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维纳指数

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维纳指数 W，记作 w (Wiener 1947)，也称为“路径数”或维纳数 (Plavšić et al. 1993)，是为节点数为 n 的图定义的图指标，定义如下

W=1/2sum_(i=1)^nsum_(j=1)^n(d)_(ij),
(1)

其中 (d)_(ij) 是图距离矩阵。

除非另有说明，否则在计算此类指标时通常会忽略氢原子，就像有机化学家通常将苯环写成六边形一样（Devillers 和 Balaban 1999, p. 25）。

图 W(G) 的维纳指数 G，其顶点数为 |G|，与图的平均无序数 A(G) 相关，关系如下

A(G)=(2W(G))/(|G|)
(2)

（Fried 2022）。

图 G_1 和 G_2 的图笛卡尔积的维纳指数由下式给出

W(G_1 square G_2)=|G_1|^2W(G_2)+|G_2|^2W(G_1)
(3)

（Yeh 和 Gutman 1994，Fried 2022）。

维纳指数的区分度不高。事实上，爪图和四个节点的方图已经无法用维纳指数区分（两者都为 8）。非维纳唯一连通图在 n=1、2、... 个节点上的数量由 0, 0, 0, 2, 16, 108, 847, 11110, 261072, ... 给出 (OEIS A193217)。

许多图的预计算值在 Wolfram Language 中实现为GraphData[g,"WienerIndex"].

下表总结了各种特殊图类的维纳指数值。

图类OEISW(G_1), W(G_2), ...
Andrásfai 图A2920181, 15, 44, 88, 147, 221, 310, 414, ...
羚羊图 n×nA2920390, infty, infty, infty, infty, infty, infty, 11548, 16660, ...
反棱柱图A002411X, X, 18, 40, 75, 126, 196, 288, ...
阿波罗网络A2890226, 27, 204, 1941, 19572, 198567, ...
黑主教图 n×nA2920510, 1, 14, 42, 124, 251, 506, 852, 1432, 2165, ...
鸡尾酒会图A001105infty, 8, 18, 32, 50, 72, 98, 128, 162, ...
完全二分图 K_(n,n)A0005671, 1, 5, 73, 2069, 95401, 6487445, ...
完全三部图 K_(n,n,n)A0941591, 11, 1243, 490043, 463370491, ...
完全图 K_nA0002170, 1, 3, 6, 10, 15, 21, 28, 36, ...
2n-交叉棱柱图A292022X, 48, 132, 288, 540, 912, 1428, ...
皇冠图 K_2 square K_n^_A033428X, X, 27, 48, 75, 108, 147, 192, 243, ...
立方体连接循环图A292028X, X, 888, 9472, 76336, 559584, 3594952, ...
圈图 C_nA034828X, X, 3, 8, 15, 27, 42, 64, 90, ...
斐波那契立方体图A2384191, 4, 16, 54, 176, 548, 1667, 4968, ...
五跳图 n×nA2920400, infty, infty, infty, infty, infty, infty, 6364, 9888, 15216, ...
折叠立方体图A292029X, 1, 6, 40, 200, 1056, 4928, 23808, ...
齿轮图A049598X, X, 36, 72, 120, 180, 252, 336, 432, ...
网格图 P_n square P_nA1439450, 8, 72, 320, 1000, 2520, 5488, 10752, ...
网格图 P_n square P_n square P_nA2920450, 48, 972, 7680, 37500, 136080, 403368, ...
半立方体图A2920440, 1, 6, 32, 160, 768, 3584, 16384, ...
河内图A2900043, 72, 1419, 26580, 487839, 8867088, ...
超立方体图 Q_nA0026971, 8, 48, 256, 1280, 6144, 28672, ...
Keller 图A292056infty, 200, 2944, 43392, 650240, 9889792, ...
国王图 n×nA2920530, 6, 52, 228, 708, 1778, 3864, 7560, ...
骑士图 n×nA2920540, infty, infty, 288, 708, 1580, 3144, 5804, 9996, ...
门格海绵图A292036612, 794976, 954380016, ...
莫比乌斯梯子A180857X, X, 21, 44, 85, 138, 217, 312, 441, ...
Mycielski 图A2920550, 1, 15, 90, 435, 1926, 8175, 33930, ...
奇图 O_nA1363280, 3, 75, 1435, 25515, 436821, ...
平底锅图A1808618, 16, 26, 42, 61, 88, 119, 160, 206, 264, ...
路径图 P_nA0002920, 1, 4, 10, 20, 35, 56, 84, 120, ...
排列星图 PS_nA2840390, 1, 27, 744, 26520, 1239840, ...
棱柱图 Y_nA138179X, X, 21, 48, 85, 144, 217, 320, 441, ...
皇后图 n×nA2920570, 6, 44, 164, 440, 970, 1876, 3304, 5424, ...
车图 K_n square K_nA085537X, 8, 54, 192, 500, 1080, 2058, 3584, 5832, ...
车补图 K_n square K_n^_A2920580, infty, 54, 168, 400, 810, 1470, 2464, ...
谢尔宾斯基地毯图A29202564, 13224, 2535136, 485339728, ...
谢尔宾斯基垫片图A2901293, 21, 246, 3765, 64032, 1130463, 20215254, ...
谢尔宾斯基四面体图A2920266, 66, 1476, 42984, 1343568, 42744480, ...
星图 S_nA0002900, 1, 4, 9, 16, 25, 36, 49, 64, ...
太阳图A180863X, X, 21, 44, 75, 114, 161, 216, 279, 350, ...
日瓣图 C_n circledot K_1A180574X, X, 27, 60, 105, 174, 259, 376, 513, 690, ...
四面体约翰逊图A292061X, X, X, X, X, 300, 1050, 2940, 7056, 15120, ...
环面网格图 C_n square C_nA12265754, 256, 750, 1944, 4116, 8192, 14580, 25000, ...
转置图A2920620, 1, 21, 552, 19560, 920160, 55974240, ...
三角形图A0060110, 3, 18, 60, 150, 315, 588, 1008, 1620, ...
三角形网格图A1128513, 21, 81, 231, 546, 1134, 2142, 3762, 6237, ...
网络图A180576X, X, 69, 148, 255, 417, 616, 888, 1206, 1615, ...
轮图 W_nA002378X, X, X, X, 12, 20, 30, 42, 56, 72, ...
白主教图 n×nA292059X, 1, 8, 42, 104, 251, 464, 852, 1360, 2165, ...

下表总结了闭合形式。圈图由 Plavšić 等人 (1993) 和 Babić 等人 (2002) 考虑，路径图由 Plavšić 等人 (1993) 考虑。

Andrásfai 图1/2(3n-1)(5n-1)
反棱柱图1/2(n+1)n^2
鸡尾酒会图 K_(n×2)2n^2
完全图 K_n1/2n(n-1)
交叉棱柱图4n(n^2+2)
皇冠图 S_n^03n^2
圈图 C_n, n>=3{1/8n^3 for n even; 1/8(n-1)n(n+1) for n odd
齿轮图6n(n-1)
网格图 P_n square P_n1/3n^3(n-1)(n+1)
网格图 P_n square P_n square P_n1/2n^5(n-1)(n+1)
半立方体图{0 for n=1; n·2^(2n-5) for n>=2
超立方体图 Q_nn·4^(n-1)
莫比乌斯梯子1/4n[2n(n+2)-3-(-1)^n]
Mycielski 图 M_n1/(144)(216-272^(3+n)-563^n+814^n)
路径图 P_n1/6n(n^2-1)
车图 K_n square K_nn^3(n-1)
星图 S_n(n-1)^2
太阳图n(4n-5)
日瓣图 C_n circledot K_11/4n[(-1)^n+2n(n+4)-5]
三角形图1/4n(n-1)^2(n-2)
轮图 W_n(n-2)(n-1)

另请参阅

平均无序数, 巴拉班指数, 图距离矩阵, 基尔霍夫指数, 电阻距离, 拓扑指数, 维纳和指数

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参考文献

Babić, D.; Klein, D. J.; Lukovits, I.; Nikolić, S.; and Trinajstić, N. "Resistance-Distance Matrix: A Computational Algorithm and Its Applications." Int. J. Quant. Chem. 90, 166-176, 2002.Devillers, J. and Balaban, A. T. (Eds.). Topological Indices and Related Descriptors in QSAR and QSPR. Amsterdam, Netherlands: Gordon and Breach, pp. 26 and 108-109, 1999.Entringer, R. C.; Jackson, D. E.; and Snyder, D. "Distance in Graphs." Czech. Math. J. 26, 283-296, 1976.Fried, S. "The Disorder Number of a Graph." 7 Aug 2022. https://arxiv.org/abs/2208.03788/.Hosoya, H. "Topological Index. A Newly Proposed Quantity Characterizing the Topological Nature of Structural Isomers of Saturated Hydrocarbons." Bull. Chem. Soc. Japan 44, 2322-2239, 1971.Plavšić, D.; Nikolić, S.; Trinajstić, N.; and Mihalić, Z. "On the Harary Index for the Characterization of Chemical Graphs." J. Math. Chem. 12, 235-250, 1993.Sloane, N. J. A. Sequence OEIS A193217 in "The On-Line Encyclopedia of Integer Sequences."Wiener, H. J. "Structural Determination of Paraffin Boiling Points." J. Amer. Chem. Soc. 69, 17-20, 1947.Wiener, H. "Influence of Interatomic Forces on Paraffin Properties." J. Chem. Phys. 15, 766, 1947.Wiener, H. "Vapor Pressure-Temperature Relationships Among the Branched Paraffin Hydrocarbons." J. Phys. Chem. 52, 425-430, 1948.Wiener, H. "Relation of the Physical Properties of the Isomeric Alkanes to Molecular Structure. Surface Tension, Specific Dispersion, and Critical Solution Temperature in Aniline." J. Phys. Chem. 52, 1082-1089, 1948.Yeh, Y.-N. and Gutman, I. "On the Sum of All Distances in Composite Graphs." Disc. Math. 135, 359-365, 1994.Zerovnik, J. "Szeged Index of Symmetric Graphs." J. Chem. Inf. Comput. Sci. 39, 77-80, 1999.

在 Wolfram|Alpha 中被引用

维纳指数

引用为

Weisstein, Eric W. "维纳指数。" 来自 MathWorld-- Wolfram Web 资源。 https://mathworld.net.cn/WienerIndex.html

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