According to my teacher, you can use this expression without the のいかん.
Also, the word before にかかわらず needs to have at least two meanings built into the words, for example: 性別, 国籍, 晴雨 etc. If not have to have to include at least two words with opposite meaning.
This expression can also be used with verbs, with dic + nai , for example:
There's another page with the shortened form: http://www.renshuu.org/index.php?page=grammar/individual&id=263 (They are linked as similar entries at the top of the page).
As for the noun restriction, it's not that I doubt what your teacher said, but can you ask her/him to comment on these sentences (taken from a grammar text): 国民の賛意のいかんにかかわらず、その提案へ国会で可決されるだろう。 理由のいかんによらず、殺人は許されないことだ。
