<td style="width:62px" class="deletesuppliernetwork '.$rows["id"].'"><img src="/image/delete.png" style="margin:0 0 0 17px" ></td>
de index van uw ID is 1, dat is de tweede index. niet 2.
$.ajax({
type: "POST",
url: "suppliernetwork/delete.php",
data: "rowid="+arr[1]+"&supplier="+supplier,
success: function(data){
$('.ajax').html($('.ajax input').val());
$('.ajax').removeClass('ajax');
}});