Send private message to my friend(s) using my Android application

It is not possible to send private messages on the behalf of the user using the graph api.

You should however be able to use the Send Dialog, though I haven't tried it on android, but it should be something like:

Bundle params = new Bundle();
params.putString("to", "USER_ID");
params.putString("name", "TITLE HERE");
params.putString("link", "A URL"); // this link param is required

facebook.dialog(context, "send", params, new DialogListener() {
    @Override
    public void onComplete(Bundle values) {
       ....
    }

    @Override
    public void onFacebookError(FacebookError error) {}

    @Override
    public void onError(DialogError e) {}

    @Override
    public void onCancel() {}
});

Another approach you can use is the Chat API with which you can send messages on the behalf of the user, it requires the xmpp_login permission and you to implement an xmpp client.

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Edit

Since this dialog is not supported yet in android, you have 3 options:

1. Wait for facebook to implement the dialog for android.
2. Try to open the dialog in a browser (the url for that is in the docs) in the mobile device.
3. Ask for the xmpp_login and add a xmpp client (i.e.: asmack) and with that you can implement your own "Send Message" dialog.