# Show that if $G$ is connected then $L(G)$ is connected

Well, the exercise it's as the title says. I know that if $$G$$ is connected then for every pair of vertex $$u,v$$ in $$G$$ there's a walk between them. So when the Line Graph $$L(G)$$ is constructed those edges in $$G$$ where the walk between $$u$$ and $$v$$ is formed will be adjacent vertex in the Line Graph $$L(G)$$. Since $$G$$ is connected therefore $$L(G)$$ must be connected.

I'm not sure about if the argument is valid, also I'm looking for a more formal/detailed proof, because I think there's a lot of holes in this argument.

The argument given is correct, though a bit informal. Here's an elaboration:

To be more precise, suppose $$G$$ is connected. Then for any two vertices $$u,v\in G$$, there is a path $$P=u\to x_1\to x_2\to\cdots\to x_n\to v$$ where the $$x_i$$'s are some other vertices of $$G$$.

By the existence of such a path $$P$$, we can say that $$u$$ is adjacent to $$x_1$$, $$x_i$$ is adjacent to $$x_{i+1}$$ (for $$1\leq i\lt n$$) and $$x_n$$ is adjacent to $$v$$

By the definition of the line graph $$L(G)$$, this means $$(u,x_1),(x_i,x_{i+1})$$ and $$(x_n,v)$$ are vertices of $$L(G)$$ [for $$1\leq i\lt n$$]

Now, since $$(u,x_1)$$ and $$(x_1,x_2)$$ share a common endpoint $$x_1$$, they must be adjacent in $$L(G)$$. Similarly, $$(x_i,x_{i+1})$$ is adjacent to $$(x_{i+1},x_{i+2})$$ for $$1\leq i\lt n-1$$ and $$(x_{n-1},x_n)$$ is adjacent to $$L(G)$$

So, we have a path $$L_P$$ in $$L(G)$$ as $$L_p=(u,x_1)\to (x_1,x_2)\to\cdots\to(x_{n-1},x_n)\to(x_n,v)$$

Hence, for every path $$P$$ in $$G$$, there is a path $$L_P$$ in $$L(G)$$.

Suppose $$(a,b)$$ and $$(A,B)$$ are two vertices of $$L(G)$$. We show that there is a path between them: By definition of $$L(G)$$, all four of $$a,b,A,B$$ are vertices of $$G$$ with $$a$$ adjacent to $$b$$, ie, $$a\to b$$ and $$A$$ adjacent to $$B$$, ie, $$A\to B$$; since $$G$$ is connected, there must be a path $$P_{bA}$$ from $$b$$ to $$A$$ which translates to a path $$L_{\large P_{bA}}$$ in $$L(G)$$ which gives us a path $$a\to b\to P_{bA}\to A\to B$$ in $$G$$ and a path $$(a,b)\to L_{\large P_{bA}}\to (A,B)$$ in $$L(G)$$