论文标题
在二分图$ d（k，q）$的周期周期中
On the girth cycles of the bipartite graph $D(k,q)$
论文作者
论文摘要
对于Integer $ k \ geq2 $和Prime Power $ Q $，由Lazebnik和Ustimenko（1995）提出的代数二分$ D（K，Q）$不仅在极端图理论中有意义，而且在编码理论和密码学中也是有意义的。该图是$ q $ - 定型，边缘传递，至少$ k+4 $。为了确切的围墙$ g = g（d（k，q））$， Füredi等。 （1995）猜想$ g = k+5 $，用于奇数$ k $和$ q \ geq4 $。 当$（k+5）/2 $是$ Q-1 $的任意因素和$ \ mathbb {f} _q $的特征的任意功率时，该猜想在2016年有效。 在本文中，我们确定$ d（k，q）$的所有围绕$ 3 \ leq k \ leq 5 $，$ q> 3 $，以及以$ 3 \ leq k \ leq8 $，$ q = 3 $的价格。
For integer $k\geq2$ and prime power $q$, the algebraic bipartite graph $D(k,q)$ proposed by Lazebnik and Ustimenko (1995) is meaningful not only in extremal graph theory but also in coding theory and cryptography. This graph is $q$-regular, edge-transitive and of girth at least $k+4$. For its exact girth $g=g(D(k,q))$, Füredi et al. (1995) conjectured $g=k+5$ for odd $k$ and $q\geq4$. This conjecture was shown to be valid in 2016 when $(k+5)/2$ is the product of an arbitrary factor of $q-1$ and an arbitrary power of the characteristic of $\mathbb{F}_q$. In this paper, we determine all the girth cycles of $D(k,q)$ for $3\leq k\leq 5$, $q>3$, and those for $3\leq k\leq8$, $q=3$.
