Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Can someone tell me the answer to these?

  1. Determine $|V|$ given that $G(V,E)$ is a regular graph with $E=12$.
  2. If $G(V,E)$ be connected undirected graph what is largest value of $|V|$, if $|E|=19$ and $\deg(v)\ge 4$ for all $v\in V$.
share|improve this question
We're here to help you come up with answers, not just give them to you. What progress have you made so far? –  Alexander Gruber Dec 29 '12 at 18:08
(-1) for "Can someone tell me the answer". –  TMM Dec 29 '12 at 18:21

1 Answer 1

up vote 3 down vote accepted


(1) A $12$-cycle is a $2$-regular graph with $12$ edges and $12$ vertices. The union of two copies of $K_4$ is a $3$-regular graph with $12$ edges and $8$ vertices. Thus, $|V|$ cannot be uniquely determined from the given data. The best you can hope to do is limit the possibilities. Since $G$ is regular, there is a $d$ such that $\deg(v)=d$ for all $v\in V$. Then $$\sum_{v\in V}\deg(v)=d|V|\;.$$ By the handshaking formula we know that $$\sum_{v\in V}\deg(v)=2|E|=2\cdot 12=24\;.$$ So what are the possibilities for $|V|$?

(2) You know that $$38=2\cdot19=2|E|=\sum_{v\in V}\deg(v)\ge 4|V|\;.$$ This immediately gives you an upper bound on $V$, and you have to determine whether there is a connected graph with that number of vertices and $19$ edges.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.