Paths::Graph has a debug method that runs the gamut of the entire algorithm. Here's an example:

#!/usr/bin/perl 

BEGIN {
          $| = 1;
          $^W = 1;
}

use autodie;
use strictures 1;

my %graph = (
          A => { 
                  B => '10.0.0.128', 
                  C => '10.0.0.132', 
                  G => '10.0.0.129',
          },
          C => { 
                  A => '10.0.0.135', 
                  B => '10.0.0.128',
                  F => '10.0.0.134',
          },
);

use Paths::Graph;
use strict qw/refs/;
use warnings FATAL => 'syntax';

my $obj = Paths::Graph->new( 
        -origin   => "A", 
        -destiny => "B", 
        -graph   => \%graph,
);
$obj->debug();

my @paths = $obj->shortest_path();
foreach my $path (@paths) {
        print "Shortest Path:"
        .  join( "->", @{$path} )
        .  " Cost:"
        .  $obj->get_path_cost( @{$path} );
        print "\n";
}
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Please note that your use of @${path} is incorrect. The correct way is @{$path}.