You are not using strict and warnings for starters. Next:

my $root= <>;
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You should chomp it, otherwise $root will contain newline in the end.

my %graph= %graph;
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Oh, really?

$dist{$n2} = $dist{$n} + $graph{$n1}{$n2};
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And this is why you should use strict, note $dist{$n}, shouldn't it be $dist{$n1}?