正则图 Tutte 多项式的渐近估计
组合数学
2021-05-17 v1
摘要
设 T G ( x , y ) T_G(x,y) T G ( x , y ) G G G ( G n ) n (G_n)_n ( G n ) n g ( G n ) → ∞ g(G_n)\to \infty g ( G n ) → ∞ d d d x ≥ 1 x\geq 1 x ≥ 1 0 ≤ y ≤ 1 0\leq y\leq 1 0 ≤ y ≤ 1 lim n → ∞ T G n ( x , y ) 1 / v ( G n ) = t d ( x , y ) , \lim_{n\to \infty}T_{G_n}(x,y)^{1/v(G_n)}=t_d(x,y), n → ∞ lim T G n ( x , y ) 1/ v ( G n ) = t d ( x , y ) , t d ( x , y ) = { ( d − 1 ) ( ( d − 1 ) 2 ( d − 1 ) 2 − x ) d / 2 − 1 \mbox i f x ≤ d − 1 , x ( 1 + 1 x − 1 ) d / 2 − 1 \mbox i f x > d − 1. t_d(x,y)=\left\{\begin{array}{lc} (d-1)\left(\frac{(d-1)^2}{(d-1)^2-x}\right)^{d/2-1}&\ \ \mbox{if}\ x\leq d-1,\\ x\left(1+\frac{1}{x-1}\right)^{d/2-1} &\ \ \mbox{if}\ x> d-1. \end{array}\right. t d ( x , y ) = ⎩ ⎨ ⎧ ( d − 1 ) ( ( d − 1 ) 2 − x ( d − 1 ) 2 ) d /2 − 1 x ( 1 + x − 1 1 ) d /2 − 1 \mbox i f x ≤ d − 1 , \mbox i f x > d − 1. 0 ≤ y ≤ 1 0\leq y\leq 1 0 ≤ y ≤ 1 y y y ( G n ) n (G_n)_n ( G n ) n d d d x = 1 , y = 1 x=1,y=1 x = 1 , y = 1 d d d x = 1 , y = 1 x=1,y=1 x = 1 , y = 1 d d d T G ( 2 , 1 ) T_G(2,1) T G ( 2 , 1 ) T G ( 2 , 0 ) T_G(2,0) T G ( 2 , 0 )
引用
@article{arxiv.2105.06798,
title = {Evaluations of Tutte polynomials of regular graphs},
author = {Ferenc Bencs and Péter Csikvári},
journal= {arXiv preprint arXiv:2105.06798},
year = {2021}
}