论文标题
带有参数（3250,57,0,1）的摩尔图不存在
Moore graph with parameters (3250,57,0,1) does not exist
论文作者
论文摘要
如果定期图$ k $和直径$ d $具有$ v $顶点，则$ v \ le 1+k+k+k（k-1）+\ dots+k（k-1）^{d-1}。$ v = 1+k+k+k+k（k-1）+\ dots+\ dots+k（k-1） Damerell证明了Moore $ k \ ge 3 $的摩尔图$ 2 $。如果$γ$是直径$ 2 $的摩尔图，那么$ v = k^2+1 $，$γ$非常规是常规的，$λ= 0 $ and $μ= 1 $ = 1 $，并且以下一项陈述中的一个含有{\ rm：} $ k = 2 $ k = 2 $ and $γ$，$ k = 3 $ and $ k = 3 $和$γ$ k = k = k = k = k = k = k = k = k = k = k = k = k = k = k = Hoffman-singleton图，或$ k = 57 $。摩尔学位$ 57 $的存在是未知的。 Jurishich和Vidali已经证明，摩尔图的存在$ k> 3 $等于存在带有距离的与距离的距离，带有相交阵列$ \ {k-3,2，k-3,2; 1,1，k-3 \} $ $ \ {55,54,2; 1,1,54 \} $）。在本文中，我们证明了一个带有相交数组$ \ {55,54,2; 1,1,54 \} $的距离定型图。作为推论，我们证明了摩尔的学位$ 57 $不存在。
If a regular graph of degree $k$ and diameter $d$ has $v$ vertices then $$v\le 1+k+k(k-1)+\dots+k(k-1)^{d-1}.$$ Graphs with $v=1+k+k(k-1)+\dots+k(k-1)^{d-1}$ are called Moore graphs. Damerell proved that a Moore graph of degree $k\ge 3$ has diameter $2$. If $Γ$ is a Moore graph of diameter $2$, then $v=k^2+1$, $Γ$ is strongly regular with $λ=0$ and $μ=1$, and one of the following statements holds{\rm:} $k=2$ and $Γ$ is the pentagon, $k=3$ and $Γ$ is the Petersen graph, $k=7$ and $Γ$ is the Hoffman-Singleton graph, or $k=57$. The existence of a Moore graph of degree $57$ was unknown. Jurishich and Vidali have proved that the existence of a Moore graph of degree $k>3$ is equivalent to the existence of a distance-regular graph with intersection array $\{k-2,k-3,2;1,1,k-3\}$ (in the case $k=57$ we have a distance-regular graph with intersection array $\{55,54,2;1,1,54\}$). In this paper we prove that a distance-regular graph with intersection array $\{55,54,2;1,1,54\}$ does not exist. As a corollary, we prove that a Moore graph of degree $57$ does not exist.
