This should help you over the first hurdle. At the end of the following snippet you have two hashes to start with. One hash describes the edges and the other the vertices found. Putting these informations into hashes let you access them by node/vertex number quite easily:

#!/usr/bin/perl
use strict;
use warnings;

use 5.010;
use Data::Dumper;

my $VERY_BIG_NUM = 2_000_000_000;

my @problem = grep { $_ !~ /^\s*#/ and $_ !~ /^\s*$/ } map { chomp; $_
+ } <DATA>;
die "ERROR: Problem description incomplete."
    unless(@problem >= 3);

my ($num_of_vertices, $num_of_edges) = split / /, shift @problem;
die "ERROR: Not enough vertices given."
    unless($num_of_vertices and $num_of_vertices > 1);
die "ERROR: No edge given."
    unless($num_of_edges);

die "ERROR: Number of edges not equal to declare numberd $num_of_edges
+ of edges."
    if(@problem - 1 != $num_of_edges);

# Get the problem to solve
my ($start_vertex, $end_vertex) = split / /, pop @problem;
say "Problem with $num_of_vertices vertices and $num_of_edges edges";
say "Determine shortes path from vertex $start_vertex to $end_vertex";

# read the remaining edges
my %vertices;
my %edges;

foreach my $edge (@problem) {
    my ($start, $end) = split / /, $edge;
    $edges{$start}->{$end} = {
        'weight' => 1,
    };
    $edges{$end}->{$start} = $edges{$start}->{$end};

    foreach my $node ($start, $end) {
        $vertices{$node} = {
            'predecessor' => undef,
            'distance' => $node == $start_vertex ? 0 : $VERY_BIG_NUM,
        };
    }
}

say "Vertices: " . Dumper(\%vertices);
say "Edges: " . Dumper(\%edges);


__DATA__
5 6
1 2
2 3
2 4
4 5
1 3
3 5
1 5
[download]

We all - at least me - will be interested to see how your algorithms will be implemented.

Regards
McA