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\begin{flashcard}[eigenvalue-facts-prop]
\begin{proposition}[Eigenvalue facts]
  \label{eval_facts}
  Let $G$ be a \gls{graph}. Then:
  \begin{enumerate}[(i)]
    \item \cloze{$\lambda$ is an \gls{eval} $\implies$
      $|\lambda| \le \maxdeg$.}
    \item \cloze{For $G$ \gls{connected}: $\maxdeg$ is an
      \gls{eval} $\iff$ $G$ \gls{regular}.}
    \item \cloze{For $G$ \gls{connected}: $-\maxdeg$ is
      an \gls{eval} $\iff$ $G$ \gls{regular} and
      \gls{bipartite}.}
    \item \cloze{$\lambda_{\max} \ge \mindeg$.}
  \end{enumerate}
\end{proposition}

\begin{proof}
  \phantom{}
  \begin{enumerate}[(i)]
    \item \cloze{Let $\lambda$ be an \gls{eval} with
      \gls{evec} $x$. Choose $i$ with $|x_i|$
      maximal -- without loss of generality $x_i =
      1$. Then $(Ax)_i = \sum_{j \in \nbd(i)}
      x_j$, so
      \[ |(Ax)_i| \le \sum_{j \in \nbd(i)}
      |x_j|\ le \maxdeg \cdot 1 .\]
      Then $|\lambda| \le \maxdeg$.}
    \item \cloze{\begin{enumerate}[$\Leftarrow$]
        \item[$\Leftarrow$] Let $x = (1, \ldots, 1)$ -- then $Ax =
          (\maxdeg, \maxdeg, \ldots, \maxdeg)$.
        \item[$\Rightarrow$] Choose $i$ with
          $|x_i|$ maximal. Without loss of
          generality $x_i = 1$. Then as we have
          equality in the inequality of (i) above,
          we must have $\vdeg(i) = \maxdeg$ and
          $x_j = 1$ for all $j \in \nbd(i)$. We
          can repeat at each $j \in \nbd(i)$ (as
          $x_j = 1$) to obtain $\vdeg(j) =
          \maxdeg$ and $x_{j' = 1}$ for all $j'
          \in \nbd(j)$. Continue: we obtain
          $\vdeg(j) = 1$ for all $j$ (as $G$
          \gls{connected}).
      \end{enumerate}}
    \item \cloze{\begin{enumerate}[$\Leftarrow$]
        \item[$\Leftarrow$] Let $x = 1$ on $X$,
          $-1$ on $Y$ ($X, Y$ our bipartition).
          Then $Ax = (-\maxdeg) x$.
        \item[$\Rightarrow$] Choose $i$ with
          $|x_i|$ maximal. Without loss of
          generality $x_i = 1$. Then since we have
          equality in the inequality of (i), we
          must have $\vdeg(i)=  \maxdeg$ and $x_j =
          -1$ for all $j \in \nbd(i)$. So can
          repeat at each $j \in \nbd(i)$ to obtain
          $\vdeg(j) = \maxdeg$ and $x_{j'} = 1$
          for all $j' \in \nbd(j)$. Continue: we
          get that $x_j = \pm 1$ for all $j$, and
          $jj' \in E$ implies $x_j = 1$, $x_{j'} =
          -1$ or vice versa (as $G$
          \gls{connected}). So $G$ has no odd
          cycle, hence is bipartite (and we've
          already seen that it is \kreg{\maxdeg}).
      \end{enumerate}}
    \item \cloze{By \eqref{lambda_minmax_eq}, enough to
      find $x$, $\|x\| = 1$, with $\langle Ax, x
      \rangle \ge \delta$. Let $x = (1, 1, \ldots,
      1)$. Then $(Ax)_i \ge \delta$ for all $i$.
      So $\langle Ax, x \rangle \ge \delta n$,
      with $\langle x, x \rangle = n$.\qedhere}
  \end{enumerate}
\end{proof}
\end{flashcard}

\begin{remark*}
  Proof of (ii) actually gives that if $G$ is
  \gls{connected} then \gls{eval} $k$ has
  multiplicity $1$.
\end{remark*}
\vspace{-1em}

\glsref[eval]{Eigenvalues} can relate to other
graph parameters. For example, we know
$\chromnum(G) \le \maxdeg + 1$. Can strengthen
this to:

\begin{flashcard}[chromnum-le-lambdamax-plus-1-prop]
\begin{proposition}
  For any \gls{graph} $G$, have $\chromnum(G) \le
  \lambda_{\max} + 1$.
\end{proposition}

\begin{proof}
  \cloze{Without loss of generality $|G| \ge 2$ ($|G| =
  1$ trivial). Choose $v \in \sq[n] = \vset(G)$
  with $\vdeg(v) = \mindeg$, and let $G' = G - v$.

  \textbf{Claim:} $\lambda_{\max}(G') \le
  \lambda_{\max}$.

  Once we prove this we're done,
  because we can colour $G'$ in $\lambda_{\max} +
  1$ colours using induction, and now $\vdeg(v) =
  \mindeg \le \lambda_{\max}$ (by
  \cref{eval_facts}(iv)), so can colour $v$ as
  well.

  \textit{Proof of claim:} Adjacency matrix of
  $G'$, $B$ say, is formed from $A$ by deleting
  $v$-th row and column: say $n$-th row and
  column. By \eqref{lambda_minmax_eq}, enough to
  show that
  \[ \max_{\substack{x \in \RR^{n - 1} \\ \|x\| =
  1}} \langle Bx, x \rangle \le \max_{\substack{x
  \in \RR^n \\ \|x\| = 1}} \langle Ax \rangle x .\]
  But if $x \in \RR^{n - 1}$, say $x = (x_1,
  \ldots, x_{n - 1})$, then $y = (x_i, \ldots,
  x_{n - 1}, 0) \in \RR^n$ with $\|y\| = \|x\|$
  and $\langle B x, x \rangle = \langle A y,
  y\rangle$.}
\end{proof}
\end{flashcard}

\subsection{Towards Moore Graphs}

\begin{flashcard}[strongly-regular-defn]
\begin{definition*}[Strongly regular]
  \glsadjdefn{streg}{strongly regular}{\gls{graph}}
  \glssymboldefn{streg}{$(k, a, b)$}{$(k, a, b)$}
  \cloze{Say a \gls{graph} $G$ is \emph{strongly regular}
  with \emph{parameters} $(k, a, b)$ if $G$ is
  \kreg{k}, with any two \gls{adj_v} \glspl{vertex}
  having $a$ common \glspl{neighbour} and any two
  non-\gls{adj_v} \glspl{vertex} having $b$ common
  \glspl{neighbours}.}
\end{definition*}
\end{flashcard}
\vspace{-1em}

``One step up from being \gls{regular}''.

\begin{example*}
  \phantom{}
  \begin{enumerate}
    \item $C_4$:
      \begin{tsqx}
        ! size(0.5cm);
        A .= dir 45
        B .= dir 135
        C .= dir 225
        D .= dir 315
        A--B--C--D--A
      \end{tsqx}
      has $\strp(2, 0, -2)$.
    \item $C_5$:
      \begin{tsqx}
        ! size(0.7cm);
        A .= dir 0
        B .= dir 72
        C .= dir 144
        D .= dir 216
        E .= dir 288
        A--B--C--D--E--A
      \end{tsqx}
      has $\strp(2, 0, 1)$.
    \item And in general, if $G$ is a
      \gls{moore_graph} of degree $k$, then $G$ is
      \gls{streg}, with parameters $\strp(k, 0,
      1)$.
    \item Triangle $\times$ triangle:
      \begin{center}
        \includegraphics[width=0.4\linewidth]{images/b1a9ed33e1de4f35.png}
      \end{center}
      has $\strp(4, 1, 2)$.
  \end{enumerate}
\end{example*}
\vspace{-1em}

Seems that `\gls{streg}' is a bit more than
\gls{regular}, but\ldots