Currently reading Algebraic Graph Theory by Godsil and Royle, and I came across the following statement:
If $X$ has no triangles (cliques of size three), then any vertex of $L(X)$ with at least two neighbours in one of these cliques must be contained in that clique. Hence the cliques determined by the vertices of X are all maximal.
$L(X)$ in this case is the line graph of $X$.
I'm having trouble interpreting this statement. Say that $X$ indeed has no triangles. Then I'm thinking that if $L(X)$ has two vertices such that the edges that they represent are part of the same clique in $X$, then it must either be the same vertex or there is a triangle since the third edge is adjacent to both of them. But we assumed that $X$ has no triangles. What am I missing here?