  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 $\Gamma$ is a Moore graph
of diameter $2$, then $v=k^2+1$, $\Gamma$ is strongly regular with $\lambda=0$
and $\mu=1$, and one of the following statements holds{\rm:} $k=2$ and $\Gamma$
is the pentagon, $k=3$ and $\Gamma$ is the Petersen graph, $k=7$ and $\Gamma$
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.
