How can we express the "intentional ignoring" or someone in one word? Particularly if someone is ignoring others by his/her behaviour, showing that he/she isn't interested in talking or communicating, etc.
It's not one word, but the idiom cold shoulder fits nicely.
TFD defines it as:
cold-shoulder - pay no attention to, disrespect : She cold-shouldered her ex-fiance
Collins mentions it can be used as a verb or a noun:
the cold shoulder (noun) a show of indifference; a slight
cold-shoulder (verb) to treat with indifference
cold-shoulder (v.) to be unfriendly toward someone you know
If a single idiom won't suffice, and you absolutely must have a single word, you could try some of the synonyms listed at TFD, such as dismiss, disregard, or ignore. I really like cold shoulder, though, because that idiom is used much more personally than its synonyms. In other words, you might dismiss, disregard, or ignore my advice, but you wouldn't cold-shoulder my advice; you'd only cold-shoulder me. Giving someone the cold shoulder is very much directed at the person you are ignoring, so it might be an ideal way to describe what you are trying to say.
To ostracize, “To exclude (a person) from society or from a community, by not communicating with them or by refusing to acknowledge their presence; to refuse to talk to or associate with; to shun” may be a good choice. Note, to shun is “To avoid, especially persistently”.
Also consider snub, “To slight, ignore or behave coldly toward someone” and proscribe, “To banish or exclude”. Banish means “To send someone away and forbid that person from returning”; the banishment aspect of proscription is less important than the exclusion aspect, for the current question.
The expression send to Coventry means giving the silent treatment to someone; for example, when Mr. Grainger says (in the “Forward Mr. Grainger” episode of Are You Being Served?) “You're not sending me to Coventry, are you?” and gets no reply from his coworkers.
As suggested in Hellion's comment, silent treatment (“A form of social sanction that consists of ignoring a particular individual, neither speaking to that person nor responding to his or her words”) is worthy of notice. Wiktionary shows the following example:
Finally we gave him the silent treatment, and for weeks before he died we neither spoke to him nor did he speak to us. – 1917, Jack London, “That Dead Men Rise Up Never”
I think to blank would fit the bill quite well too, although I like snub.
3 British informal deliberately ignore (someone):
I just blanked them and walked out
There's a somewhat old-fashioned usage of the word "cut" that means to ignore someone deliberately in a social situation.
For example, see the last section (heading: The "Cut Direct") of this page from an etiquette manual.
Shine [someone] on as in:
I've been trying all day to talk to him about the money but he just keeps shining me on.