using graph theory .It is possible that in a group of 9 people each Is one friend with exactly 5 of the others? [closed]

I'm having problems resolving this question. I need to use grapgh theory

As mentioned by @JMoravitz, we can use the handshaking lemma for this question. The handshaking lemma claims that every finite undirected graph (in your question, friendships aren't directed - they're $2$-way so that's fine) has an even number of vertices with odd degree. In our case, this can be translated as "the amount of people with an odd number of friendships within our group is even."

Clearly, that violates our lemma, as there is an odd amount of vertices with odd degree (or, odd number of people with an odd number of friendships). Therefore it's not possible for everyone in a group of 9 friends to be friends with exactly friends with 5 of the others.

You can read more about the lemma here.