What is the number of edges in a 2-regular graph that has 7 vertices?

Question

I've just started learning about graph theory and I am doing some exercises online.

This is the question I'm currently on:

What is the number of edges in a 2-regular graph that has 7 vertices?

And

What is the smallest number of edges in a connected graph with 6 vertices?

Answer 1

For the first, remember your hand-shaking lemma:

$$\sum\limits_{v\in V}\deg(v) = 2|E|$$

Next, remember that it means to be $2$-regular and notice that the problem tells you there are seven vertices.

The graph being $2$-regular means that $\deg(v)=2$ for all $v\in V$, so the above simplifies as $7\cdot 2 = 2|E|$, and so $|E|=7$

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For the second, recall that any connected graph has a subgraph which is a spanning tree and that trees are the graphs with the smallest number of edges such that the graph remains connected.

How many edges are there in a tree over six vertices?

A tree on $n$ vertices has $n-1$ edges, so there are at least $5$ edges for a connected graph on six vertices.