 53741276891 |
(1, 2, 11) | 313227115950 | (1, 2, 12) | 1825621418809 | (1, 2, 13) | 10640501396904 |
(1, 2, 14) | 62017386962615 | (1, 2, 15) | 361463820378786 | (1, 2, 16) | 2106765535310101 |
(1, 2, 17) | 12279129391481820 | (1, 3, 3) | 1370880 | (1, 3, 4) | 7990074 |
(1, 3, 5) | 46569564 | (1, 3, 6) | 271427310 | (1, 3, 7) | 1581994296 |
(1, 3, 8) | 9220538466 | (1, 3, 9) | 53741236500 | (1, 3, 10) | 313226880534 |
(1, 3, 11) | 1825620046704 | (1, 3, 12) | 10640493399690 | (1, 3, 13) | 62017340351436 |
(1, 3, 14) | 361463548708926 | (1, 3, 15) | 2106763951902120 | (1, 3, 16) | 12279120162703794 |
(1, 4, 4) | 46569563 | (1, 4, 5) | 271427304 | (1, 4, 6) | 1581994261 |
(1, 4, 7) | 9220538262 | (1, 4, 8) | 53741235311 | (1, 4, 9) | 313226873604 |
(1, 4, 10) | 1825620006313 | (1, 4, 11) | 10640493164274 | (1, 4, 12) | 62017338979331 |
(1, 4, 13) | 361463540711712 | (1, 4, 14) | 2106763905290941 | (1, 4, 15) | 12279119891033934 |
(1, 5, 5) | 1581994260 | (1, 5, 6) | 9220538256 | (1, 5, 7) | 53741235276 |
(1, 5, 8) | 313226873400 | (1, 5, 9) | 1825620005124 | (1, 5, 10) | 10640493157344 |
(1, 5, 11) | 62017338938940 | (1, 5, 12) | 361463540476296 | (1, 5, 13) | 2106763903918836 |
(1, 5, 14) | 12279119883036720 | (1, 6, 6) | 53741235275 | (1, 6, 7) | 313226873394 |
(1, 6, 8) | 1825620005089 | (1, 6, 9) | 10640493157140 | (1, 6, 10) | 62017338937751 |
(1, 6, 11) | 361463540469366 | (1, 6, 12) | 2106763903878445 | (1, 6, 13) | 12279119882801304 |
(1, 7, 7) | 1825620005088 | (1, 7, 8) | 10640493157134 | (1, 7, 9) | 62017338937716 |
(1, 7, 10) | 361463540469162 | (1, 7, 11) | 2106763903877256 | (1, 7, 12) | 12279119882794374 |
(1, 8, 8) | 62017338937715 | (1, 8, 9) | 361463540469156 | (1, 8, 10) | 2106763903877221 |
(1, 8, 11) | 12279119882794170 | (1, 9, 9) | 2106763903877220 | (1, 9, 10) | 12279119882794164 |
(2, 2, 2) | 235200 | (2, 2, 3) | 1370845 | (2, 2, 4) | 7989870 |
(2, 2, 5) | 46568375 | (2, 2, 6) | 271420380 | (2, 2, 7) | 1581953905 |
(2, 2, 8) | 9220303050 | (2, 2, 9) | 53739864395 | (2, 2, 10) | 313218883320 |
(2, 2, 11) | 1825573435525 | (2, 2, 12) | 10640221729830 | (2, 2, 13) | 62015756943455 |
(2, 2, 14) | 361454319930900 | (2, 2, 15) | 2106710162641945 | (2, 2, 16) | 12278806655920770 |
(2, 3, 3) | 7989864 | (2, 3, 4) | 46568339 | (2, 3, 5) | 271420170 |
(2, 3, 6) | 1581952681 | (2, 3, 7) | 9220295916 | (2, 3, 8) | 53739822815 |
(2, 3, 9) | 313218640974 | (2, 3, 10) | 1825572023029 | (2, 3, 11) | 10640213497200 |
(2, 3, 12) | 62015708960171 | (2, 3, 13) | 361454040263826 | (2, 3, 14) | 2106708532622785 |
(2, 3, 15) | 12278797155472884 | (2, 4, 4) | 271420164 | (2, 4, 5) | 1581952645 |
(2, 4, 6) | 9220295706 | (2, 4, 7) | 53739821591 | (2, 4, 8) | 313218633840 |
(2, 4, 9) | 1825571981449 | (2, 4, 10) | 10640213254854 | (2, 4, 11) | 62015707547675 |
(2, 4, 12) | 361454032031196 | (2, 4, 13) | 2106708484639501 | (2, 4, 14) | 12278796875805810 |
(2, 5, 5) | 9220295700 | (2, 5, 6) | 53739821555 | (2, 5, 7) | 313218633630 |
(2, 5, 8) | 1825571980225 | (2, 5, 9) | 10640213247720 | (2, 5, 10) | 62015707506095 |
(2, 5, 11) | 361454031788850 | (2, 5, 12) | 2106708483227005 | (2, 5, 13) | 12278796867573180 |
(2, 6, 6) | 313218633624 | (2, 6, 7) | 1825571980189 | (2, 6, 8) | 10640213247510 |
(2, 6, 9) | 62015707504871 | (2, 6, 10) | 361454031781716 | (2, 6, 11) | 2106708483185425 |
(2, 6, 12) | 12278796867330834 | (2, 7, 7) | 10640213247504 | (2, 7, 8) | 62015707504835 |
(2, 7, 9) | 361454031781506 | (2, 7, 10) | 2106708483184201 | (2, 7, 11) | 12278796867323700 |
(2, 8, 8) | 361454031781500 | (2, 8, 9) | 2106708483184165 | (2, 8, 10) | 12278796867323490 |
(2, 9, 9) | 12278796867323484 | (3, 3, 3) | 46568304 | (3, 3, 4) | 271419960 |
(3, 3, 5) | 1581951456 | (3, 3, 6) | 9220288776 | (3, 3, 7) | 53739781200 |
(3, 3, 8) | 313218398424 | (3, 3, 9) | 1825570609344 | (3, 3, 10) | 10640205257640 |
(3, 3, 11) | 62015660936496 | (3, 3, 12) | 361453760361336 | (3, 3, 13) | 2106706901231520 |
(3, 3, 14) | 12278787647027784 | (3, 4, 4) | 1581951421 | (3, 4, 5) | 9220288566 |
(3, 4, 6) | 53739779975 | (3, 4, 7) | 313218391284 | (3, 4, 8) | 1825570567729 |
(3, 4, 9) | 10640205015090 | (3, 4, 10) | 62015659522811 | (3, 4, 11) | 361453752121776 |
(3, 4, 12) | 2106706853207845 | (3, 4, 13) | 12278787367125294 | (3, 5, 5) | 53739779940 |
(3, 5, 6) | 313218391074 | (3, 5, 7) | 1825570566504 | (3, 5, 8) | 10640205007950 |
(3, 5, 9) | 62015659481196 | (3, 5, 10) | 361453751879226 | (3, 5, 11) | 2106706851794160 |
(3, 5, 12) | 12278787358885734 | (3, 6, 6) | 1825570566469 | (3, 6, 7) | 10640205007740 |
(3, 6, 8) | 62015659479971 | (3, 6, 9) | 361453751872086 | (3, 6, 10) | 2106706851752545 |
(3, 6, 11) | 12278787358643184 | (3, 7, 7) | 62015659479936 | (3, 7, 8) | 361453751871876 |
(3, 7, 9) | 2106706851751320 | (3, 7, 10) | 12278787358636044 | (3, 8, 8) | 2106706851751285 |
(3, 8, 9) | 12278787358635834 | (4, 4, 4) | 9220288362 | (4, 4, 5) | 53739778751 |
(4, 4, 6) | 313218384144 | (4, 4, 7) | 1825570526113 | (4, 4, 8) | 10640204772534 |
(4, 4, 9) | 62015658109091 | (4, 4, 10) | 361453743882012 | (4, 4, 11) | 2106706805182981 |
(4, 4, 12) | 12278787087215874 | (4, 5, 5) | 313218383940 | (4, 5, 6) | 1825570524889 |
(4, 5, 7) | 10640204765394 | (4, 5, 8) | 62015658067475 | (4, 5, 9) | 361453743639456 |
(4, 5, 10) | 2106706803769261 | (4, 5, 11) | 12278787078976110 | (4, 6, 6) | 10640204765190 |
(4, 6, 7) | 62015658066251 | (4, 6, 8) | 361453743632316 | (4, 6, 9) | 2106706803727645 |
(4, 6, 10) | 12278787078733554 | (4, 7, 7) | 361453743632112 | (4, 7, 8) | 2106706803726421 |
(4, 7, 9) | 12278787078726414 | (4, 8, 8) | 12278787078726210 | (5, 5, 5) | 1825570523700 |
(5, 5, 6) | 10640204758260 | (5, 5, 7) | 62015658025860 | (5, 5, 8) | 361453743396900 |
(5, 5, 9) | 2106706802355540 | (5, 5, 10) | 12278787070736340 | (5, 6, 6) | 62015658024671 |
(5, 6, 7) | 361453743389766 | (5, 6, 8) | 2106706802313925 | (5, 6, 9) | 12278787070493784 |
(5, 7, 7) | 2106706802312736 | (5, 7, 8) | 12278787070486650 | (6, 6, 6) | 361453743382836 |
(6, 6, 7) | 2106706802272345 | (6, 6, 8) | 12278787070251234 | (6, 7, 7) | 12278787070244304 |
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Gong, H., Jin, X. & Zhang, F. Tutte polynomials for benzenoid systems with one branched hexagon. J Math Chem 54, 1057–1071 (2016). https://doi.org/10.1007/s10910-016-0601-3
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DOI: https://doi.org/10.1007/s10910-016-0601-3