Let $G=(V,E)$ with $V=\{1,$dots,$n\}$ and $E=\{21,$dots,$n1\}.$ This is called a

star graph and is shown below for $n=5.$ Note: the $n=5$ picture is just an example;

your answers should be for general $n.$ Answer the following questions:

A$.$ What do you expect the hub and authority scores to be$?$ Why? $(2$ points$)$

B$.$ Let $A$ be the adjacency matrix of $G.$ Find length$-n$ vectors $x$ and $y$ such that

$A=x{y}^T,$ and compute $A{A}^T$ in terms of $x$ and $y.(5$ points$)$

C$.$ Using your answer to Part B$,$ prove that $x$ is an eigenvector of $A{A}^T.$ What is the

associated eigenvalue? $($Your answer should be in terms of $n.)(5$ points$)$

D$.$ Prove that all other eigenvalues of $A{A}^T$ are zero. $(5$ points$)$

E$.$ Using your answers above, what are the hub and authority scores? Justify your

answer. Is this what you expected in Part A$?(8$ points$)$