Exercise 2.3.19

For the following questions, I would like to prove them in the language of Graph Theory.
Let $G=G(B)$ be the graph associated with the symmetric matric $B$. And ${(B^3)}_{\mathit{ii}}$ is the number of walks of length $3$ from $i$ to $i$. If $i$ is in some clique, then there must be a walk of length $3$ from $i$ back to $i$ since a clique must have a number of vertexes greater than $3$. Conversely, if ${(B^3)}_{\mathit{ii}}$ is greater than zero, this means there is at least one walk of length $3$ from $i$ to $i$, say $i\to j\to k\to i$. Note that $i$, $j$, and $k$ should be different vertices since the length is $3$ and there is no loop. So $i$, $j$, and $k$ must be a triangle, this means three vertices adjacent to each other. So $i$ is contained in $\{i,j,k\}$ and so contained in some clique.