%! TEX root = SGT.tex
% vim: tw=80
% 26/05/2025 11AM

\gls{ndlam}

$n$: number of vertices, $d$-regular, $\lambda_k(A) \in [-\lambda, \lambda]$, $k
\neq n$.

Ramanujan graph: $(n, d, 2\sqrt{d - 1})$-graph.
\begin{itemize}
  \item Petersen $(10, 3, 2\sqrt{2})$-graph.
  \item Complete (bipartite), $d = n - 1$ big :(.
  \item Paley $\Cay(\Zbb / p\Zbb, \{x^2 : x \in \Zbb / p\Zbb, x \neq 0\})$, $p
    \equiv 1 \pmod{4}$. $d = \frac{p - 1}{2}$ big :(.
\end{itemize}

Alon-Boppana: For every $\eps > 0$ fixed $d \ge 3$, there are finitely many $(n,
d, 2\sqrt{d - 1} - \eps)$-graphs.

Bipartite: $(n, d, \lambda)$-graph, $n$ vertices, $d$-regular, $\lambda_k(A) \in
[-\lambda, \lambda]$ for $k \neq n, 1$.

\begin{fcthm}[Lubotsky-Phillips-Sanak / Margoulis]
  Assuming:
  - $p$ prime
  - $d = p + 1$
  - $n$ arbitrarily large
  Then:
  there exists a $(n, d, 2\sqrt{d - 1})$-graph.
\end{fcthm}

\noproof

Goal:

\begin{fcthm}[Marcus, Spielman, Srivastava]
  \label{thm:exists bipartite raman}
  Assuming:
  - $d \ge 3$
  Then:
  there are $(n, d, 2\sqrt{d - 1})$-bipartite graphs for arbitrarily large $n$.
\end{fcthm}

Strategy: Bilu and Linial. Lifts of graphs:

\begin{fcdefn}[2-lift]
  A \emph{2-lift} of a graph $G = (V, E)$ is a graph $\hat{G} = (\hat{V},
  \hat{E})$ with
  \begin{citems}
    \item $x \in V$ $\implies$ $x_0, x_1 \in \hat{V}$.
    \item $xy \in E$ $\implies$ either $x_0 y_0, x_1 y_1 \in \hat{E}$ or $x_0
      y_1, x_1 y_0 \in \hat{E}$.
  \end{citems}
  (and no other vertices or edges).
\end{fcdefn}

\begin{center}
  \includegraphics[width=0.6\linewidth]{images/aada27d44fbc4293.png}
\end{center}

\begin{fcdefn}[Signing]
  $S : V \times V \to \Rbb$ is a \emph{signing} of $G$ if
  \[
    S(x, y)
    = \begin{cases}
      \pm 1
      & \text{if $A(x, y) = 1$} \\
      0
      & \text{if $A(x, y) = 0$}
    \end{cases}
  \]
  and $S(x, y) = S(y, x)$ (symmetric).
  So can think of $S$ as a function $E \to \{\pm 1\}$.
\end{fcdefn}

$A_S^+(x, y) = A(x, y) \indicator{S(x, y) = 1}$.
$A_S^-(x, y) = A(x, y) \indicator{S(x, y) = -1}$.
Have $A = A_S^+ + A_S^-$, $S = A_S^+ - A_S^-$.

\begin{fclemma}[]
  The eigenvalues of $\hat{G}$ \cloze{are the eigenvalues of $\adjm_G$ (old)
  together with the eigenvalues of $S$ (new).}
\end{fclemma}

\begin{proof}
  \[
    \adjm_{\hat{G}_S}
    = \begin{pmatrix}
      A_S^+ & AS_S^- \\
      A_S^- & AS_S^+
    \end{pmatrix}
  \]
  Let $A \varphi = \lambda \varphi$.
  Then
  \[
    \adjm_{\hat{G}_S}
    \begin{pmatrix}
      \varphi \\
      \varphi
    \end{pmatrix}
    = \begin{pmatrix}
      A_S^+ \varphi + A_S^- \varphi \\
      A_S^- \varphi + A_S^+ \varphi
    \end{pmatrix}
    = \begin{pmatrix}
      A \varphi \\
      A \varphi
    \end{pmatrix}
    = \begin{pmatrix}
      \lambda \varphi \\
      \lambda \varphi
    \end{pmatrix}
    = \lambda \begin{pmatrix}
      \varphi \\
      \varphi
    \end{pmatrix}
  .\]
  Let $S\eta = \mu\eta$.
  Now
  \[
    \adjm_{\hat{G}_S} \begin{pmatrix}
      \eta \\
      -\eta
    \end{pmatrix}
    = \begin{pmatrix}
      A_S^+ \eta - A_S^- \eta \\
      A_S^- \eta - A_S^+ \eta
    \end{pmatrix}
    = \begin{pmatrix}
      S \eta \\
      -S \eta
    \end{pmatrix}
    = \begin{pmatrix}
      \mu \eta \\
      -\mu \eta
    \end{pmatrix}
    = \mu \begin{pmatrix}
      \eta \\
      -\eta
    \end{pmatrix}
    .
    \qedhere
  \]
\end{proof}

\begin{conjecture}[Bilu, Linial]
  If $G$ is $d$-regular, then there exists a signing $S : E(G) \to \{\pm 1\}$
  whose eigenvalues are in $[-2\sqrt{d - 1}, 2\sqrt{d - 1}]$.
\end{conjecture}

\begin{fcthm}[Bilu, Linial]
  Can find signings $S$ with eigenvalues $\lambda$ satisfying $|\lambda| =
  O(\sqrt{d(\log d)^3})$.
\end{fcthm}

\begin{fcthm}[Marcusm, Spielman, Srivastava]
  \label{thm:exists signing}
  Assuming:
  - $G$ is a $d$-regular graph
  Then:
  there exists a signing with eigenvalues $\lambda$ with $\lambda \le 2\sqrt{d -
  1}$.
\end{fcthm}

\begin{proof}[\cref{thm:exists signing} implies \cref{thm:exists bipartite raman}]
  \leavevmode
  \begin{itemize}
    \item Start with $K_{d, d}$.
    \item Keep applying \cref{thm:exists signing} to find signing.
    \item Build lift with that signing.
    \item 2-lift of bipartite graph is bipartite.
    \item Spectrum of adjacency matrix of bipartite graph is symmetric around
      $0$.
      \qedhere
  \end{itemize}
\end{proof}

\begin{notation*}
  For $\pi \in \Sym(X)$, let $|\pi|$ be the number of inversions.
\end{notation*}

\cref{thm:exists signing}:
$S \sim \Unif(\{\pm 1\}^{E(G)})$.
\begin{align*}
  \Ebb_S \det(xI - S)
  &= \Ebb_S \left( \sum_{\pi \in \Sym(X)} (-1)^{|\pi|} \prod_{y \in V} (xI -
  S)(y, \pi(y)) \right) \\
  &= \sum_{k = 0}^n x^{n - k} (-1)^k \sum_{\substack{T \subseteq V \\ |T| = k}}
  \sum_{\pi \in \Sym(T)}
  \Ebb_S \left( (-1)^{|\pi|} \prod_{y \in T} (xI - S) (y, \pi(y)) \right)
\end{align*}
For $xy \in E$: $\Ebb S(x, y)^{2k + 1} = 0$, $\Ebb S(x, y)^{2k} = 1$.
\begin{center}
  \includegraphics[width=0.3\linewidth]{images/7939ed525e9b44a6.png}
\end{center}
\begin{align*}
  &= \sum_{\substack{k = 0 \\ \text{$k$ even}}}^n x^{n - k} (-1)^k
  \sum_{\substack{\text{$M$ matching} \\ \text{of size $\le \frac{k}{2}$}}}
  (-1)^{k/2} \ldots \\
  &= \sum_{k \ge 0} x^{n - 2k} (-1)^k M_k(G) \\
  &= \mu_G(x)
\end{align*}
where $M_k(G)$ is the number of matchings of size $k$ in $G$.

$\mu_G(x)$ is the matching polynomial of $G$.

Heilman-Lieb-Godsil:
\begin{itemize}
  \item $\mu_G(x)$ is real rooted for all $G$.
  \item If $G$ has degree $\le d$, then $\mu_G(x)$ has roots in $[-2\sqrt{d -
    1}, 2\sqrt{d - 1}]$.
\end{itemize}