hence quasiextremal black holes coupled to moduli are no heavier than an
extremal Reissner-Nordström black hole of the same charge that would result if
the moduli were artificially frozen in place at their asymptotic values.

####  A.2.2 The attractor mechanism

A smooth horizon requires $R(\rho)$ to remain finite as $\rho\rightarrow 0$.
Defining $\chi\mathrel{:=}\psi+\log z$,
$\mathrm{e}^{-\chi}=(d-3)V_{d-2}R^{d-3}$, so $\chi(z)$ must remain finite a
$z\rightarrow\infty$. Written in terms of $\chi$ and $\tau=-\log z$, the
equations of motion become:

$\displaystyle\frac{\mathrm{d}^{2}\phi^{i}}{\mathrm{d}\tau^{2}}+\Gamma^{i}_{\;jk}\frac{\mathrm{d}\phi^{j}}{\mathrm{d}\tau}\frac{\mathrm{d}\phi^{k}}{\mathrm{d}\tau}$
$\displaystyle=-\frac{\mathrm{d}\phi^{i}}{\mathrm{d}\tau}+\frac{1}{2}G^{ij}Q^{2}_{,j}\mathrm{e}^{2\chi},$
(A.24) $\displaystyle\frac{\mathrm{d}^{2}\chi}{\mathrm{d}\tau^{2}}$
$\displaystyle=-\frac{\mathrm{d}\chi}{\mathrm{d}\tau}+k_{N}\mathrm{e}^{2\chi}Q^{2}-1,$
(A.25)
$\displaystyle\biggl{[}\frac{\mathrm{d}\chi}{\mathrm{d}\tau}\biggr{]}^{2}\\!\\!\\!+k_{N}G_{ij}\frac{\mathrm{d}\phi^{i}}{\mathrm{d}\tau}\frac{\mathrm{d}\phi^{i}}{\mathrm{d}\tau}$
$\displaystyle=-2\frac{\mathrm{d}\chi}{\mathrm{d}\tau}+k_{N}\mathrm{e}^{2\chi}Q^{2}-1.$
(A.26)

(A.24) and (A.25) describe (generalized) Newtonian motion in the potential
$V(\chi,\phi^{i})=k_{N}^{-1}\chi-\frac{1}{2}\mathrm{e}^{2\chi}Q^{2}(\phi^{i})$
with metric $G_{IJ}=\mathop{\rm diag}\nolimits(k_{N}^{-1},G_{ij})$ and a
linear drag force. A smooth horizon requires $\chi(\tau)$ and $\phi^{i}(\tau)$
to approach finite values $\chi_{h}$ and $\phi_{h}^{i}$ respectively as
$\tau\rightarrow-\infty$, which can only occur at a critical point of
$V(\chi,\phi^{i})$, i.e.,

$k_{N}\mathrm{e}^{2\chi_{h}}Q^{2}(\phi_{h})=1,\qquad Q^{2}_{,i}(\phi_{h})=0,$
(A.27)

from which the constraint (A.26) automatically follows. Thus, $\phi_{h}$ is a
critical point of $Q^{2}(\phi)$ and $Q^{2}(\phi_{h})$ determines the horizon
area:

$A_{h}=V_{d-2}R_{h}^{d-2}=V_{d-2}\left[\frac{\sqrt{k_{N}}Q(\phi_{h})}{(d-3)V_{d-2}}\right]^{\frac{d-2}{d-3}}.$
(A.28)

This is the attractor mechanism [303, 304, 305, 306, 307]. The trivial
solution $\phi^{i}(z)=\phi^{i}_{h}$ is Reissner-Nordström, with mass
$M_{0}=k_{N}^{-1/2}Q(\phi_{h})$. All other solutions are strictly heavier
since
$\mathcal{W}(z)=k_{N}^{-1}\frac{\mathrm{d}}{\mathrm{d}z}(\mathrm{e}^{-\psi})$
evaluates to $M_{\text{BH}}=k_{N}^{-1}(-\dot{\psi}_{\infty})$ and
$M_{0}=k_{N}^{-1}\mathrm{e}^{-\chi_{h}}$ at $z=0$ and $z=\infty$ respectively,
and
$\dot{\mathcal{W}}=-\mathrm{e}^{-\psi}G_{ij}(\phi)\dot{\phi}^{i}\dot{\phi}^{j}\leqslant
0$ per (A.15).222222Thus, $k_{N}^{-1/2}Q(\phi_{h})\leq M\leq
k_{N}^{-1/2}Q(\phi_{\infty})$.

If $\phi_{h}^{i}$ is a local minimum of $Q^{2}(\phi)$ (the attractor point is
“stable”) then $(\chi_{h},\phi_{h}^{i})$ is a local maximum of
$V(\chi,\phi^{i})$ and we can roll off the hill in any direction, hence there
are attractor solutions for any nearby choice of $\phi_{\infty}^{i}$. This is
not necessarily the case farther from the attractor point, where the family of
solutions to (A.24–A.25) beginning at $\phi_{h}$ may encounter turning points
and/or caustics.

Unstable critical points of $Q^{2}(\phi)$ also admit attractor solutions but
by the same reasoning these do not exist for generic values of
$\phi_{\infty}^{i}$, and so play little role in the determing the extremality
bound.

####  A.2.3 Fake superpotentials and a Bogomol’nyi bound

Combining the preceeding observations, we see that there are families of
quasiextremal solutions corresponding to each stable attractor point
$\phi_{h}^{i}$, with mass $M$ determined by the choice of vacuum
$\phi_{\infty}^{i}$. The resulting mass function $M=W(\phi_{\infty})$ is also
known as the “fake superpotential” [308, 309, 310, 311, 312] associated to the
attractor point in question. Because each member of the family has the same
charge and horizon-area (entropy), we can identify $\frac{\partial
W}{\partial\phi^{i}}$ as the scalar charge $\mu_{i}$ via the first law $\delta
M=\mu_{i}\delta\phi_{\infty}^{i}+\Phi_{h}^{A}\delta
Q_{A}+\frac{1}{\kappa^{2}}g_{h}\delta A_{h}$ [313]. Therefore, due to the no-
force condition (A.23), the fake superpotential satisfies

$k_{N}W^{2}(\phi)+G^{ij}(\phi)\partial_{i}W(\phi)\partial_{j}W(\phi)=Q^{2}(\phi),$
(A.29)

where $W(\phi)$ has a global minimum at $\phi_{h}^{i}$. Solutions to the non-
linear first-order differential equation (A.29) are in general highly non-
unique. However, the condition that $W(\phi)$ has a minimum at $\phi=\phi_{h}$
is enough to fix this ambiguity, at least locally.

This can be shown using a Bogomol’nyi bound, as follows. Consider any black
hole solution $\psi(r),\phi^{i}(r),f(r)$ (not necessarily quasiextremal). The
functional

$\displaystyle
I[\psi,\phi,f]\mathrel{:=}\int_{0}^{z_{h}}\biggl{[}\frac{1}{2k_{N}}(f\dot{\psi}+\dot{f})^{2}+\frac{1}{2}f^{2}G_{ij}(\phi)\dot{\phi}^{i}\dot{\phi}^{j}+\frac{1}{2}\mathrm{e}^{2\psi}Q^{2}(\phi)\biggr{]}\textrm{d}z,$
(A.30)

evaluates to the black hole mass upon imposing the equations of motion:

$\displaystyle
I[\psi,\phi,f]=k_{N}^{-1}\biggl{[}\int_{0}^{z_{h}}\frac{\mathrm{d}}{\mathrm{d}z}\left(\frac{1+f}{2}f\dot{\psi}\right)\textrm{d}z+\frac{1}{2z_{h}}\biggr{]}=k_{N}^{-1}\biggl{[}-\dot{\psi}_{\infty}+\frac{1}{2z_{h}}\biggr{]}=M_{\text{BH}},$
(A.31)

where we use $f=0$ at $z=z_{h}$ in the nonextremal ($r_{h}>0$) case to
eliminate the boundary term at the horizon. In the quasiextremal case $(f=1)$
a smooth horizon requires $\dot{\psi}\rightarrow-\frac{1}{z}$ as
$z\rightarrow\infty$ (since $\dot{\chi}=\dot{\psi}+\frac{1}{z}\rightarrow 0$),
which has the same effect.

Given any function $W(\phi)$, $I$ can be factored as follows:

$\displaystyle I[\psi,\phi,f]=$
$\displaystyle\int_{0}^{z_{h}}\biggl{(}\frac{k_{N}}{2}\Bigl{[}\frac{f\dot{\psi}+\dot{f}}{k_{N}}+\mathrm{e}^{\psi}W(\phi)\Bigr{]}^{2}+\frac{1}{2}G_{ij}[f\dot{\phi}^{i}+\mathrm{e}^{\psi}G^{ik}W_{,k}][f\dot{\phi}^{j}+\mathrm{e}^{\psi}G^{jl}W_{,l}]\biggr{)}\mathrm{d}z$
$\displaystyle+\frac{1}{2}\int_{0}^{z_{h}}\mathrm{e}^{2\psi}[Q^{2}(\phi)-k_{N}W(\phi)^{2}-G^{ij}W_{,i}W_{,j}]\mathrm{d}z+W(\phi_{\infty}),$
(A.32)

where we use
$\int_{0}^{z_{h}}\frac{\mathrm{d}}{\mathrm{d}z}[f\mathrm{e}^{\psi}W(\phi)]\mathrm{d}z=-W(\phi_{\infty})$
with vanishing horizon boundary term due to $f(z_{h})=0$ in the nonextremal
case and because a smooth horizon requires
$\mathrm{e}^{\psi}\propto\frac{1}{z}\rightarrow 0$ as $z\rightarrow\infty$ in
the quasiextremal case. Since the terms on the first line of (A.32) are
positive-definite, provided that $k_{N}W(\phi)^{2}+G^{ij}W_{,i}W_{,j}\leqslant
Q^{2}(\phi)$ along the entire trajectory $\phi^{i}(r)$, we can conclude that

$M_{\text{BH}}\geqslant W(\phi_{\infty}).$ (A.33)

Saturating the bound requires (A.29) along the trajectory together with $f=1$
(quasiextremality)232323The Bogomol’nyi equations
$f\dot{\psi}+\dot{f}=-\mathrm{e}^{\psi}k_{N}W$,
$f\dot{\phi}^{i}=-\mathrm{e}^{\psi}G^{ij}W_{,j}$ and (A.29) together imply
$\frac{\mathrm{d}}{\mathrm{d}z}[f\mathrm{e}^{\psi}W]=-\mathrm{e}^{2\psi}Q^{2}(\phi)$,
whereas
$\frac{\mathrm{d}}{\mathrm{d}z}[\mathrm{e}^{\psi}W]=-k_{N}^{-1}\frac{\mathrm{d}}{\mathrm{d}z}[f\dot{\psi}+\dot{f}]=-\mathrm{e}^{2\psi}Q^{2}$
using (A.11). Thus
$\frac{\mathrm{d}}{\mathrm{d}z}[(1-f)\mathrm{e}^{\psi}W]=0$, implying
$(1-f)\mathrm{e}^{\psi}W=0$ by integrating from $z=0$. Since
$\mathrm{e}^{\psi}W>0$, $f=1$ follows. and the Bogomol’nyi equations

$\displaystyle\dot{\psi}=-\mathrm{e}^{\psi}k_{N}W(\phi),~{}~{}~{}~{}~{}~{}\dot{\phi}^{i}=-\mathrm{e}^{\psi}G^{ij}W_{,j}.$
(A.34)

These equations, implying (A.10–A.12), have a unique solution for each choice
of $\phi_{\infty}^{i}$ provided that the $W(\phi)$ gradient flow remains
entirely within the region $R_{0}$ where (A.29) is satisfied, and the solution
has a smooth horizon provided that the gradient flow ends at a critical point
of $W(\phi)$ with $W\left(\phi_{\text{crit}}\right)>0$ (ensuring
$\mathrm{e}^{\psi}\propto\frac{1}{z}$ as $z\rightarrow\infty$). Thus, given
$W(\phi)$ and $\phi_{\infty}^{i}$, the black hole solution saturating (A.33)
is unique if it exists.

If both $W_{1}(\phi)$ and $W_{2}(\phi)$ satisfy (A.29) in a region $R_{0}$
encompassing coincident local minima at $\phi=\phi_{h}$,242424This implies
that $Q^{2}(\phi)$ also has a local minimum at $\phi_{h}$. $W_{1}$ gradient
flows ending at $\phi_{h}$ produce quasiextremal solutions of mass
$M_{\text{BH}}=W_{1}(\phi_{\infty})$, which must satisfy
$M_{\text{BH}}=W_{1}(\phi_{\infty})\geqslant W_{2}(\phi_{\infty})$ per (A.33).
By the same token $W_{2}(\phi_{\infty})\geqslant W_{1}(\phi_{\infty})$ hence
$W_{1}(\phi_{\infty})=W_{2}(\phi_{\infty})$ for all $\phi_{\infty}$ in $R_{0}$
flowing to $\phi_{h}$.

Thus, the fake superpotential $W(\phi)$ associated to a given stable attractor
point $\phi^{i}_{h}$ is uniquely fixed near the attractor point by (A.29) and
the condition that $W(\phi)$ has a minimum at $\phi_{h}^{i}$, and the
corresponding attractor solutions can be obtained from $W(\phi)$ by solving
the gradient flow equations (A.34). (The family of attractor solutions
beginning at $\phi_{h}^{i}$ may have turning points and/or caustics farther
away from the attractor point, so $W(\phi)$ does not necessarily extend
uniquely throughout moduli space.)

####  A.2.4 Asymptotic attractors

A special case of the attractor mechanism occurs when $\phi_{h}^{i}$ lies at
infinite distance in the moduli space, usually in a direction where
$Q^{2}(\phi)\rightarrow 0$.252525In principle, $\phi_{h}^{i}$ could lie at an
infinite distance point where $Q^{2}(\phi_{h})>0$, but this does not occur
anywhere in the landscape to our knowledge. Such “asymptotic attractors”
technically do not lead to smooth black hole solutions (e.g.,
$Q^{2}(\phi_{h})=0$ implies vanishing horizon area), but can be understood as
the limit of a family of smooth nonextremal solutions, and therefore play a
role in determining the extremality bound.

Asymptotic attractors are typically also characterized by a fake
superpotential. Heuristically, because modifying $Q^{2}(\phi)$ very far out in
the moduli space can turn the asymptotic attractor into a standard attractor
at finite distance with an associated unique fake superpotential, taking a
limit where the new attractor point is sent off to infinity while restoring
$Q^{2}(\phi)$ to its original form should yield a fake superpotential for the
original asymptotic attractor. This argument could fail in several ways when
the asymptotic behavior of $Q^{2}(\phi)$ is sufficiently strange, but within
the realm of actual quantum gravities, we know of no issue with it.

###  A.3 The extremality bound

Cosmic censorship requires extremal black holes to be quasiextremal, and
therefore for a given choice of $Q_{A}$ and $\phi_{\infty}^{i}$ the lightest
quasiextremal black hole should be extremal. Combined with the discussion of
fake superpotentials in the previous section, this suggests that the
extremality bound is given by

$M_{\text{BH}}\geqslant W(\phi)\mathrel{:=}\min_{\\{a\\}}W_{a}(\phi),$ (A.35)

where $W_{a}(\phi)$ are the fake superpotentials associated to the various
stable or asymptotic attractors and the minimum is taken amongst all the fake
superpotentials defined at the point in question.

To verify (A.35), we assume that the “global fake superpotential” $W(\phi)$
specified in (A.35) is defined and continuous everywhere in moduli space.
Thus, the moduli space is partitioned into different “attractor basins”
associated to the various stable or asymptotic attractors, with $W(\phi)$
equal to the corresponding fake superpotential $W_{a}(\phi)$ within each
basin. Since each constituent fake superpotential $W_{a}(\phi)$ satisfies
(A.29), $W(\phi)$ also satisfies (A.29) except possibly at the boundaries
between attractor basins. However, as these boundaries are sets of measure
zero and the continuity of $W(\phi)$ precludes delta-function contributions to
$\nabla_{i}W$, the argument leading to the Bogomol’nyi bound (A.33) is
unaffected, hence $M_{\text{BH}}\geqslant W(\phi)$. As this bound can be
saturated by construction, it is indeed the extremality bound.

Naviely, to construct $W(\phi)$ we must first classify _all_ quasiextremal
solutions. Fortunately, this is not the case: given any (possibly incomplete)
collection of local fake superpotentials $W_{a}(\phi)$ such that
$\hat{W}(\phi)\mathrel{:=}\min_{\\{a\\}}W_{a}(\phi)$ is everywhere defined and
continuous, the same reasoning as above implies that
$M_{\text{BH}}\geqslant\hat{W}(\phi)$ is the extremality bound, hence
$\hat{W}(\phi)=W(\phi)$.

###  A.4 Examples

A simple example that has played an outsized role in the development of the
WGC is Einstein-Maxwell-Dilaton theory, with the effective action:

$S=\int\textrm{d}^{d}x\sqrt{-g}\left[\frac{1}{2\kappa^{2}}\left(R-\frac{1}{2}(\nabla\phi)^{2}\right)-\frac{1}{2\hat{e}^{2}}\mathrm{e}^{-\alpha\phi}|F_{2}|^{2}\right].$
(A.36)

Then (A.29) becomes:

$\xi\kappa^{2}[W(\phi)]^{2}+2\kappa^{2}[W^{\prime}(\phi)]^{2}=\mathrm{e}^{\alpha\phi}(\hat{e}Q)^{2},\qquad\xi\mathrel{:=}\frac{d-3}{d-2}.$
(A.37)

There is an asymptotic attractor at $\phi_{h}=-\infty$. Guessing a solution of
the form $W(\phi)=\hat{M}\mathrm{e}^{\alpha\phi/2}$, we obtain

$\displaystyle\left[\xi+\frac{\alpha^{2}}{2}\right](\kappa\hat{M})^{2}=(\hat{e}Q)^{2}\qquad\Rightarrow\qquad\kappa\hat{M}=\gamma^{-\frac{1}{2}}\hat{e}|Q|,\qquad\gamma=\xi+\frac{\alpha^{2}}{2}.$
(A.38)

Since $W(\phi)$ is globally defined, positive, and satisfies (A.29)
everywhere, it defines a global fake superpotential, and the extremality bound
is

$\kappa M_{\text{BH}}\geqslant\kappa
W(\phi_{\infty})=\gamma^{-\frac{1}{2}}e|Q|,$ (A.39)

where $e=\hat{e}\,\mathrm{e}^{\alpha\phi_{\infty}/2}$ is the vacuum gauge
coupling. The same result applies to the corresponding $(p-1)$-brane theory
with $\xi\rightarrow\xi_{(p)}=\frac{p(d-p-2)}{d-2}$, as in (3.10), (3.11).

As a somewhat less trivial example, consider two gauge fields with different
dilaton couplings:

$\displaystyle
S=\int\textrm{d}^{d}x\sqrt{-g}\biggl{[}\frac{1}{2\kappa^{2}}\left(R-\frac{1}{2}(\nabla\phi)^{2}\right)-\frac{1}{2\hat{e}^{2}_{1}}\mathrm{e}^{-\alpha_{1}\phi}|F_{2}|^{2}-\frac{1}{2\hat{e}^{2}_{2}}\mathrm{e}^{\alpha_{2}\phi}|H_{2}|^{2}\biggr{]}.$
(A.40)

Then (A.29) becomes

$\xi\kappa^{2}[W(\phi)]^{2}+2\kappa^{2}[W^{\prime}(\phi)]^{2}=\mathrm{e}^{\alpha_{1}\phi}(\hat{e}_{1}Q_{1})^{2}+\mathrm{e}^{-\alpha_{2}\phi}(\hat{e}_{2}Q_{2})^{2}.$
(A.41)

Provided that $\alpha_{1}\alpha_{2}>0$ and $Q_{1,2}\neq 0$, there is a stable
attractor point at
$\phi_{h}=\frac{1}{\alpha_{1}+\alpha_{2}}\log\left[\frac{\alpha_{2}(\hat{e}_{2}Q_{2})^{2}}{\alpha_{1}(\hat{e}_{1}Q_{1})^{2}}\right]$.

Guessing a solution of the form
$W(\phi)=\hat{M}_{1}\mathrm{e}^{\alpha_{1}\phi/2}+\hat{M}_{2}\mathrm{e}^{-\alpha_{2}\phi/2}$,
the left-hand-side of (A.41) has cross terms proportional to
$\mathrm{e}^{(\alpha_{1}-\alpha_{2})\phi}$ whose cancellation requires
$\alpha_{1}\alpha_{2}=2\xi$. With this condition, we obtain

$\displaystyle\kappa\hat{M}_{1,2}=\gamma_{1,2}^{-\frac{1}{2}}\hat{e}_{1,2}|Q_{1,2}|,~{}~{}~{}~{}~{}~{}\gamma_{1,2}=\xi+\frac{\alpha_{1,2}^{2}}{2},$
(A.42)

so the extremality bound in this case is

$\kappa
M_{\text{BH}}\geqslant\gamma_{1}^{-\frac{1}{2}}e_{1}|Q_{1}|+\gamma_{2}^{-\frac{1}{2}}e_{2}|Q_{2}|,$
(A.43)

where $e_{1}=\hat{e}_{1}\mathrm{e}^{\alpha_{1}\phi_{\infty}/2}$ and
$e_{2}=\hat{e}_{2}\mathrm{e}^{-\alpha_{2}\phi_{\infty}/2}$ are the gauge
couplings in the vacuum in question. When $\alpha_{1}\alpha_{2}\neq 2\xi$, the
fake superpotential solving (A.41) is not known in closed form (apart from
some special cases) but it is easily found by numerical integration.

So far, we have considered examples with a single attractor basin. A simple
(if contrived) example that exhibits multiple attractor basins is

$\displaystyle
Q^{2}(\phi)=k_{N}M_{0}^{2}\Bigl{(}1+\bigl{[}(\phi/\phi_{0})^{2}-\lambda\bigr{]}^{2}\Bigr{)},\quad
G_{\phi\phi}=k_{N}^{-1}\quad$ $\displaystyle\Rightarrow\quad
W(\phi)^{2}+W^{\prime}(\phi)^{2}=M_{0}^{2}\Bigl{(}1+\bigl{[}(\phi/\phi_{0})^{2}-\lambda\bigr{]}^{2}\Bigr{)},$
(A.44)

with attractor points at $\phi=\pm\sqrt{\lambda}\phi_{0}$. The associated fake
superpotentials can be found by numerical integration, see figures 18 and 19.
Note that for some values of $\phi_{0},\lambda$ there are quasi-extremal
solutions that are not extremal due to finite overlap between the domains of
the local fake superpotentials $W_{\pm}(\phi)$ associated to the attractor
points $\phi=\pm\sqrt{\lambda}\phi_{0}$.

Figure 18: The fake superpotentials for the two stable attractor points of
(A.44) with $\lambda=4/3$ and $\phi_{0}=\sqrt{3}/4$. Because $W_{\pm}(\phi)$
cross over each other at $\phi=0$, $W_{-}(\phi)$ gradient flow solutions with
$\phi_{\infty}>0$ are quasiextremal but not extremal, as are $W_{+}(\phi)$
gradient flow solutions with $\phi_{\infty}<0$. Figure 19: The fake
superpotentials for the two stable attractor points of (A.44) with
$\lambda=4/3$ and $\phi_{0}=2$. In this case, all quasiextremal solutions are
extremal.

Some examples where numerical integration was used to determine the
extremality bound in an actual theory of quantum gravity are discussed in
[46].

###  A.5 Closing comments

To derive the extremality bound (A.35) we assumed that
$W(\phi)\mathrel{:=}\min_{\\{a\\}}W_{a}(\phi)$ was everywhere defined and
continuous. This is easily proved for a one-dimensional moduli space, since
(1) the domain $\mathcal{D}_{a}$ of the fake superpotential $W_{a}(\phi)$
associated to a minimum $\phi_{(a)}$ of $Q^{2}(\phi)$ is at least as large as
the interval between the two adjacent maxima of $Q^{2}(\phi)$ and (2)
$W_{a}(\phi)=|Q(\phi)|/\sqrt{k_{N}}$ at the boundary of this domain (see,
e.g., figure 18), saturating the upper bound
$W_{b}(\phi)\leq|Q(\phi)|/\sqrt{k_{N}}$ on all fake superpotentials, where
these two properties ensure (1) the existence and (2) the continuity of
$W(\phi)$ at each point. While we do not know a general proof for a higher-
dimensional moduli spaces, the existence and continuity of $W(\phi)$ can be
checked on a case-by-case basis.

Phase transitions in the moduli space can create additional subtleties.
Firstly, while $t_{AB}(\phi)$ and $G_{ij}(\phi)$ need not be analytic at a
phase transition, in itself this has little effect on the foregoing analysis.
More importantly, different branches of moduli space can meet at a phase
transition, opening up the possibility of black hole solutions that cross from
one branch to another. This is a rather complicated question that has not been
worked out in the literature to our knowledge, but it seems probable that some
version of fake superpotentials will still be applicable. Yet more
drastically, the moduli space could have finite-distance boundaries where a
strongly-coupled CFT appears, as in some examples from [46]. Black hole
solutions that reach this CFT boundary outside their event horizon lie outside
the regime of validity of the weakly-coupled EFT that our analysis is based
on, and require a separate analysis.

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# Note on the product of the largest
and the smallest eigenvalue of a graph

A. Abiad***Department of Mathematics and Computer Science, Eindhoven
University of Technology, The Netherlands<EMAIL_ADDRESS>C.
Dalfó†††Departament de Matemàtica, Universitat de Lleida, Igualada
(Barcelona), Catalonia<EMAIL_ADDRESS>M. A. Fiol‡‡‡Departament de
Matemàtiques, Universitat Politècnica de Catalunya, Barcelona Graduate School
of Mathematics, Institut de Matemàtiques de la UPC-BarcelonaTech (IMTech),
Barcelona, Catalonia<EMAIL_ADDRESS>
###### Abstract

In this note, we use eigenvalue interlacing to derive an inequality between
the maximum degree of a graph and its maximum and minimum adjacency
eigenvalues. The case of equality is fully characterized.

##  1 Introduction

Spectral graph theory seeks to deduce the structural properties of a graph
from its spectrum. Eigenvalue interlacing provides a powerful tool for
obtaining inequalities and regularity results concerning the structure of
graphs in terms of eigenvalues of the adjacency matrix; see, for instance,
Brouwer and Haemers [4] and Haemers [10]. In addition, studying the cases of
equality in such inequalities often provides interesting information on the
structure of the graph.

Let $G$ be a (connected) graph with $n$ vertices and adjacency matrix having
eigenvalues $\lambda_{1}\geq\cdots\geq\lambda_{n}$. Not many bounds involving
the product of the largest and smallest eigenvalue, $\lambda_{1}$ and
$\lambda_{n}$, of a graph seem to exist. In fact, we are only aware of an
inequality obtained by Haemers [10, Theorem 3.3], who provides an upper bound
for the independence number $\alpha$ of a non-regular graph, extending the
celebrated Hoffman ratio bound for regular graphs $\alpha\leq
n/[1-(\lambda_{1}/\lambda_{n})]$ (see [10, Theorem 3.2]). Namely, Haemers
proved that, if $G$ is a graph with $n$ vertices, minimum degree $\delta$, and
independence number $\alpha$, then

$-\lambda_{1}\lambda_{n}\geq\frac{\alpha\delta^{2}}{n-\alpha}.$ (1)

Several papers, such as the one by Gregory, Hershkowitz, and Kirkland [9], are
on the graph spread, that is, $\lambda_{1}-\lambda_{n}$. There is also quite
some work on bounding the sum $\lambda_{1}+\lambda_{n}$, see, for instance,
Brandt [3], Csikvári [5] and Balogh, Clemen, Lidický, Norin, and Volec [2].
Moreover, bounds on $\lambda_{1}$ knowing $\lambda_{n}$, or the other way
around, have been obtained by Rojo, Rojo, and Soto [11].

The structure of this note is as follows. We start Section 2 by providing some
basic definitions and some known results, such as the interlacing theorem. In
Section 3, we present our main result, that is, an inequality involving only
the maximum degree $\Delta$ of a graph and the product of its maximum and
minimum adjacency eigenvalues. Moreover, the case of equality is fully
characterized, and some infinite families satisfying it are provided. Finally,
in Section 4 we show that our bound can outperform the other known bound of a
similar nature by Haemers [10, Theorem 3.3].

##  2 Preliminaries

Throughout this note, $G=(V,E)$ denotes a (simple and connected) graph with
$n=|V|$ vertices, and adjacency matrix $A$ with eigenvalues

$\lambda_{1}(G)\geq\lambda_{2}(G)\geq\cdots\geq\lambda_{n}(G).$

Moreover, $G(u)$ stands for the set of vertices adjacent to a given vertex
$u\in V$.

Given two square matrices $A$ and $B$ with eigenvalues
$\lambda_{1}\geq\cdots\geq\lambda_{n}$ and $\mu_{1}\geq\cdots\geq\mu_{m}$
(with $m<n$), we say that the second sequence interlaces the first if, for all
$i=1,\ldots,m$, it follows that $\lambda_{i}\geq\mu_{i}\geq\lambda_{n-m+i}$.

A basic result about interlacing is the following (see Haemers [10]).

######  Theorem 1.

Let $S$ be a real $n\times m$ matrix such that
$\mbox{\boldmath$S$}^{T}\mbox{\boldmath$S$}=\mbox{\boldmath$I$}$, and let $A$
be an $n\times n$ matrix with eigenvalues
$\lambda_{1}\geq\cdots\geq\lambda_{n}$. Define
$\mbox{\boldmath$B$}=\mbox{\boldmath$S$}^{T}\mbox{\boldmath$A$}\mbox{\boldmath$S$}$,
and call its eigenvalues $\mu_{1}\geq\cdots\geq\mu_{m}$.

  1. $(i)$

The eigenvalues of $B$ interlace those of $A$.

  2. $(ii)$

If $\mu_{i}=\lambda_{i}$ or $\mu_{i}=\lambda_{n-m+i}$, then there is an
eigenvector $v$ of $B$ for $\mu_{i}$ such that $S$$v$ is an eigenvector of $A$
for $\mu_{i}$.

  3. $(iii)$

If there is an integer $k\in\\{0,\ldots,m\\}$ such that $\lambda_{i}=\mu_{i}$
for $1\leq i\leq k$, and $\mu_{i}=\lambda_{n-m+i}$ for $k+1\leq i\leq m$
$($tight interlacing$)$, then
$\mbox{\boldmath$S$}\mbox{\boldmath$B$}=\mbox{\boldmath$A$}\mbox{\boldmath$S$}$.

Let $A$ be the adjacency matrix of a graph $G=(V,E)$. We distinguish two
interesting cases of eigenvalue interlacing depending on the matrix $B$:

  * 1.

If $B$ is a principal submatrix of $A$, then $B$ corresponds to the adjacency
matrix of an induced subgraph $G^{\prime}$ of $G$.

  * 2.

If for a given partition of the vertices of $G$, say $V=U_{1}\cup\cdots\cup
U_{m}$, $B$ is the so-called quotient matrix of $A$. The entries of $B$, which
are denoted by $b_{ij}$ for $i,j=1,\ldots,m$, are the average row sums of the
corresponding blocks $\mbox{\boldmath$A$}_{ij}$ of $A$.

Moreover, if the interlacing is tight, Theorem 1$(iii)$ reflects that $S$
corresponds to a regular (or equitable) partition of $A$, that is, each block
of the partition has constant row and column sums. Then, the bipartite induced
subgraphs $G_{ij}$, with adjacency matrices $\mbox{\boldmath$A$}_{ij}$, for
$i\neq j$, are biregular, and the subgraphs $G_{ii}$ are regular.

The cone graph over a graph $H$ is obtained by adding a new vertex $v$ and
joining it to all vertices of $H$. The following result is well known; see,
for instance, Brouwer and Haemers [4, Ch. 1, Ex. 6].

######  Lemma 2.

Let $G=H^{\prime}+v$ be the cone graph on $n$ vertices, over the
$\delta$-regular (connected) graph $H^{\prime}$ with $n-1$ vertices and
eigenvalues $\theta_{1}(=\delta)>\theta_{2}\geq\cdots\geq\theta_{n-1}$. Then,
$G$ has eigenvalues

$\theta_{2}\geq\cdots\geq\theta_{n-1},\quad\mbox{together
with}\quad\lambda_{1,2}=\frac{1}{2}\left(d\pm\sqrt{d^{2}-4(n-1}\right).$

Notice that $\lambda_{1}(>\delta)$ is the spectral radius of $G$, whereas, in
principle, $\lambda_{2}$ is not necessarily the minimum eigenvalue of $G$.

##  3 The new bound

In this section we present our main result: an upper bound on the product of
the maximum and minimum eigenvalues of a graph. This bound is very useful when
we only know information about the maximum degree of the graph.

######  Theorem 3.

Let $G$ be a graph on $n$ vertices with maximum degree $\Delta=\Delta(G)$ and
eigenvalues
$\lambda_{\max}(G)=\lambda_{1}(G)\geq\cdots\geq\lambda_{n}(G)=\lambda_{\min}(G)$.
Then,

$-\lambda_{\min}(G){\lambda_{\max}(G)}\geq\Delta.$ (2)

Moreover, equality holds if and only if $G=H^{\prime}+u$ is a cone graph over
a $\delta$-regular graph $H^{\prime}$ on $\Delta=n-1$ vertices (the degree of
$u$) and minimum eigenvalue satisfying

$\lambda_{\min}(H^{\prime})\geq\frac{1}{2}\left(\delta-\sqrt{\delta^{2}+4\Delta}\right).$
(3)

###### Proof.

Let $u$ be a vertex of $G$ with maximum degree $\Delta$, and consider the
graph $H$ induced by the vertex set $\\{u\\}\cup G(u)$. Let us first prove
that $H$ satisfies (2). If $\mbox{\boldmath$A$}^{\prime}$ is the adjacency
matrix of the graph $H^{\prime}=\langle G(u)\rangle$ induced by $G(u)$, the
adjacency matrix $A$ of $H$ is of the form

$\mbox{\boldmath$A$}=\left(\begin{array}[]{cc}0&\mbox{\boldmath$j$}\\\
\mbox{\boldmath$j$}^{\top}&\mbox{\boldmath$A$}^{\prime}\\\
\end{array}\right),$ (4)

where $j$ is the all-1 vector with $\Delta$ entries. Then, its quotient matrix
(where each block is replaced by the average sum of its arrows) is

$\mbox{\boldmath$Q$}^{\prime}=\left(\begin{array}[]{cc}0&\Delta\\\
1&\overline{\delta}\\\ \end{array}\right),$

where $\overline{\delta}$ is the mean degree of $H^{\prime}$. Since the
eigenvalues of $\mbox{\boldmath$Q$}^{\prime}$ are

$\lambda_{1,2}=\frac{1}{2}\left(\overline{\delta}\pm\sqrt{\overline{\delta}^{2}+4\Delta}\right)$

we have, by interlacing,

$\lambda_{\max}(H)\geq\lambda_{1}>\lambda_{2}\geq\lambda_{\min}(H).$ (5)

Then, noting that

$\frac{-\Delta}{\lambda_{1}}=\frac{-2\Delta}{\overline{\delta}+\sqrt{\overline{\delta}^{2}+4\Delta}}=\frac{1}{2}\left(\overline{\delta}-\sqrt{\overline{\delta}^{2}+4\Delta}\right)=\lambda_{2},$

we have

$-\frac{\Delta(H)}{\lambda_{\max}(H)}\geq-\frac{\Delta}{\lambda_{1}}=\lambda_{2}\geq\lambda_{\min}(H),$
(6)

and (2) follows. In fact, as pointed out by Haemers (personal communication,
2024), this can be proved more directly by noting that

$-\lambda_{\max}(H)\lambda_{\min}(H)\geq-\lambda_{1}\lambda_{2}=-\det(\mbox{\boldmath$Q$})=\Delta.$

Now, since $H$ is an induced subgraph of $G$, we have, again by interlacing,

$\lambda_{\max}(G)\geq\lambda_{\max}(H)>\lambda_{\min}(H)\geq\lambda_{\min}(G).$

Thus, using (6),

$-\frac{\Delta(G)}{\lambda_{\max}(G)}\geq-\frac{\Delta}{\lambda_{\max}(H)}\geq\lambda_{\min}(H)\geq\lambda_{\min}(G),$
(7)

and (2) holds.

Now, the equality in (2) first implies equalities in (6), that is
$\lambda_{\max}(H)=\lambda_{1}$ and $\lambda_{\min}(H)=\lambda_{2}$.
Consequently, we have tight interlacing, and (4) corresponds to a regular
partition of $H$. That is, $H=H^{\prime}+v$ is a cone graph over the regular
graph $H^{\prime}$ with degree $\overline{\delta}=\delta$. Second, assuming
that $H$ is a proper subgraph of $G$, the equalities in (7) imply that
$\lambda_{\max}(G)=\lambda_{\max}(H)$. However, this is impossible since the
spectral radius of a proper subgraph of a graph $G$ is known to be smaller
than the spectral radius of $G$ (see, for instance, Cvetković, Rowlinson and
Simić [6, Prop. 3.1.10]). Therefore, $G=H=H^{\prime}+u$ is a cone graph, as
claimed. Conversely, if $G=H^{\prime}+v$ is a cone graph over a
$\delta$-regular graph $H^{\prime}$ with minimum eigenvalue satisfying (3), by
Lemma 2, we have that
$\lambda_{\min}(H)=\lambda_{\min}(G)=\lambda_{2}=\frac{1}{2}\left(\delta-\sqrt{\delta^{2}+4\Delta}\right)$
and, hence, we get the equality in (2). ∎

For example, (3) is satisfied when $G$ is a cone graph of the following type:

$(i)$

$G=K_{n}=K_{n-1}+u$, the complete graph on $n$ vertices.

$(ii)$

$G=S_{n-1}=(n-1)K_{1}+u$, the star graph with $n-1$ rays.

$(iii)$

$G=W_{n-1}=C_{n-1}+u$, the wheel graph with $n-1$ spokes with $n\geq 8$.

$(iv)$

$G=O(k)+v$, where $O(k)$ is the odd graph of degree $k$.

$(v)$

$Q_{k}+u$, where $Q_{k}$ is the $k$-cube of dimension $k$ with $k\geq 7$.

Indeed, let
$\phi(\delta,\Delta)=\frac{1}{2}\left(\delta-\sqrt{\delta^{2}+4\Delta}\right)$,
and let us check that the condition (3) on $H^{\prime}$ holds. (Recall that
$\Delta$ is used both for the degree of $u$, or maximum degree of $G$, and the
number of vertices of $H^{\prime}$.)

$(i)$

The complete graph $H^{\prime}=K_{n-1}$ on $\Delta=n-1$ vertices, has degree
$\delta=n-2$ and its smallest eigenvalue is
$\lambda_{\min}(H^{\prime})=-1=\phi(n-2,n-1)$.

$(ii)$

The graph $H^{\prime}=(n-1)K_{1}$ with $\Delta=n-1$ isolated vertices has
degree $\delta=0$, and its smallest eigenvalue is
$\lambda_{\min}(H^{\prime})=0\geq\phi(0,n-1)=-\sqrt{n-1}$.

$(iii)$

The cycle graph $H^{\prime}=C_{n-1}$ on $n-1$ vertices has eigenvalues
$2\cos\left(\frac{2k\pi}{n-1}\right)$ for $k=0,\ldots,n-2$. Since
$W_{3}=H^{\prime}+v\cong K_{4}$, we can assume that $n\geq 5$. For these
values, the minimum of $2\cos\left(\frac{2k\pi}{n-1}\right)$ is when
$k=(n/2)-1,n/2$ for even $n$, and when $k=(n-1)/2$ for odd $n$. In the fist
case the minimum eigenvalue
$\lambda_{\min}(H^{\prime})=2\cos\left(\frac{(n-2)\pi}{n-1}\right)\geq\phi(2,n-1)=1-\sqrt{n}$,
excepting the case for $n=6$. In the second case,
$\lambda_{\min}(H^{\prime})=-2\geq 1-\sqrt{n}$ excepting when $n=5,7$.

$(iv)$

The Odd graph $H^{\prime}=O(k)$ on $\Delta={{2k-1}\choose{k-1}}$ vertices, has
degree $\delta=k$ and its smallest eigenvalue is
$\lambda_{\min}(H^{\prime})=-(k-1)\geq\phi(k,\Delta)$. Thus, the successive
values of $\phi(k,\Delta)$ for $k=3,4,5,6,\ldots$ are (approximately) $-2$,
$-4.245$, $-9$, $-18.702,\ldots$ Then, for instance, $G=P+v$, with $P=O(3)$
being the Petersen graph, has maximum degree $\Delta=10$ and different
eigenvalues $5,1,-2$, so satisfying equality in (2).

$(v)$

The $k$-cube $H^{\prime}=Q_{k}$ has $\Delta=2^{k}$ vertices, degree
$\delta=k$, and its smallest eigenvalue is
$\lambda_{\min}(H^{\prime})=-k\geq\phi(k,2^{k})$ when $k\geq 7$.

From Theorem 3, we obtain the following straightforward consequences.

######  Corollary 4.

  * $(i)$

If $G$ is a (non-trivial) regular graph, then its minimum eigenvalue satisfies
$\lambda_{\min}\leq-1$, with equality if and only if $G$ is a complete graph.

  * $(ii)$

If $G$ is a bipartite graph with maximum degree $\Delta$ and spectral radius
$\lambda_{\max}$, then $\Delta\leq\lambda_{\max}^{2}$, and equality holds if
and only if $G=S_{\Delta}$.

##  4 Bounds comparison

When equality does not hold in (2), it is interesting to compare our bound to
the one by Haemers (1). Apart from the fact that (2) is more ‘economical’ in
the sense that it uses only the maximum degree, in general, (2) gives a better
bound for $-\lambda_{1}\lambda_{n}$ when $\alpha$ is not very large. Namely,
from $\Delta\geq\alpha\delta^{2}/(n-\alpha)$, we obtain that (2) outperforms
(1) when

$\alpha\leq\frac{n}{1+\frac{\delta^{2}}{\Delta}}.$

Order | Proportion
---|---
4 | 2/4=0.5
5 | $14/19\approx 0.737$
6 | $79/107\approx 0.738$
7 | $692/849\approx 0.815$
8 | $9489/11100\approx 0.855$
Table 1: Proportion of small irregular graphs for which the new bound (2)
outperforms the known bound (1).

In contrast, Haemers’ bound (1) is better than (2) for the case of bipartite
biregular graphs. In fact, we will show that, for such graphs, equality holds
in (1). A bipartite graph $G=(V,E)$ with $V=V_{1}\cup V_{2}$ is biregular if
all the vertices of the stable set $V_{1}$ have degree $k_{1}$, whereas all
the vertices of $V_{2}$ have degree $k_{2}$. Then, notice that by counting in
two ways the number of edges of $G$, we have

$k_{1}n_{1}=k_{2}n_{2},$ (8)

where $n_{i}=|Vi|$, for $i=1,2$. Since $G$ is bipartite and biregular, its
maximum and minimum eigenvalues are
$\lambda_{1}=-\lambda_{n}=\sqrt{k_{1}k_{2}}$ (see, for example, [1] by the
authors). Moreover, assuming that $n_{1}\geq n_{2}$, its minimum degree is
$\delta=k_{1}$, and its independence number is $\alpha(G)=n_{1}$. Therefore,
using (8), the bound in (1) become

$\frac{\alpha\delta^{2}}{n-\alpha}=\frac{n_{1}k_{1}^{2}}{n-n_{1}}=\frac{n_{1}k_{1}}{n_{2}}k_{1}=k_{2}k_{1}=-\lambda_{1}\lambda_{n},$

as claimed. Thus, since the maximum degree is $\Delta=k_{2}$, Haemers’ bound
(1) outperforms (2) except when $k_{1}=1$, which corresponds to the mentioned
star graph $S_{k_{2}}$, as a special case in Theorem 3.

In fact, this can be seen as a particular case of the following result by the
authors [1, Proposition 3.1], which can be seen as the bipartite version of
Haemers’ bound.

######  Proposition 5 ([1]).

Let $G=(V_{1}\cup V_{2},E)$ be a bipartite graph with maximum and minimum
eigenvalues $\lambda_{1}$ and $\lambda_{n}=-\lambda_{1}$. Let
$\overline{\delta}_{i}$ be the mean degree of the vertices in $V_{i}$, for
$i=1,2$. Then,

$-\lambda_{1}\lambda_{n}=\lambda_{1}^{2}\geq\overline{\delta}_{1}\overline{\delta}_{2},$
(9)

with equality if, and only if, $G$ is biregular with degrees
$k_{1}=\overline{\delta_{1}}$ and $k_{2}=\overline{\delta}_{2}$.

Then, if $G$ has minimum degree $\delta$ and maximum degree $\Delta$, (9) (or
(1) with $\alpha=|V_{1}|\geq|V_{2}|$) and (2) imply that, for a bipartite
graph,

$-\lambda_{1}\lambda_{n}=\lambda_{1}^{2}\geq\max\\{\delta^{2},\Delta\\}.$

### Acknowledgments

The authors thank the useful comments made by Willem Haemers, which helped to
improve this paper. A. Abiad is supported by the Dutch Research Council (NWO)
through the grant VI.Vidi.213.085. The research of C. Dalfó and M. A. Fiol is
supported by AGAUR from the Catalan Government via the project 2021SGR00434
and MICINN from the Spanish Government via the project PID2020-115442RB-I00.
The research of M. A. Fiol is also supported by the Universitat Politècnica de
Catalunya via the grants AGRUPS-2022 and AGRUPS-2023.

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