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\homework{2}{Week 3}

Attempt the problems that seem interesting!  Easier exercises are marked with $(-)$ signs; harder ones are marked by $(*)$.  Open questions are denoted by writing $(**)$, as they are presumably quite hard.  Oh!  Also, typos build character: if you find any (not that there ever could be such things in my problem sets,) correct them to the most reasonable thing you can think of and proceed from there!

\begin{enumerate}
\item In class, we found the eigenvectors/values of the directed cycle $D_n$.  Use these to find the eigenvectors/values of the undirected cycle $C_n$.
\item Using our results on the cycle graph, find the spectrum of the path graph $P_n$.
\item Answer the question we asked yesterday on the HW/today in class: if $G_1$ and $G_2$ are a pair of graphs with the same spectrum, are $G_1$ and $G_2$ isomorphic?
\item Prove or disprove: There is no graph with eigenvalue -1/2.
\item $(-)$ What happens to a graph when you add additional vertices that aren't connected to anything?
\item $(-)$ In terms of the graphs $G_1$ and $G_2$, what's the spectrum of the graph given by the disjoint union of $G_1$ and $G_2$?
\item Find the spectrum of the Petersen graph.  For extra style points, find it without ever actually looking at an adjacency matrix.
\addtocounter{enumi}{1} 
\item Prove the series of linear algebra propositions we stated in class, should you not believe them:
\begin{enumerate}
\item The area of a parallelogram spanned by the two vectors $(a,b)$ and $(c,d)$ is $|ad-bc|$; similarly, the area of a parallelepiped spanned by $(a,b,c), (d,e,f),(x,y,z)$ is $|aez - afy + bfx - bdz + cdy - cex |$.
\item If $I_n$ is the $n \times n$ identity matrix, then $\det(I_n) = 1$.
\item Suppose that $A$ is a $n \times n$ matrix.  If $A'$ is the matrix acquired by multiplying the $k$-th row of $A$ by some constant $\lambda$, then $\det(A') = \lambda \det(A)$.
\item For any pair of $n \times n$ matrices $A, B$, $\det(AB) = \det(A)\cdot \det(B)$.  In particular, this tells us that $\det(A^{-1}) = 1/\det(A)$, whenever $A$ is an invertible matrix.
\end{enumerate}
\end{enumerate}


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