# Early Dark Energy During Big Bang Nucleosynthesis

David McKeen<EMAIL_ADDRESS>TRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T
2A3, Canada  Afif Omar<EMAIL_ADDRESS>TRIUMF, 4004 Wesbrook Mall, Vancouver,
BC V6T 2A3, Canada  Department of Physics and Astronomy, University of
Victoria, Victoria, BC V8P 5C2, Canada

###### Abstract

We study the impact of an early dark energy component (EDE) present during big
bang nucleosynthesis (BBN) on the elemental abundances of deuterium D/H, and
helium $Y_{p}$, as well as the effective relativistic degrees of freedom
$N_{\rm eff}$. We consider a simple model of EDE that is constant up to a
critical time. After this critical time, the EDE transitions into either a
radiation component that interacts with the electromagnetic plasma, a dark
radiation component that is uncoupled from the plasma, or kination that is
also uncoupled. We use measured values of the abundances and $N_{\rm eff}$ as
determined by CMB observations to establish limits on the input parameters of
this EDE model. In addition, we explore how those parameters are correlated
with BBN inputs; the baryon to photon ratio $\eta_{b}$, neutron lifetime
$\tau_{n}$, and number of neutrinos $N_{\nu}$. Finally, we study whether this
setup can alleviate the tension introduced by recent measurements of the
primordial helium abundance.

††preprint: APS/123-QED

##  I Introduction

The earliest direct probe of the universe’s history is the formation of light
elements when it was a few minutes old, during a process called big bang
nucleosynthesis (BBN). The successful predictions of light element abundances,
particularly ${}^{2}{\rm H}$ and ${}^{4}{\rm He}$, in a standard cosmological
history places strong bounds on physics beyond the standard model (SM).

Big bang nucleosynthesis takes place as the universe cools from a temperature
of around a few MeV to tens of keV. In the standard $\Lambda$CDM cosmological
history, at this time most of the energy of the universe is stored in
relativistic species in the form of photons, three flavors of neutrinos, and
electrons and positrons until they become nonrelativistic and annihilate.
Because of the relatively low density of baryons, nucleosynthesis proceeds via
two body reactions, with the main impediment being the formation of deuterium
due to its low binding energy–this deuterium bottleneck is not cleared until
temperatures of order $T_{\rm bn}\equiv 70~{}\rm keV$, an order of magnitude
below its binding energy because of the large photon-to-baryon ratio. Once
deuterium is produced heavier nuclei such as tritium and helium can be formed.

Since, in the standard cosmological picture, the energy budget of the universe
is “radiation dominated” during this epoch, the temperature evolves in time as
$T\propto t^{-1/2}$. The successful predictions of BBN rely on this
relationship which governs the opening of the deuterium bottleneck and the
subsequent formation of heavier nuclei, which is determined by complicated
nuclear physics cross sections that depend on the temperature of the universe.

Periods where vacuum energy played an important role have been motivated both
observationally and theoretically. As is well known, observations of the
accelerating expansion rate of the universe today tell us that the current
energy budget is predominantly in such a (different) component today which
behaves like a cosmological constant. The cosmological horizon and flatness
problems are alleviated if the expansion during earliest moments after the big
bang was driven by another form of dark energy, a process called inflation,
that transformed into a different form during “reheating.” Recently, the
possibility that a non-negligible fraction of the universe’s energy density
transformed as “early dark energy” (EDE) around the time of recombination has
been studied as a potential solution to the tension between early- and late-
time measurements of the expansion rate of the universe (see, e.g. Karwal and
Kamionkowski (2016); Mörtsell and Dhawan (2018); Poulin  _et al._ (2019);
Smith  _et al._ (2021); Schöneberg  _et al._ (2022)). The enhanced absorption
of 21-cm radiation reported by the EDGES collaboration Bowman  _et al._ (2018)
can be explained by a period of EDE, although not without increasing tension
with CMB observations Hill and Baxter (2018). More speculatively, the zero-
point energy density in the cores of neutron stars could differ substantially
from that in the vacuum due to a phase transition in hadronic matter,
potentially leading to observable effects in mergers Csáki  _et al._ (2018).

Big bang nucleosynthesis Schramm and Wagoner (1977); Bernstein  _et al._
(1989); Walker  _et al._ (1991); Cyburt  _et al._ (2016) has long been a test
of cosmology and particle physics Sarkar (1996); Iocco  _et al._ (2009);
Jedamzik and Pospelov (2009); Pospelov and Pradler (2010). In Ref. Kaplinghat
_et al._ (2000), the BBN effects of “power law cosmologies” with differing
expansion histories were studied, showing that the standard radiation-
dominated picture was favored by data on light element abundances. In this
paper, we explore the impact of variations of this picture of radiation
domination during BBN. In particular, we study the effect of EDE that briefly
becomes significant when compared to radiation energy density during BBN. This
could be achieved by considering dynamic models of dark energy as a scalar
field like quintessence or perhaps a scalar field sitting in a potential that
undergoes a phase transition (e.g. Caldwell  _et al._ (1998); Doran and
Robbers (2006); Gómez-Valent  _et al._ (2022); Wetterich (2004); Poulin  _et
al._ (2019)) . We parameterize the EDE by the energy density during its dark
energy phase as well as the time at which it transitions from dark energy to a
new form. We solve the equations that govern the production of light elements
and compare to observations, constraining parts of this parameter space. We
also consider recent measurement of the primordial ${}^{4}{\rm He}$ abundance
that differs from what is expected in standard BBN, and show that this tension
could be somewhat alleviated with EDE.

The paper is outlined as follows: In Section II, we present the equations
governing $\Lambda$CDM cosmology and some of the simplifying assumptions
taken. In Section III, we write down the system of equations that describes
the thermal evolution of the universe necessary for solving the Boltzman
equations for the BBN nuclear reaction network. In Section IV, we detail how
the nuclear reaction network is solved to produce standard BBN. In Section V
we introduce EDE into the model and explore three different scenarios for how
the excess energy redshifts away. We establish regions in the parameters space
of the model that are ruled out by observations. Section VI explores varying
some BBN parameters with one of the model’s two input parameters, while
keeping the other fixed. In Section VII, we discuss the recent anomalous
measurement of primordial 4He, and how EDE could alleviate this tension.
Finally, we summarize our conclusions in Section VIII.

##  II Cosmological Equations for $\Lambda$CDM

The standard $\Lambda$CDM model of cosmology is characterized by an expanding,
flat, homogeneous and isotropic universe which is described by a Friedmann-
Lemaître-Robertson-Walker (FLRW) metric. The conservation equation given an
FLRW metric is

$D_{\mu}T^{\mu}_{\ \nu}=\dot{\rho}+3\left(\frac{\dot{a}}{a}\right)(\rho+p)=0,$
(1)

where $\rho$ is the energy density, $P$ is the pressure, and $a$ is the scale
factor. Applying this stress energy tensor and the FLRW metric to the Einstein
field equations results in the Friedmann equation

$H^{2}\equiv\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi}{3m_{\rm
pl}^{2}}\rho,$ (2)

where $H$ is the Hubble parameter and $m_{\rm pl}=1.22\times 10^{19}~{}\rm
GeV$ is the Planck mass. The conservation equation Eq. 1, can be applied to
each stress-energy component (radiation, matter, dark energy, etc.), where
each component exerts a different pressure according to its equation of state,
resulting in a different scaling of energy with the scale factor. The terms in
the equation can be rearranged to give

$a^{-3}\frac{\partial[\rho a^{3}]}{\partial t}=-3\frac{\dot{a}}{a}P.$ (3)

From this relation one can infer that non-relativistic matter scales as
$\rho_{m}\propto a^{-3}$, relativistic matter (radiation) scales as
$\rho_{r}\propto a^{-4}$, and dark energy (or a cosmological constant) is
constant, $\rho_{\Lambda}\propto a^{0}$. Since each energy component scales
differently with $a$, the dominant component is different at each epoch
through the cosmological evolution of the universe. In particular, during the
period of BBN, the universe was radiation dominated in $\Lambda$CDM. In this
work, we propose an EDE component during BBN.

In standard BBN (SBBN), we assume $\Lambda$CDM cosmology and the standard
model of particle physics with the number of neutrino species fixed at
$N_{\nu}=3$. The only parameter in the theory is the baryon to photon ratio
$\eta_{b}$, which is determined to better than $1\%$ accuracy from independent
CMB measurements at later times Aghanim  _et al._ (2020); Cyburt  _et al._
(2016). To simplify the treatment here, we assume an instantaneous neutrino
decoupling somewhat before the point when electron-positron annihilation
happens (when the temperature of the primordial plasma is $T\sim 0.5$ MeV).
BBN is also sensitive to the neutron lifetime $\tau_{n}$, nuclear reaction
rates, and the effective relativistic degrees of freedom $N_{\rm eff}$

$N_{\rm eff}\equiv\frac{8}{7}\left(\frac{11}{4}\right)^{4/3}\frac{\rho_{\rm
rad}-\rho_{\gamma}}{\rho_{\gamma}},$ (4)

where $\rho_{\rm rad}$ is the total radiation energy density

$\displaystyle\rho_{\rm rad}=$ $\displaystyle\
\rho_{\gamma}+N_{\nu}\rho_{\nu}$ $\displaystyle+(\rm other\
radiation\text{-}like\ components).$ (5)

Those quantities are either very well measured or set by SM calculations (e.g.
see Ref. de Salas and Pastor (2016) for $N_{\rm eff}$, the average from the
particle data group for $\tau_{n}$ Zyla  _et al._ (2020), and Serpico  _et
al._ (2004); Nagai  _et al._ (2006) for nuclear reaction rates).

The abundances of elements during BBN are governed by the Boltzmann equation
in an expanding homogeneous and isotropic universe

$\frac{\partial f}{\partial t}-Hp\frac{\partial f}{\partial p}=C[f],$ (6)

where $f$ is the distribution function of some species, and $C[f]$ is the
collision term which depends on the nuclear reaction rates. Assuming kinetic
equilibrium with Maxwell Boltzmann distributions and ignoring Pauli blocking
and Bose enhancement factors (see Escudero (2019); Dodelson (2003) for why
those assumptions are justified), the Boltzmann equation for a reaction of the
form $1+2\rightarrow 3+4$ can be simplified to

$a^{-3}\frac{d(n_{1}a^{3})}{dt}=n_{1}^{(0)}n_{2}^{(0)}\langle\sigma
v\rangle\left[\frac{n_{3}n_{4}}{n_{3}^{(0)}n_{4}^{(0)}}-\frac{n_{1}n_{2}}{n_{1}^{(0)}n_{2}^{(0)}}\right],$
(7)

where $\langle\sigma v\rangle$ is the thermally averaged cross section,
$n_{s}$ is the number density of some species given by

$n_{s}=g_{s}\int\frac{d^{3}p}{(2\pi)^{3}}f,$ (8)

with $g_{s}$ being the spin degrees of freedom. The equilibrium number density
$n_{s}^{(0)}$ for $m_{s}\gg T$ (Maxwell-Boltzmann) is

$n_{s}^{(0)}=g_{s}\left(\frac{m_{s}T}{2\pi}\right)^{3/2}e^{-m_{s}/T}.$ (9)

Equation (7) can be used for each reaction considered in BBN to build the
nuclear reaction network to solve for elemental abundances. In building the
nuclear reaction network (Section IV), we consider a simple network of only
the light elements (p, n, D, 3H, 3He, and 4He). Since we are using BBN as a
probe of new physics, we avoid Be and Li due to the lithium problem where
predictions exceed observations by a factor of 2-3 Ryan  _et al._ (2000);
Olive  _et al._ (2000), although recent re-observations of halo stars suggest
possible Lithium depletion which may relax this tension Fields and Olive
(2022).

##  III The time-temperature relation

The right hand side of the Boltzmann equation has nuclear reaction cross
sections $\langle\sigma v\rangle$ which are most easily expressed as functions
of the temperature. So in order to solve the Boltzmann equation to obtain the
abundances, one needs to find a time-temperature relation that describes the
thermal evolution of the universe given its contents.

The temperature evolution in time is obtained by considering conservation of
total entropy

$\frac{dS}{dt}=0,$ (10)

where the total entropy is $S=sV\propto sa^{3}$. The entropy density is
defined as

$s\equiv\frac{\rho+P}{T},$ (11)

where we have neglected chemical potentials. For relativistic species,
$P=\rho/3$ and so

$s=s_{\rm rad}=\frac{4}{3}\frac{\rho_{\rm rad}}{T}.$ (12)

After the neutrinos have decoupled from the electromagnetic plasma, the
entropy of each sector is separately conserved. The entropy density of the
universe is

$s=\frac{2\pi^{2}}{45}g_{*s}T_{\gamma}^{3},$ (13)

where we define the effective relativistic entropy degrees of freedom

$g_{*s}\equiv\sum_{B}\left(\frac{T_{a}}{T_{\gamma}}\right)^{3}+\frac{7}{8}\sum_{F}\left(\frac{T_{a}}{T_{\gamma}}\right)^{3},$
(14)

where $T_{a}$ is the temperature of a given species, and the first sum is over
bosonic species and the second sum is over fermionic species. Key quantities
we want to keep track of are the photon temperature $T_{\gamma}$ and neutrino
temperature $T_{\nu}$. For temperatures above about $3~{}{\rm MeV}$, weak
interactions keep the neutrinos coupled to the photons and their temperatures
are the same. After this point, the neutrinos decouple and the two sectors
evolve separately with the entropy in each individually conserved. In the
photon sector, the relevant degrees of freedom for the entropy are photons,
electrons, and positrons and the effective relativistic entropy degrees of can
be written as

$g_{*s\gamma}=2\left(1+\frac{s_{e^{-}}+s_{e^{+}}}{s_{\gamma}}\right),$ (15)

where $s_{e^{\pm}}$ is the entropy density of the electrons or positrons
(computed using Fermi-Dirac statistics with the same temperature as the
photons) and $s_{\gamma}$ that of the photons. As the universe cools, $s_{e}$
decreases drastically, reheating the photons. In the neutrino sector, the
effective relativistic entropy degrees of freedom does not change during this
epoch. Plugging the entropy expressions for the two separate sectors into Eq.
10 gives a system of coupled equations,

$\displaystyle\frac{dT_{\gamma}}{dt}=$ $\displaystyle-
HT_{\gamma}\bigg{(}1+\frac{T_{\gamma}}{3g_{*s\gamma}}\frac{dg_{*s\gamma}}{dT_{\gamma}}\bigg{)}^{-1},$
(16) $\displaystyle\frac{dT_{\nu}}{dt}=$ $\displaystyle-HT_{\nu}.$ (17)

The Hubble parameter in those equations is given by the Friedmann equation

$H=\sqrt{\frac{8\pi}{3m_{\rm Pl}^{2}}\rho_{\rm tot}},$ (18)

where in SBBN, the energy density

$\rho_{\rm
tot}=\rho_{\gamma}(T_{\gamma})+\rho_{e^{-}}(T_{\gamma})+\rho_{e^{+}}(T_{\gamma})+N_{\nu}\rho_{\nu}(T_{\nu}),$
(19)

is dominated by radiation. Equations (16) and (17) can be solved numerically
to obtain $T_{\gamma}(t)$ which is necessary to solve the BBN nuclear reaction
network to obtain nuclear abundances. Finally, using the definition of the
Hubble parameter, we have the additional equation

$\frac{da(t)}{dt}=a(t)H(t),$ (20)

which we also need to solve for when considering EDE scenarios to determine
the excess energy’s decay rate as a function of $a(t)$.

##  IV The BBN nuclear reaction network

A general form of the Boltzmann equation for a reaction of the form
$k+l\rightarrow i+j$, is Wagoner  _et al._ (1967); Esposito  _et al._ (2000)

$\displaystyle\dot{X}_{i}=\sum_{j,k,l}N_{i}\bigg{(}$
$\displaystyle\Gamma_{kl\rightarrow
ij}\frac{X_{l}^{N_{l}}X_{k}^{N_{k}}}{N_{l}!N_{k}!}$ (21)
$\displaystyle\quad\quad-\Gamma_{ij\rightarrow
kl}\frac{X_{i}^{N_{i}}X_{j}^{N_{j}}}{N_{i}!N_{j}!}\bigg{)},$

where $N_{s}$ indicates how many times a species shows up in a reaction. This
equation implicitly assumes that each nuclear species follow a Maxwell-
Boltzmann distribution during nucleosynthesis, since their energies
(essentially rest masses) are much larger than the temperature.

We consider here a simplified reaction network consisting of only 6 species
(n, p, D,3H, 3He, and 4He), therefore we have a system of 6 coupled equations,
one for each species. For each reaction a species appears in, we write down a
term as shown on the right hand side of Eq. 21. The forward reaction rate
$\Gamma_{kl\rightarrow ij}$ is defined as

$\displaystyle\Gamma_{kl\rightarrow ij}$ $\displaystyle\equiv\langle\sigma
v\rangle_{kl\rightarrow ij}n_{b}$ (22)
$\displaystyle=\eta_{b}\times\frac{2\zeta(3)}{\pi^{2}}T^{3}\frac{f_{kl\rightarrow
ij}}{N_{A}^{N_{i}-1}},$

where $n_{b}$ is the baryon number density which could be expressed in terms
of $\eta_{b}$ and the photon number density as in the second equality. We
obtain the cross sections from reaction rate fits Serpico  _et al._ (2004);
Nagai  _et al._ (2006)111We adopt reaction rate fits primarily from Serpico
_et al._ (2004), except for the reaction $d+n\rightarrow\gamma+^{3}H$ which is
adopted from Nagai  _et al._ (2006). $f\equiv\langle\sigma v\rangle
N_{A}^{N_{i}-1}$, with $N_{A}$ being Avogadro’s number and $N_{i}$ the number
of incoming nuclides (2 in all the cases we consider here). The backward
reaction rate $\Gamma_{ij\rightarrow kl}$ is defined as

$\Gamma_{ij\rightarrow kl}\equiv\Gamma_{kl\rightarrow ij}R,$ (23)

where depending on the case, $R$ is calculated from

  1. (i)

$k+l\rightarrow i+j$

$R\equiv\frac{n_{i}^{(0)}n_{j}^{(0)}}{n_{k}^{(0)}n_{l}^{(0)}}=\frac{X_{i,eq}X_{j,eq}}{X_{k,eq}X_{l,eq}},$
(24)

  2. (ii)

$k+\gamma\rightarrow i+j$

$R\equiv\frac{n_{i}^{(0)}n_{j}^{(0)}}{n_{k}^{(0)}n_{b}}=\frac{X_{i,eq}X_{j,eq}}{X_{k,eq}}.$
(25)

The equilibrium distribution for any species can be expressed in terms of only
the neutron and proton abundances

$\displaystyle X_{i,eq}\equiv$
$\displaystyle\frac{n_{i}^{(0)}}{n_{b}}\simeq\frac{g_{i}}{2}\left(\zeta(3)\sqrt{\frac{8}{\pi}}\right)^{A_{i}-1}A_{i}^{3/2}$
(26)
$\displaystyle\times\left(\frac{T}{m_{p}}\right)^{\frac{3}{2}(A_{i}-1)}\eta_{b}^{A_{i}-1}X_{p}^{Z_{i}}X_{n}^{A_{i}-Z_{i}}e^{B_{i}/T},$

where $A_{i}$, $Z_{i}$ and $B_{i}\equiv\sum_{n}m_{n}-m_{i}$ are the mass
number, atomic number, and binding energy of the species respectively, and
$m_{n}$ is the mass of comprising nucleons. Here, we assume that the masses in
the prefactor are just multiples of $m_{p}$ (e.g. $m_{{}^{4}He}\simeq
4m_{p}$), but not in the exponent. In addition to determining reverse rates,
the equilibrium abundance fractions determine the abundances at early times
when the reaction rates are large enough for each species to obtain nuclear
statistical equilibrium.

We consider 10 reactions that contribute the most to the BBN network of the
light elements we’re considering here (n, p, D,3H, 3He, and 4He). The
reactions and their corresponding contributing terms in the Boltzmann equation
are shown in Table 1. Including those reactions in the Boltzmann equation Eq.
21 results in the system of equations

$\displaystyle\dot{X}_{n}=$
$\displaystyle\dot{X}_{pn}-\dot{X}_{png}-\dot{X}_{ndd}-\dot{X}_{ndt}+\dot{X}_{ptn}$
$\displaystyle-\dot{X}_{dng},$ (27) $\displaystyle\dot{X}_{p}=$
$\displaystyle-\dot{X}_{pn}-\dot{X}_{png}-\dot{X}_{dpg}-\dot{X}_{pdd}+\dot{X}_{he3dp}$
$\displaystyle-\dot{X}_{ptn}-\dot{X}_{tpg},$ (28) $\displaystyle\dot{X}_{d}=$
$\displaystyle\dot{X}_{png}-\dot{X}_{dpg}+2\dot{X}_{ndd}+2\dot{X}_{pdd}+\dot{X}_{ndt}$
$\displaystyle-\dot{X}_{he3dp}-\dot{X}_{dng},$ (29)
$\displaystyle\dot{X}_{{}^{3}H}=$
$\displaystyle-\dot{X}_{pdd}+\dot{X}_{ndt}-\dot{X}_{ptn}+\dot{X}_{dng}-\dot{X}_{tpg},$
(30) $\displaystyle\dot{X}_{{}^{3}He}=$
$\displaystyle\dot{X}_{dpg}-\dot{X}_{ndd}-\dot{X}_{he3dp}+\dot{X}_{ptn},$ (31)
$\displaystyle\dot{X}_{{}^{4}He}=$
$\displaystyle-\dot{X}_{ndt}+\dot{X}_{he3dp}+\dot{X}_{tpg}.$ (32)

Reaction | Contribution
---|---
$n\leftrightarrow p$ 222This includes the reactions $e^{+}+n\leftrightarrow p+\overline{\nu}_{e}$, $p+e^{-}\leftrightarrow n+\nu_{e}$, and $n\leftrightarrow p+e^{-}+\overline{\nu}_{e}$. |  $\dot{X}_{pn}\equiv\omega_{pn}X_{p}-\omega_{np}X_{n}$
$p+n\leftrightarrow\gamma+d$ | $\dot{X}_{png}\equiv\Gamma_{png}X_{n}X_{p}-\Gamma_{gpn}X_{d}$
$d+p\leftrightarrow\gamma+\rm^{3}He$ | $\dot{X}_{dpg}\equiv\Gamma_{dpg}X_{d}X_{p}-\Gamma_{gdp}X_{{}^{3}He}$
$d+d\leftrightarrow n+\rm^{3}He$ | $\dot{X}_{ndd}\equiv\Gamma_{ndd}X_{n}X_{{}^{3}He}-\Gamma_{ddn}\frac{X_{d}^{2}}{2}$
$d+d\leftrightarrow p+\rm^{3}H$ | $\dot{X}_{pdd}\equiv\Gamma_{pdd}X_{p}X_{{}^{3}H}-\Gamma_{ddp}\frac{X_{d}^{2}}{2}$
${\rm{}^{3}H}+d\leftrightarrow n+\rm^{4}He$ | $\dot{X}_{ndt}\equiv\Gamma_{ndt}X_{n}X_{{}^{4}He}-\Gamma_{tdn}X_{{}^{3}H}X_{d}$
${\rm{}^{3}He}+d\leftrightarrow p+\rm^{4}He$ | $\dot{X}_{he3dp}\equiv\Gamma_{he3dp}X_{d}X_{{}^{3}He}-\Gamma_{phe3d}X_{p}X_{{}^{4}He}$
${\rm{}^{3}He}+n\leftrightarrow p+\rm^{3}H$ | $\dot{X}_{ptn}\equiv\Gamma_{ptn}X_{p}X_{{}^{3}H}-\Gamma_{npt}X_{n}X_{{}^{3}He}$
$d+n\leftrightarrow\gamma+\rm^{3}H$ | $\dot{X}_{dng}\equiv\Gamma_{dng}X_{d}X_{n}-\Gamma_{gdn}X_{{}^{3}H}$
${\rm{}^{3}H}+p\leftrightarrow\gamma+\rm^{4}He$ | $\dot{X}_{tpg}\equiv\Gamma_{tpg}X_{{}^{3}H}X_{p}-\Gamma_{gtp}X_{{}^{4}He}$
Table 1: The 10 reactions considered in the BBN network along with their
contributing terms.

The initial conditions for this set of equations come from Eq. 26, and the
constants needed to find equilibrium abundances (and consequently the reverse
rates) are shown in Table 2.

$N$ | $g$ | $A$ | $Z$ | $m\ (\rm m_{u})$
---|---|---|---|---
$n$ | 2 | 1 | 0 | 1.00866
$p$ | 2 | 1 | 1 | 1.00728
$d$ | 3 | 2 | 1 | 2.01410
3H | 2 | 3 | 1 | 3.01605
3He | 2 | 3 | 2 | 3.01603
4He | 1 | 4 | 2 | 4.00260
Table 2: Nuclear properties for nuclides used, where $g$ is the spin degrees
of freedom, $A$ is the atomic mass number (number of protons + number of
neutrons), $Z$ is the atomic number (number of protons), and $m$ is the
isotope mass in atomic mass units. The listed masses are isotope masses Wang
_et al._ (2021). Subtract $Zm_{e}$ to get the nuclear mass.

We fix BBN input parameters, namely the baryon to photon ratio $\eta_{b}\times
10^{10}\equiv\eta_{10}=6.104\pm 0.058$, neutron lifetime $\tau_{n}=879.4\pm
0.6$ s, as reported by Zyla  _et al._ (2020); Aghanim  _et al._ (2020); Fields
_et al._ (2020). The simplified treatment of temperature evolution does not
take into account non-instantaneous neutrino decoupling and radiative effects
which in the standard case would result in an $N_{\rm eff}=3.045$ de Salas and
Pastor (2016). To approximate this effect, we take $N_{\nu}=3.045$, which in
the SBBN case gives the effective result of $N_{\rm eff}=3.045$ as desired.

Solving this system results in the standard picture for SBBN as shown in Fig.
1, which shows the evolution of elemental abundances, consistent with previous
more detailed BBN work in the literature Pospelov and Pradler (2010); Cyburt
_et al._ (2016); Pitrou  _et al._ (2018); Serpico  _et al._ (2004).

Figure 1: The nuclear abundances that result from solving the system of
equations. The abundances of D, T, and 3He are normalized to the hydrogen
(proton) abundance, and $Y_{p}$ is the helium mass fraction.

##  V Early Dark Energy During BBN

The treatment in Sections III and IV so far has been entirely within standard
$\Lambda$CDM and SBBN. We now modify the equations to introduce an EDE
component. We consider a simple model with a constant dark energy present at
early times which transitions into a different form at some critical time. In
particular, in order to correctly describe the subsequent formation of the
CMB, this new component must redshift like radiation or faster. Hence, this
model we consider has only two parameters; the amount of constant EDE present
$\rho_{\rm\Lambda}$, and the critical temperature at which the ‘phase
transition’ happens $T_{\rm crit}$ where it transitions into a decaying
component. These modifications directly change the time-temperature equations
in Section III, which consequently changes the outcomes of BBN, allowing us to
put constraints on the two parameters.

Most generically, the first addition to be made is to include an extra term in
Eq. 19 to account for the extra EDE. Generally, this term should be constant
up to a certain time, then it decays depending on which scenario we are
considering. The other significant physical quantity to consider is entropy.
When EDE is still a constant, there is no additional entropy since the
equation of state parameter for a constant energy density is $w=-1$, so its
entropy is zero. After the transition, which we approximate as instantaneous,
we consider three cases for the equation of state of this new component: (1)
$w=1/3$, coupled to the SM plasma with associated entropy injection (which we
dub “transition into photons” for short); (2) $w=1/3$, uncoupled to the SM
plasma (which we dub “transition into dark radiation” for short); and (3)
$w=1$, uncoupled to the SM plasma (“transition into kination”).

###  V.1 Transition into SM photons

In this case, EDE transitions into regular SM photons which are in thermal
contact with the rest of the plasma. The additional energy density takes the
form

$\rho_{\rm EDE}=\begin{cases}\rho_{\Lambda},\quad&t<t_{\rm crit}\\\
0,\quad&t>t_{\rm crit}\end{cases},$ (33)

where $t_{\rm crit}$ is the critical time when the temperature is equal to
$T_{\rm crit}$. Regardless of the scenario we’re considering, we can determine
$t_{\rm crit}$ by solving the Friedmann equation up to a sufficiently large
time and then finding the time that satisfies $T(t_{\rm crit})=T_{\rm crit}$.
In this specific scenario, to preserve energy density conservation, our
assumption of an instantaneous transition causes the photon temperature to
suddenly increase at $t_{\rm crit}$. If we denote the temperature just before
the transition as $T_{-}$, then the temperature just after the transition,
$T_{+}$, can be found by setting the total energy density before and after the
transition to be equal

$\rho_{\gamma}(T_{+})+2\rho_{e}(T_{+})=\rho_{\gamma}(T_{-})+2\rho_{e}(T_{-})+\rho_{\Lambda},$
(34)

which can be solved numerically. Therefore in this regard, the thermal
evolution of the universe is split into two steps, one before the critical
point where the total energy density includes $\rho_{\Lambda}$, and the other
after the critical point where the total energy denisty is that of standard
$\Lambda$CDM, but with the initial temperature given by $T_{+}$. The result of
those numerical solutions of Eqs. 16 and 17, are the functions $T_{\gamma}(t)$
and $T_{\nu}(t)$, which are necessary in determining the outcomes of BBN. An
example case for input parameters $\rho_{\Lambda}^{1/4}=2$ MeV, and $T_{\rm
crit}=1$ MeV is shown in Fig. 2, where the sharp increase in $T_{\gamma}$
corresponds to the critical point where EDE transitions to SM photons. This
choice of parameters is ruled out by BBN, but we pick those large numbers to
demonstrate the effect EDE would have.

Figure 2: Photon and neutrino plasma temperature evolution in the presence of
an EDE, $\rho_{\Lambda}^{1/4}=2$ MeV, that transitions to SM photons at
$T_{\rm crit}=1$ MeV. The discontinuity in the photon temperature is a result
of our simplifying assumption of an instantaneous transition.

Importantly, we note that we do not choose parameters that lead to the
$T_{\gamma}>3~{}\rm MeV$ after the transition since in such a case, the
neutrinos would recouple to the rest of the plasma and BBN would proceed as in
the standard case.

In addition to a time-temperature relation, since this scenario results in an
increased number of photons, the baryon-to-photon ratio $\eta_{b}$ is expected
to decrease after the critical point. Since the baryon number density is
proportional to $a^{-3}$, and the photon number density is proportional to
$T_{\gamma}^{3}$, then the baryon to photon ratio changes according to

$\displaystyle\frac{\eta_{b}}{\eta_{b}|_{\rm initial}}$
$\displaystyle=\left(\frac{aT_{\gamma}|_{\rm
initial}}{aT_{\gamma}}\right)^{3}$ (35)
$\displaystyle=\left(\frac{T_{\gamma}/T_{\nu}|_{\rm
initial}}{T_{\gamma}/T_{\nu}}\right)=\left(\frac{T_{\nu}}{T_{\gamma}}\right)^{3},$

where in the second equality we used the fact that the neutrino temperature is
approximately proportional to $a^{-1}$ (minimal neutrino interactions), and
the last equality comes from the fact that $T_{\gamma}=T_{\nu}$ at early
times. The ratio changes when electrons and positrons annihilate (since the
photon number density increases), and again once the EDE gets converted into
SM photons. An example of this behaviour is shown in the green curve in Fig. 3
for the example case with parameters $\rho_{\Lambda}^{1/4}=2$ MeV, and $T_{\rm
crit}=1$ MeV as before. The initial ratio is calculated such that we obtain a
final baryon to photon ratio $\eta_{b}=6.104\times 10^{-10}$, as observed by
Planck 2018 Aghanim  _et al._ (2020); Fields  _et al._ (2020).

Figure 3: The change in the baryon to photon ratio $\eta_{b}$ with time in
each of the scenarios considered here with example input parameters $T_{\rm
crit}=1$ MeV and $\rho_{\Lambda}^{1/4}=2$ MeV. The smooth transition is due to
electron-positron annihilation.

Each pair of points in the parameter space of $\rho_{\Lambda}$ and $T_{\rm
crit}$ results in a different outcome for the abundances of elements at the
end of BBN. Here we consider constraints from D and 4He abundances as they are
well measured Zyla  _et al._ (2020)

$\displaystyle\rm D/H\times 10^{5}$ $\displaystyle=2.547\pm 0.025,$ (36)
$\displaystyle Y_{p}$ $\displaystyle=0.245\pm 0.003,$ (37)

in addition to constraints from $N_{\rm eff}$ as measured by Planck Aghanim
_et al._ (2020)

$\displaystyle N_{\rm eff}$ $\displaystyle=2.99\pm 0.17.$ (38)

We establish exclusion regions that result in abundances and $N_{\rm eff}$
within 1$\sigma$ from the observed values, taking into account theoretical
uncertainties on the abundance coming from uncertainties on the input
parameters $\eta_{b},\tau_{n}$ as well as the nuclear uncertainties as
parameterized in Serpico  _et al._ (2004). Those regions for the two input
parameters in this scenario are shown in Fig. 4, where the EDE transitions
into SM photons.

The yellow curve indicates the limit where $\chi^{2}<1$ for the D/H abundance
(D abundance normalized by H abundance), where

$\chi^{2}=\frac{\left(\rm
D/H_{calculated}-D/H_{observed}\right)^{2}}{\sigma_{\rm obs}^{2}+\sigma_{\rm
theoretical}^{2}},$ (39)

for one degree of freedom. The region above the curve is ruled out. Similarly,
the green curve is the $\chi^{2}<1$ limit for helium abundance $Y_{p}$, and
the red curve is the limit for $N_{\rm eff}$. The thick black curve is the
combined limit given all three observations (where $\chi^{2}<3.506$ for 3
degrees of freedom). The axes are cutoff at 3 MeV because we do not consider
transitions that recouple the neutrinos since, as mentioned above, that
results in standard BBN.

At any given time after the critical point, this scenario results in a
temperature higher than the corresponding temperature at the same time in the
case of a $\Lambda$CDM cosmology as shown in Fig. 5 (green curve) with example
input parameters as before. This is effectively equivalent to a universe that
has expanded at a slower rate than $\Lambda$CDM, and so results in lower final
abundances since fewer neutrons were available at the time of nucleosynthesis.
The lower abundance of neutrons available at the start of BBN is caused by
more time elapsing in this scenario, thus allowing more neutrons to convert to
protons via free neutron decay. In addition, $N_{\rm eff}$ is lowered here
since the photon energy density has increased.

The analysis in this section could also be applied for a case of negative
initial EDE density $\rho_{\Lambda}<0$. Doing so, we find similar limits on
$\left|\rho_{\Lambda}\right|$. This scenario is not discussed further in this
work.

Figure 4: Limits on the parameter space of $\rho_{\Lambda}$ and $T_{\rm crit}$
from D/H abundance (blue curve), $Y_{p}$ (green curve), and $N_{\rm eff}$ (red
curve). The combined limit from all three observables is shown in the thick
black curve. The region above the curves is excluded by this model. (a) is the
case of a decay of EDE into SM photons, (b) is a decay into dark radiation,
and (c) is a decay into kination.

###  V.2 Transition into dark radiation

At the transition point in this case, we assume that the EDE instantaneously
becomes ‘dark radiation’. In other words it starts to redshift $\propto
a^{-4}$, and does not interact with the SM plasma (no entropy injection).
Since neutrinos also redshift as $\propto a^{-4}$ and do not interact with the
plasma, this scenario is roughly equivalent to an EDE transition into
neutrinos. The additional energy density takes the form

$\rho_{\rm EDE}=\begin{cases}\rho_{\Lambda},\quad&t<t_{\rm crit}\\\
\rho_{\Lambda}\left(\frac{a_{\rm crit}}{a(t)}\right)^{3(1+w)},\quad&t>t_{\rm
crit}\end{cases},$ (40)

where $a_{\rm crit}$ is the value of the scale factor at the critical time,
and $w$ is the equation of state parameter which is equal to $1/3$ in this
case. Unlike the previous case, there is no jump in the photon temperature or
$\eta_{b}$ at the critical point since there is no entropy injection
($T_{-}=T_{+}$). Owing to the added energy density, this scenario results in a
faster expansion of the universe and thus a lower temperature of the plasma at
any given time when compared to $\Lambda$CDM as can be seen in Fig. 5 (blue
curve). The small bump is due to the time shift of electron-positron
annihilation which alters the photon temperature (solid) relative to the
neutrino temperature (dashed). As in the previous case, the baryon to photon
ratio also changes with time, however it does not exhibit the jump (blue curve
for this case, versus green curve for the previous case in Fig. 3), since no
entropy is injected and so the ratio isn’t changed due to EDE. The smooth
change due to electron-positron annihilation however is still present. Since
the expansion is faster in this scenario, annihilation occurs earlier than
$\Lambda$CDM (black curve), and so the change in $\eta_{b}$ shifts earlier as
shown in Fig. 3.

Figure 5: Photon (solid) and neutrino (dashed) temperature ratios of a given
scenario relative to $\Lambda$CDM in the three scenarios we consider.

The limits on the parameter space in this scenario are shown in Fig. 4. Since
no entropy is injected into the plasma, and consequently $\eta_{b}$ doesn’t
change, the limit on $\rho_{\Lambda}$ from elemental abundances plateaus at
late-time transitions (low $T_{\rm crit}$). This is because when the
transition occurs at a critical temperature well below $T_{\rm bn}$, BBN
processes have already largely finished. Therefore, the maximum allowed EDE
does not depend on the critical point, and is rather just an upper limit on
the allowed excess energy during BBN. This shows up in Fig. 4 as the plateau
at low $T_{\rm crit}$ (especially visible for limits from 4He since its
synthesis finished relatively quickly, compared to D which exhibits late-time
depletion). This is in contrast to the previous case where a plateau did not
occur since there is also entropy injection in that scenario. The direct
result of entropy injection is to force the initial value of $\eta_{b}$ to
change (see SM photons case in Fig. 3) so that its late-time value agrees with
CMB measurements. However, the value of $\eta_{b}$ during BBN does affect the
outcome of elemental abundances. Limits from $N_{\rm eff}$ do not plateau (in
either scenario) since they depend on the much later measurement during the
CMB epoch, and the amount of allowed extra radiation continues to be related
to the time it is injected.

Unlike the previous case, the abundances of D and 4He in this scenario are
higher than in SBBN. This is because more neutrons are available at the time
of nucleosynthesis since the universe expanded more quickly leaving less time
for them to get converted into protons (or decay). More available neutrons in
the reaction network result in higher abundances for D and 4He (among others).
In addition, since the extra added energy transitions into (dark) radiation,
this scenario results in an increased $N_{\rm eff}$.

###  V.3 Transition into kination

The last scenario we consider is EDE that transitions into a component that
redshifts $\propto a^{-6}$, called kination Joyce (1997). The energy density
here is characterized by a scalar field whose dynamics are mainly dominated by
its kinetic energy. A scalar field can be treated as a perfect fluid, thus its
equation of state is given by

$w=\frac{\rm KE-PE}{\rm KE+PE}\simeq 1.$ (41)

This is the maximum EOS allowed by causality, i.e. the speed of sound is equal
to the speed of light, and has the fastest possible redshifting of energy
density ($\rho\propto a^{-6}$). Therefore a kination component of the energy
density becomes sub-dominant quickly compared to other components (e.g.
radiation, matter, etc.), or it leads to a slowed expansion of the universe if
it is the dominant component present. A kination period could be induced by a
fast-rolling inflaton field at the end of inflation where the field has a
steep potential Gouttenoire  _et al._ (2021); Spokoiny (1993), analogous to
the brief period where the energy density is dominated (or nearly dominated)
by $\rho_{\Lambda}$ in our treatment here.

Since the equation of state parameter is set to $1$, the EDE density is given
by

$\rho_{\rm EDE}=\begin{cases}\rho_{\Lambda},\quad&t<t_{\rm crit}\\\
\rho_{\Lambda}\left(\frac{a_{\rm crit}}{a(t)}\right)^{6},\quad&t>t_{\rm
crit}\end{cases}.$ (42)

The extra energy density redshifts very quickly after the transition, bringing
the temperature of the plasma approximately back to its $\Lambda$CDM value at
late times as shown in the red curve in Fig. 5. This scenario results in a
faster expansion of the universe (similar to the previous case), and so the
change in $\eta_{b}$ due to annihilation is also shifted earlier as shown in
the green curve in Fig. 3. The shift is less dramatic as in the previous
scenario since expansion is not as fast due to the quick disappearance of the
EDE.

The limits on the parameter space in this scenario are shown in Fig. 4. As in
the case of dark radiation, no entropy is injected here, and so the limits
from elemental abundances plateau at low critical temperatures, whereas limits
from $N_{\rm eff}$ do not. The limits from $N_{\rm eff}$ are much weaker here
compared to other scenarios since kination results in only a small change of
the expansion rate of the universe.

##  VI Varying other BBN parameters

In the previous section, we put constraints on the input parameters
$\rho_{\Lambda}$ and $T_{\rm crit}$, given constant SBBN parameters

$\displaystyle\tau_{n}$ $\displaystyle=879.4\pm 0.6\ \rm s,$ (43)
$\displaystyle\eta_{b}$ $\displaystyle=6.104\pm 0.058\times 10^{-10},$ (44)
$\displaystyle N_{\nu}$ $\displaystyle=3.045.$ (45)

We consider here the dependence of the EDE input parameters on each of those
for all the scenarios considered in the previous section.

Figure 6: Dependence of $\rho_{\Lambda}$ on BBN parameters with a fixed
$T_{\rm crit}=0.5$ MeV, for a transition of EDE into SM photons (bottom row),
dark radiation (middle row), and kination (top row). Space inside the contours
results in outcomes that are within $1\sigma$ from observed D/H (blue),
$Y_{p}$ (red), and $N_{\rm eff}$ (dashed) Zyla  _et al._ (2020); Aghanim  _et
al._ (2020). Thick contours are the combined $1\sigma$ limits, and thin
vertical lines are the $1\sigma$ errors on the observed $\tau_{n}$ and
$\eta_{10}$.

To explore the dependence of $\rho_{\Lambda}$ on the BBN parameters in a 2D
parameter space, we set $T_{\rm crit}=0.5$ MeV. Since the effect of adding
extra energy ultimately results in altering the temperature evolution, we
expect a correlation between the amount of EDE and the neutron lifetime. This
is indeed visible in the first column of panels in Fig. 6. The region inside
the blue and red contours results in D/H and $Y_{p}$ (respectively) within
$1\sigma$ of the observed values. The dashed line (or contour), is the limit
from $N_{\rm eff}$, where regions above the line are excluded. It is
independent of $\tau_{n}$ as it is unrelated to nucleosynthesis. The thin grey
lines indicate the measured value of neutron lifetime. The thick contour is
the combined $1\sigma$ limit calculated by requiring $\chi^{2}<3.51$ (for 3
degrees of freedom), or $\chi^{2}<2.28$ (for 2 degrees of freedom) for cases
where limits from $N_{\rm eff}$ cannot be applied. The direction of the
correlation depends on whether the expansion is decreased (SM case, bottom
row), or increased (dark radiation and kination cases, middle and top rows
respectively). A decay of EDE into SM photons results in an increased plasma
temperature compared to $\Lambda$CDM. This is equivalent to a relative
decrease in the expansion of the universe, and consequently more time elapsed
when $T_{\rm bn}$ is achieved, leaving fewer neutrons available at
nucleosynthesis. This effect is counteracted by an increased neutron lifetime,
hence the positive correlation seen in the bottom left panel of Fig. 6. The
reverse effect is visible for a decay of EDE into dark radiation or kination
resulting in a negative correlation as shown in the middle and top left
panels. The correlation is weaker for the case of kination since the expansion
increase isn’t as strong as the case of dark radiation.

By a similar argument, the baryon to photon ratio is correlated to the amount
of EDE since nuclear abundances are sensitive to $\eta_{b}$ Schramm and Turner
(1998); Fields  _et al._ (2020); Cyburt  _et al._ (2016). Since $Y_{p}$ is
logarithmically sensitive to $\eta_{b}$, the correlation is weak, as shown by
the red lines with a small slope in the middle column of panels. An increase
in either $\eta_{b}$ or $\rho_{\Lambda}$ for a decay of EDE into SM photons
results in lowered D/H, giving a negative correlation as shown in the bottom
panel of the middle column. For the other two cases, increasing
$\rho_{\Lambda}$ increases D/H abundance, and so the correlation is positive
(with kination being weaker as before).

Finally, the dependence of $\rho_{\Lambda}$ on the number of neutrinos
$N_{\nu}$ is shown in the third column. Unlike the other two BBN parameters,
$N_{\nu}$ does affect the thermodynamics and therefore changes the ratio
$T_{\gamma}/T_{\nu}$, resulting in an altered $N_{\rm eff}$ (and thus limits
on it). The limits are also present in the case of kination (top right panel),
although the correlation is very weak.

Figure 7: Dependence of $T_{\rm crit}$ on BBN parameters for a fixed
$\rho_{\Lambda}^{1/4}=1$ MeV. Layout and contours are the same as Fig. 6.

To see the dependence of $T_{\rm crit}$ on the BBN parameters in a 2D
parameter space, we similarly choose $\rho_{\Lambda}^{1/4}=1$ MeV and vary the
other parameters as shown in Fig. 7. The correlations follow a similar pattern
as above but with the vertical axis flipped, where EDE’s impact is weaker the
larger $T_{\rm crit}$ is.

##  VII Impact of a smaller $Y_{p}$

Recently, the EMPRESS collaboration has included new data in their
determination of the primordial 4He abundance, resulting in $Y_{p}^{\rm
EMPRESS}=0.237\pm 0.003$ Matsumoto  _et al._ (2022). This is around
$1.9\sigma$ below the value in Eq. (37) that we use in our analysis above and
$3.4\sigma$ below the value we predict with our SBBN code. This has motivated
the possibility that BBN was modified from the standard picture, e.g. varying
the Higgs vacuum expectation value Burns  _et al._ (2024), to potentially
alleviate this tension. In this section we study whether EDE during BBN can
explain a smaller $Y_{p}$.

In light of the discussion in the previous section, we note that EDE that
transitions into SM photons with $T_{\rm crit}>T_{\rm bn}$ can reduce $Y_{p}$.
We quantify this in Fig. 8, showing the $1$ and $2\sigma$ limits in the
$\rho_{\Lambda}^{1/4}$-$T_{\rm crit}$ plane which we determine by computing
$\Delta\chi^{2}$ with respect to the best-fit point. The best-fit point at
$\rho_{\Lambda}^{1/4}\simeq 0.540~{}\rm MeV$ and $T_{\rm crit}\simeq
0.864~{}\rm MeV$ is indicated with a red cross. We note that this scenario
leads to values of D/H and $N_{\rm eff}$ that are in worse agreement with the
data than the SBBN predictions but provides an improvement in the overall fit
by reducing $Y_{p}$. Also note that the hard cutoff of the $2\sigma$ contour
comes from the fact that reheating the SM plasma above this temperature just
reproduces SBBN.

Figure 8: The best fit regions (solid for $1\sigma$ and dashed for $2\sigma$)
away from the best fit point in the EDE model to relieve the primordial 4He
abundance tension between EMPRESS and the PDG.

##  VIII Conclusion

The early history of the universe is still largely a mystery. The earliest
direct observation of the universe comes from BBN. This picture is largely
consistent with an expanding radiation-dominated universe. However, the
possibility of a non-negligible component of the universe’s energy density
that redshifts like dark energy is motivated on several grounds: (i) if this
happens around recombination it can alleviate the Hubble tension, (ii) such a
component of the universe exists today in the form of dark energy and likely
did so shortly after the big bang during inflation, and (iii) the effective
field theory picture of vacuum energy indicates that its value changes during
phase transitions generally.

In this work, we have studied the impact of an early dark energy component on
the outcomes of BBN, the earliest direct probe of such a scenario that we
have. Since primordial elemental abundances, as well as $N_{\rm eff}$, are
well measured quantities, we have used observations to put constraints on the
parameters of this EDE model: the amount of EDE initially present before the
transition $\rho_{\Lambda}$, and the temperature at which it transitions
$T_{\rm crit}$. The model consists of an EDE that begins as a constant
component of energy density (similar to the cosmological constant), and then
it decays at $T_{\rm crit}$.

We considered three different scenarios, namely a transition into SM photons,
dark radiation, or kination. Each scenario impacts the thermal evolution of
the universe differently, resulting in different BBN outcomes and consequently
different constraints on $\rho_{\Lambda}$ and $T_{\rm crit}$. We have also
considered the relationships between the EDE input parameters $\rho_{\Lambda}$
and $T_{\rm crit}$ and the BBN input parameters $\tau_{n}$, $N_{\nu}$, and
$\eta_{b}$ in the predicted light element abundances. We find correlations
consistent with the general notion that in the case of a transition into SM
photons, the universe expands slower than in $\Lambda$CDM, whereas in the case
of a transition into dark radiation or kination it expands faster. Lastly, we
studied the possibility that recent hints of a smaller-than-expected
primordial helium abundance can be explained in this setup, finding that a
transition into photons before the deuterium bottleneck is able to improve the
overall BBN fit in this case.

###### Acknowledgements.

We thank John Coffey, Carlos de Lima, David Morrissey, Michael Shamma, and
Tsung-Han Yeh for useful discussions and comments. We also thank Adam Ritz for
valuable feedback on the manuscript and helpful discussions. This work is
supported in part by the Natural Sciences and Engineering Research Council of
Canada (NSERC). TRIUMF receives federal funding via a contribution agreement
with the National Research Council (NRC) of Canada.

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# Graphs for which the second largest distance eigenvalue is less than
$-\frac{1}{2}$

Haiyan Guo1111E-mail<EMAIL_ADDRESS>Bo Zhou2222E-mail:
<EMAIL_ADDRESS>
1School of Computer Science, South China Normal University,
Guangzhou 510631, P. R. China
2School of Mathematical Sciences, South China Normal University,
Guangzhou 510631, P. R. China

###### Abstract

Let $G$ be a connected graph with vertex set $V(G)$. The distance,
$d_{G}(u,v)$, between vertices $u$ and $v$ in $G$ is defined as the length of
a shortest path between $u$ and $v$ in $G$. The distance matrix of $G$ is the
matrix $D(G)=(d_{G}(u,v))_{u,v\in V(G)}$. The second largest distance
eigenvalue of $G$ is the second largest one in the spectrum of $D(G)$. We show
that any connected graph with the second largest distance eigenvalue less than
$-\frac{1}{2}$ is chordal, and characterize those bicyclic graphs and split
graphs with the second largest distance eigenvalue less than $-\frac{1}{2}$.

Keywords: second largest distance eigenvalue, chordal graphs, bicyclic graphs,
split graphs

##  1 Introduction

The adjacency eigenvalues (commonly called the eigenvalues) of a graph are the
eigenvalues of its adjacency matrix. The second largest eigenvalue of graphs
has been widely studied by many mathematicians, see, e.g. [3, Chapter 4], [7,
Subsection 3.11.2] and reference therein. For example, Cao and Hong [4]
characterized the simple graphs with the second largest eigenvalue less than
$\frac{1}{3}$. Wu et al. [22] determined the simple connected graphs with the
second largest eigenvalue less than $\frac{1}{2}$. Cheng et al. [5] considered
graphs with three eigenvalues and second largest eigenvalue at most $1$. Liu
et al. [18] determined all connected $\\{K_{1,3},K_{5}-e\\}$-free graphs whose
second largest eigenvalue does not exceed $1$. Zhang et al. [27] classified
the $2$-partially distance-regular graphs whose each local graph has second
largest eigenvalue at most $1$. Recently, multiplicity of the second largest
eigenvalue of graphs has also received much attention, see [6, 13].

The distance eigenvalues of a connected graph are the eigenvalues of its
distance matrix. The distance eigenvalues of graphs were first studied by
Graham and Pollak [11]. They established a relationship between the number of
negative distance eigenvalues of trees and the addressing problem in data
communication systems. Thereafter and in particular, in recent 15 years, the
distance eigenvalues attracted much more attention. However, the focus was
more on the largest distance eigenvalue (also known as distance spectral
radius), see the survey of Aouchiche and Hansen [2] and [8, 15, 17, 20, 24,
21].

As far as we know, the only studies on the second largest distance eigenvalue
of graphs are as follows. Lin [16] showed that the second largest distance
eigenvalue of a graph $G$ is less than the number of triangles in $G$ when the
independent number is less than or equal to two, confirming a conjecture in
[9]. Xing and Zhou [25] characterized all connected graphs with the second
largest distance eigenvalue less than $-2+\sqrt{2}$ and all trees with the
second largest distance eigenvalue less than $-\frac{1}{2}$. Besides, they
also considered unicyclic graphs with a few exceptions whose second largest
distance eigenvalue less than $-\frac{1}{2}$. In [26], they obtained sharp
lower bounds for the second largest distance eigenvalue of the $k$-th power of
a connected graph and determined all trees and unicyclic graphs $G$ such that
the second largest distance eigenvalue of the squares less than
$\frac{\sqrt{5}-3}{2}$. Liu et al. [19] proved that the graphs with the second
largest distance eigenvalue less than $\frac{17-\sqrt{329}}{2}\approx-0.5692$
are determined by their distance spectra, among other results. Xue et al. [23]
characterized all block graphs whose second largest distance eigenvalue less
than $-\frac{1}{2}$. Alhevaz et al. [1] gave some upper and lower bounds for
the second largest eigenvalue of the generalized distance matrix of graphs in
terms of some graph parameters.

A graph is chordal if every cycle of length at least four has a chord, where a
chord is an edge joining two non-adjacent vertices of the cycle. A connected
graph on $n$ vertices with $n+1$ edges is called a bicyclic graph. A graph $G$
is a split graph if both $G$ and $\overline{G}$ are chordal. In this paper, we
show that any connected graphs whose second largest distance eigenvalue is
less than $-\frac{1}{2}$ must be chordal and characterize all bicyclic graphs
and split graphs with the second largest distance eigenvalue less than
$-\frac{1}{2}$.

##  2 Preliminaries

All graphs considered in this paper are simple and connected. Let $G$ be a
graph with vertex set $V(G)$ and edge set $E(G)$.

The distance between vertices $u$ and $v$ in $G$ is defined as the length of a
shortest path connecting $u$ and $v$ in $G$. Assume that
$V(G)=\\{v_{1},\dots,v_{n}\\}$ with $n\geq 2$. The distance matrix of $G$ is
defined as the $n\times n$ matrix $D(G)=(d_{G}(v_{i},v_{j}))$. The eigenvalues
of $D(G)$ are called the distance eigenvalues of $G$. Since $D(G)$ is
symmetric, the distance eigenvalues of $G$ are all real, so they may be
ordered as $\lambda_{1}(G)\geq\dots\geq\lambda_{n}(G)$. Then $\lambda_{2}(G)$
is the second largest distance eigenvalue of $G$.

For a graph $G$ with $v\in V(G)$, we use $N_{G}(v)$ to denote the neighborhood
of $v$ in $G$, and let $d_{G}(v)=|N_{G}(v)|$ be the degree of $v$ in $G$. For
a nonempty vertex subset $S$, let $G[S]$ be the subgraph of $G$ induced by
$S$. For a graph $G$ with $n$ vertices and $m$ edges, if $m=n+c-1$, then $G$
is called a $c$-cyclic graph. Specially, a $c$-cyclic graph with $c=0,1,2$ is
known as a tree, a unicyclic graph, a bicyclic graph, respectively.

As usual, we denote by $K_{n}$ the complete graph on $n$ vertices, and
$K_{r,s}$ the complete bipartite graph with bipartite sizes $r$ and $s$. Let
$S_{n}=K_{1,n-1}$. Denote by $C_{n}$ the cycle on $n$ vertices. Let
$\overline{G}$ be the complement of a graph $G$.

A path $u_{0}\dots u_{r}$ with $r\geq 1$ in a graph $G$ is called a pendant
path of length $r$ at $u_{0}$ if $\mbox{d}_{G}(u_{0})\geq 3$, the degrees of
$u_{1},\dots,u_{r-1}$ (if any exists) are all equal to $2$ in $G$, and
$\mbox{d}_{G}(u_{r})=1$. In this case, we also say that $G$ is obtained from
$G-\\{u_{1},\dots,u_{r}\\}$ by attaching a pendant path of length $r$ at
$u_{0}$. For $v\in V(G)$, the graph obtained from $G$ by attaching a pendant
path of length $0$ in $G$ is itself. A pendant path of length $1$ at $u_{0}$
is called a pendant edge at $u_{0}$.

A clique of a graph is a set of pairwise adjacent vertices, and a maximum
clique is a clique with maximum cardinality. An independent set is a set of
pairwise non-adjacent vertices.

A graph is a split graph if its vertex set can be partitioned into a clique
and an independent set. A graph $G$ is a split graph if and only if both $G$
and $\overline{G}$ are chordal, or equivalently, it does not have an induced
$C_{4},\overline{C_{4}}$, or $C_{5}$.

A block of a given graph is a maximal connected subgraph that has no cut
vertex. A connected graph $G$ is called a block graph (also known as clique
tree) if all of its blocks are cliques. A block star is a block graph whose
blocks contain a common vertex. A block graph $G$ is loose if for each vertex
$v\in V(G)$, the number of blocks of $G$ which contain the vertex $v$ is at
most $2$. Let $BG(p,q,3,2,2)$ with $p,q\geq 2$ and $BGA$ be two block graphs
as shown in Fig. 1.

Figure 1: Graphs $BG(p,q,3,2,2)$ and BGA.

######  Lemma 2.1.

[23] Let $G$ be a block graph with the second largest distance eigenvalue
$\lambda_{2}(G)$. Then $\lambda_{2}(G)<-\frac{1}{2}$ if and only if

  * •

$G$ is a block star, or

  * •

$G$ is a loose block graph, or

  * •

$G$ is a nontrivial connected induced subgraph of $BG(p,q,3,2,2)$, or

  * •

$G$ is a nontrivial connected induced subgraph of $BGA$.

Let $\mathcal{G}$ denote the set of unicyclic graphs with at least $5$
vertices obtained from $C_{3}=uvw$ by attaching a pendant path at $u,v,w$,
respectively. Denote by $S_{n}^{+}$ the $n$-vertex unicyclic graph obtained by
adding an edge to the star $S_{n}$, where $n\geq 3$.

######  Lemma 2.2.

[25] Let $G$ be a unicyclic graph, where $G\notin\mathcal{G}$. Then
$\lambda_{2}(G)<-\frac{1}{2}$ if and only if $G\cong S_{n}^{+}$ for $n\geq 3$,
or one of the six graphs $U_{1},\dots,U_{6}$ shown in Fig. 2.

Note that any $G$ in $\mathcal{G}$ is a loose block graph, so we have
$\lambda_{2}(G)<-\frac{1}{2}$ by Lemma 2.1. Combining Lemmas 2.1 and 2.2, we
have the following.

######  Corollary 2.1.

Let $G$ be a unicyclic graph. Then $\lambda_{2}(G)<-\frac{1}{2}$ if and only
if $G\cong S_{n}^{+}$ for $n\geq 3$, $G\in\mathcal{G}$, or one of the six
graphs $U_{1},\dots,U_{6}$ shown in Fig. 2.

Figure 2: Graphs $U_{1},\dots,U_{6}$.

For an $n\times n$ real symmetric matrix $M$, let
$\rho_{1}(M)\geq\ldots\geq\rho_{n}(M)$ be the eigenvalues of $M$. The
following lemma is the classical Cauchy’s interlacing theorem, see [14,
Theorem 4.3.28] or [12].

######  Lemma 2.3 (Cauchy’s interlacing theorem).

Let $A$ be an $n\times n$ symmetric matrix. If $B$ is an $m\times m$ principal
submatrix of $A$, then $\rho_{i}(A)\geq\rho_{i}(B)\geq\rho_{n-m+i}(A)$ for
$1\leq i\leq m$.

If $H$ is a connected induced subgraph of $G$ and $d_{H}(u,v)=d_{G}(u,v)$ for
all $\\{u,v\\}\subseteq V(H)$, then we say $H$ is a distance-preserving
subgraph of $G$. In this case, $D(H)$ is a principal submatrix of $D(G)$, so
$\lambda_{2}(G)\geq\lambda_{2}(H)$ by Lemma 2.3. Specially, if $G$ is a
connected graph with $\lambda_{2}(G)<-\frac{1}{2}$, then
$\lambda_{2}(H)<-\frac{1}{2}$ for any distance-preserving subgraph $H$ of $G$.
In this paper, we are concerned with the graphs $G$ with
$\lambda_{2}(G)<-\frac{1}{2}$. So, we call a graph $H$ a forbidden subgraph of
$G$ if $H$ is a distance-preserving subgraph of $G$ but
$\lambda_{2}(H)\geq-\frac{1}{2}$. We show some forbidden subgraphs
$F_{1},\dots,F_{13}$ in Fig. 3 that we need in the proofs, where the second
largest distance eigenvalue is also listed below the corresponding graph.

Figure 3: Forbidden subgraphs $F_{1},\dots,F_{13}$.

The following lemma may be checked easily.

######  Lemma 2.4.

[10] For the cycle $C_{n}$ on $n\geq 3$ vertices,

$\lambda_{2}(C_{n})=\begin{cases}0&\text{if $n$ is even,}\\\
-\frac{1}{4}\sec^{2}\frac{\pi}{n}&\text{if $n$ is odd.}\end{cases}$

Let $J_{s\times t}$ be the $s\times t$ matrix of all $1$’s, and $I_{s}$ the
identity matrix of order $s$. For convenience, let $J_{s}=J_{s\times s}$.

##  3 Any graph $G$ with $\lambda_{2}(G)<-\frac{1}{2}$ must be chordal

In this section, we show that any graph whose second largest distance
eigenvalue is less than $-\frac{1}{2}$ must be chordal.

######  Theorem 3.1.

Suppose that $G$ is a graph with $\lambda_{2}(G)<-\frac{1}{2}$, then $G$ is
chordal.

###### Proof.

We prove the theorem by contradiction. Suppose that it is not true. Then there
exists a graph $G$ with $\lambda_{2}(G)<-\frac{1}{2}$ but $G$ is not chordal.
So there is an induced cycle in $G$ with length at least four. We choose such
an induced cycle $C_{k}$ of $G$ so that its length $k$ is minimum. By Lemmas
2.3 and 2.4, $C_{k}$ is not a distance-preserving subgraph of $G$, and in
particular, $k\geq 6$. So there are two non-adjacent vertices, say $u$ and
$v$, on $C_{k}$ such that $d_{C_{k}}(u,v)>d_{G}(u,v)$. Let

$\mathcal{C}=\\{\\{u,v\\}:d_{C_{k}}(u,v)>\max\\{d_{G}(u,v),1\\},u,v\in
V(C_{k})\\}.$

Assume that $\\{x,y\\}\in\mathcal{C}$ so that $d_{G}(x,y)$ is minimum. Let
$s=d_{G}(x,y)$ and $t=d_{C_{k}}(x,y)$. Since $C_{k}$ is an induced cycle of
$G$, we have $2\leq s<t\leq\frac{k}{2}$.

Let $P=x_{0}x_{1}\dots x_{s}$ be a shortest path between $x$ and $y$ in $G$,
where $x=x_{0}$ and $y=x_{s}$. Let $Q=y_{0}y_{1}\dots y_{t}$ be the path on
$C_{k}$ from $x$ to $y$ with length $t$, where $x=y_{0}$ and $y=y_{t}$. Let
$G^{\prime}=G[V(P)\cup V(Q)]$. Note that $G^{\prime}$ contains a Hamiltonian
cycle $C=x_{0}x_{1}\dots x_{s}y_{t-1}\dots y_{1}x_{0}$.

Claim 1. $G^{\prime}$ is chordal.

Note that $|V(G^{\prime})|=s+t<2t\leq k$. Thus, if $G^{\prime}$ contains an
induced cycle, then its length is less than $k$, so its length must be three
by the choice of $C_{k}$. Thus $G^{\prime}$ is chordal. Claim 1 follows.

By the choice of $C_{k}$ and $P$, any chord of $C$ joins a vertex from
$V(Q)\setminus\\{x,y\\}=\\{y_{1},\dots,y_{t-1}\\}$ and a vertex from
$V(P)\setminus\\{x,y\\}=\\{x_{1},\dots,x_{s-1}\\}$.

Claim 2. $x_{1}y_{1}\in E(G^{\prime})$.

If $x_{1}y_{1}\notin E(G^{\prime})$, then there is an induced cycle of length
at least four (and less than $k$) containing $x$ in $G^{\prime}$,
contradicting Claim 1. This proves Claim 2.

By Claim 2, $G^{\prime}-x_{0}$ contains a Hamiltonian cycle
$C^{\prime}=x_{1}\dots x_{s}y_{t-1}\dots y_{1}x_{1}$. By Claims 1 and 2,
$G^{\prime}-x_{0}$ is chordal.

Case 1. $C^{\prime}$ has a chord incident with $x_{1}$.

Since $G^{\prime}-x_{0}$ is chordal, it has no induced cycle of length at
least four, so $x_{1}y_{2}\in E(G^{\prime})$. If $x_{1}y_{3}\in
E(G^{\prime})$, then $G^{\prime}[\\{x_{0},x_{1},y_{1},y_{2},y_{3}\\}]\cong
F_{11}$, which is a forbidden subgraph, a contradiction. This shows that
$t\geq 4$ and $x_{1}y_{3}\notin E(G^{\prime})$. Then, step by step, we have
$x_{1}y_{i}\notin E(G^{\prime})$ for $i=4,\dots,t-1$, as, otherwise, there is
an induced cycle of length at least four in $G^{\prime}$, contradicting Claim
1. If $s=2$, then $y_{2}y_{3}\dots y_{t}x_{1}y_{2}$ is an induced cycle of
length at least four in $G^{\prime}$, contradicting Claim 1. So $s\geq 3$.

Case 1.1 $x_{2}y_{1}\in E(G^{\prime})$.

If $x_{2}y_{2}\notin E(G^{\prime})$, then
$G^{\prime}[\\{x_{0},x_{1},x_{2},y_{1},y_{2}\\}]\cong F_{12}$, which is a
forbidden subgraph, a contradiction. So $x_{2}y_{2}\in E(G^{\prime})$.

If $x_{2}y_{3}\in E(G^{\prime})$, then
$G^{\prime}[\\{x_{0},x_{1},x_{2},y_{1},y_{2},y_{3}\\}]\cong F_{13}$, which is
a forbidden subgraph, also a contradiction. So $x_{2}y_{3}\notin
E(G^{\prime})$. Step by step, we have $x_{2}y_{i}\notin E(G^{\prime})$ for
$i=4,\dots,t-1$ if $t\geq 5$. So $x_{2}y_{i}\notin E(G^{\prime})$ for $3\leq
i\leq t-1$.

Next, we claim that $x_{3}y_{i}\notin E(G^{\prime})$ for $1\leq i\leq t-1$ if
$s\geq 4$. Otherwise, the subgraph of
$G^{\prime}[\\{x_{0},x_{1},x_{2},x_{3},y_{1}\\}]\cong F_{11}$ if
$x_{3}y_{1}\in E(G^{\prime})$, the subgraph of
$G^{\prime}[\\{x_{0},x_{1},x_{2},x_{3},y_{1},y_{2}\\}]\cong F_{13}$ if
$x_{3}y_{2}\in E(G^{\prime})$, and there is an induced cycle of length at
least four in $G^{\prime}$ if $x_{3}y_{i}\in E(G^{\prime})$ for $3\leq i\leq
t-1$, so we have a contradiction in any case. Step by step, we have
$x_{i}y_{j}\notin E(G^{\prime})$ for $4\leq i\leq s-1$ and $1\leq j\leq t-1$.
So $G^{\prime}[V(G^{\prime})\setminus\\{x_{0},x_{1},y_{1}\\}]$ is an induced
cycle of $G^{\prime}$ with length at least four, a contradiction.

Case 1.2 $x_{2}y_{1}\notin E(G^{\prime})$.

In this case, $x_{2}y_{2}\notin E(G^{\prime})$. Otherwise,
$G^{\prime}[\\{x_{0},x_{1},x_{2},y_{1},y_{2}\\}]\cong F_{11}$, which is a
forbidden subgraph, a contradiction. Step by step, we have $x_{2}y_{i}\notin
E(G^{\prime})$ for $3\leq i\leq t-1$, and $x_{i}y_{j}\notin E(G^{\prime})$ for
$3\leq i\leq s-1$ and $1\leq j\leq t-1$ if $s\geq 4$. So
$G^{\prime}[V(G^{\prime})\setminus\\{x_{0},y_{1}\\}]$ is an induced cycle of
$G^{\prime}$ with length at least four, a contradiction.

Case 2. $C^{\prime}$ has no chord incident with $x_{1}$.

Suppose that $t=3$. Then $s=2$. In the case of $t=3$ or $s=2$,
$G^{\prime}[V(G^{\prime})\setminus\\{x_{0}\\}]$ is an induced cycle with
length at least four, a contradiction. So $s\geq 3$ and $t\geq 4$.

Since $G^{\prime}-x_{0}$ is chordal, $x_{2}y_{1}\in E(G^{\prime})$.

We claim that $x_{2}y_{i}\notin E(G^{\prime})$ for $2\leq i\leq t-1$.
Otherwise, $G^{\prime}[\\{x_{0},x_{1},x_{2},y_{1},y_{2}\\}]\cong F_{11}$ if
$x_{2}y_{2}\in E(G^{\prime})$, and there is an induced cycle of length at
least four if $x_{2}y_{i}\in E(G^{\prime})$ for $3\leq i\leq t-1$, so we have
a contradiction in either case. Step by step, we have $x_{i}y_{j}\notin
E(G^{\prime})$ for $3\leq i\leq s-1$ and $1\leq j\leq t-1$. So
$G^{\prime}[V(G^{\prime})\setminus\\{x_{0},x_{1}\\}]$ is an induced cycle of
length at least four, a contradiction.

Combing case 1 and 2, we complete the proof. ∎

##  4 Bicyclic graphs $G$ with $\lambda_{2}(G)<-\frac{1}{2}$

For integers $p$, $q$ and $s$ with $p\geq 3$, $q\geq 3$ and $s\geq 0$, let
$\infty(p,q,s)$ be the bicyclic graph obtained from the cycles
$C_{p}=u_{1}u_{2}\dots u_{p}u_{1}$, $C_{q}=v_{1}v_{2}\dots v_{q}v_{1}$ and the
path $P_{s+1}=p_{0}p_{1}\dots p_{s}$ by identifying $u_{1}$ with $p_{0}$ and
identifying $v_{1}$ with $p_{s}$. In particular, $\infty(p,q,0)$ consists of
two cycles of lengths $p$ and $q$ respectively with precisely one vertex in
common.

For positive integers $p$, $q$ and $s$, where at most one of $p$, $q$, $s$ is
equal to one, let $\theta(p,q,s)$ be the bicyclic graph obtained from three
paths $P_{p+1}=u_{0}u_{1}\dots u_{p-1}u_{p}$, $P_{q+1}=v_{0}v_{1}\dots
v_{q-1}v_{q}$ and $P_{s+1}=w_{0}w_{1}\dots w_{s-1}w_{s}$ by identifying
$u_{0}$, $v_{0}$ and $w_{0}$ to a new vertex $x$ and identifying $u_{p}$,
$v_{q}$ and $w_{s}$ to a new vertex $y$.

Graphs $\infty(p,q,s)$ and $\theta(p,q,s)$ are depicted in Fig. 4.

Figure 4: Graphs $\infty(p,q,s)$ and $\theta(p,q,s)$.

For a bicyclic graph $G$, if $\infty(p,q,s)$ is an induced subgraph of $G$ for
some $p,q,s$, then we say $G$ is a $\infty$-bicyclic graph. Otherwise, $G$
contains $\theta(p^{\prime},q^{\prime},s^{\prime})$ as an induced subgraph for
some $p^{\prime},q^{\prime},s^{\prime}$, so we say $G$ is a $\theta$-bicyclic
graph.

To state the results, we define several families of bicyclic graphs.

Let $u_{1}$, $u_{2}$, $u_{3}$, $u_{4}$ be the four vertices with degree two in
the cycles of $\infty(3,3,s)$ with $u_{1}u_{2},u_{3}u_{4}\in
E(\infty(3,3,s))$, where $s\geq 0$. We use $B(s;h_{1},h_{2},h_{3},$ $h_{4})$
to denote the graph obtained from $\infty(3,3,s)$ by attaching a pendant path
of length $h_{i}$ at $u_{i}$, respectively, where $h_{i}\geq 0$ for $1\leq
i\leq 4$, see Fig. 5.

Figure 5: Graph $B(s;h_{1},h_{2},h_{3},h_{4})$.

Let $B^{\infty}_{t}$ be the graph obtained from $\infty(3,3,0)$ by attaching
$t$ pendant edges at the vertex with maximum degree, where $t\geq 0$, as
depicted in Fig. 6 left. We use $B^{\theta}_{k}$ to denote the graph obtained
from $\theta(2,2,1)$ by attaching $k$ pendant edges at a vertex of degree
three in $\theta(2,2,1)$, where $k\geq 0$, as depicted in Fig. 6 right.

Figure 6: Graphs $B^{\infty}_{t}$ and $B^{\theta}_{k}$.

######  Theorem 4.1.

Let $G$ be a bicyclic graph. Then $\lambda_{2}(G)<-\frac{1}{2}$ if and only if

  * •

$G\cong B_{1},\dots,B_{7}$ as displayed in Fig. 7, or

  * •

$G\cong B(s;h_{1},h_{2},h_{3},h_{4})$, where $s\geq 0$ and $h_{i}\geq 0$ for
$1\leq i\leq 4$, or

  * •

$G\cong B^{\infty}_{t}$ with $t\geq 0$, or

  * •

$G\cong B^{\theta}_{k}$, where $k\geq 0$.

Figure 7:  Graphs $B_{1},\dots,B_{7}$.

Note that a bicyclic graph is either a $\infty$-bicyclic graph or a
$\theta$-bicyclic graph. To prove Theorem 4.1, it suffices to show the
following two lemmas.

######  Lemma 4.1.

Let $G$ be a $\infty$-bicyclic graph. Then $\lambda_{2}(G)<-\frac{1}{2}$ if
and only if $G\cong B_{1},\dots,B_{5}$, $G\cong B(s;h_{1},h_{2},h_{3},h_{4})$,
where $s\geq 0$ and $h_{i}\geq 0$ for $1\leq i\leq 4$, or $G\cong
B^{\infty}_{t}$ with $t\geq 0$.

######  Lemma 4.2.

Let $G$ be a $\theta$-bicyclic graph. Then $\lambda_{2}(G)<-\frac{1}{2}$ if
and only if $G\cong B_{6},B_{7}$, or $G\cong B^{\theta}_{k}$, where $k\geq 0$.

Firstly, we give the proof of Lemma 4.1.

###### Proof of Lemma 4.1.

Let $G$ be a $\infty$-bicyclic graph with $\lambda_{2}(G)<-\frac{1}{2}$. By
Theorem 3.1, $G$ is chordal. So the two cycles of $G$ are both of length
three. Suppose without loss of generality that $C_{p}=u_{1}u_{2}u_{3}u_{1}$
and $C_{q}=v_{1}v_{2}v_{3}v_{1}$ are the two cycles of $G$, and
$P_{s+1}=p_{0}p_{1}\dots p_{s}$ is the shortest path connecting a vertex of
$C_{p}$ and a vertex of $C_{q}$, say $u_{1}=p_{0}$ and $p_{s}=v_{1}$.

Case 1. $s\geq 2$.

In this case, we have $d_{G}(p_{i})=2$ for $1\leq i\leq s-1$. Otherwise
$F_{3}$ is an induced distance-preserving subgraph of $G$, contradicting that
$F_{3}$ is a forbidden subgraph of $G$. Suppose without loss of generality
that $d_{G}(u_{1})\geq d_{G}(v_{1})$. Since $F_{1}$ is a forbidden subgraph,
we have $d_{G}(u_{1})\leq 4$. That is, $d_{G}(u_{1})=3,4$.

Suppose first that $d_{G}(u_{1})=4$. Then $s\leq 2$ since $F_{4}$ is a
forbidden subgraph. So $s=2$. Let $w$ be the unique vertex in
$N_{G}(u_{1})\setminus\\{u_{2},u_{3},p_{1}\\}$. We have $d_{G}(w)=1$ as
$F_{3}$ is a forbidden subgraph. Similar argument leads to
$d_{G}(u_{2})=d_{G}(u_{3})=2$. Since $F_{4}$ is a forbidden subgraph, we have
$d_{G}(v_{2})=d_{G}(v_{3})=2$. Thus, if $d_{G}(v_{1})=4$, then $G\cong B_{2}$,
and if $d_{G}(v_{1})=3$, then $G\cong B_{3}$.

Suppose next that $d_{G}(u_{1})=3$. As $d_{G}(u_{1})\geq d_{G}(v_{1})\geq 3$,
one gets $d_{G}(v_{1})=3$. Since $F_{4}$ is a forbidden subgraph, we have
$\max\\{d_{G}(u_{2}),d_{G}(u_{3})\\}\leq 3$, and if $d_{G}(u_{i})=3$ for
$i=2,3$, then there is a pendant path at $u_{i}$. Similarly,
$\max\\{d_{G}(v_{2}),d_{G}(v_{3})\\}\leq 3$, and if $d_{G}(v_{i})=3$ for
$i=2,3$, then there is a pendant path at $v_{i}$. Hence, $G\cong
B(s;h_{1},h_{2},h_{3},h_{4})$, where $s\geq 2$ and $h_{i}\geq 0$ for $1\leq
i\leq 4$.

Case 2. $s=1$.

Since $F_{5}$ is a forbidden subgraph, we have $d_{G}(u_{1})=d_{G}(v_{1})=3$.
Assume that $\max\\{d_{G}(u_{2}),$
$d_{G}(u_{3}),d_{G}(v_{2}),d_{G}(v_{3})\\}=d_{G}(u_{2})$. Since $F_{1}$ is a
forbidden subgraph, we have $d_{G}(u_{2})\leq 4$.

Suppose that $d_{G}(u_{2})=4$. Let
$N_{G}(u_{2})\setminus\\{u_{1},u_{3}\\}=\\{x_{1},x_{2}\\}$. We have
$d_{G}(x_{1})=d_{G}(x_{2})=1$ as $F_{3}$ is a forbidden subgraph. Since
$F_{6}$ is a forbidden subgraph, we have $d_{G}(u_{3})=2$. Since $F_{4}$ is a
forbidden subgraph, we have $d_{G}(v_{2})=d_{G}(v_{3})=2$. Hence, $G\cong
B_{1}$.

If $d_{G}(u_{2})=3$, then for any $w\in
V(G)\setminus\\{u_{1},u_{2},u_{3},v_{1},v_{2},v_{3}\\}$, one has
$d_{G}(w)=1,2$ since $F_{4}$ is a forbidden subgraph. Hence, $G\cong
B(1;h_{1},h_{2},h_{3},h_{4})$, where $h_{1}\geq 1$ and $h_{i}\geq 0$ for
$2\leq i\leq 4$.

If $d_{G}(u_{2})=2$, then $G\cong\infty(3,3,1)\cong B(1;0,0,0,0)$.

Case 3. $s=0$.

If $d_{G}(u_{1})=4$, then
$\max\\{d_{G}(u_{2}),d_{G}(u_{3}),d_{G}(v_{2}),d_{G}(v_{3})\\}\leq 3$ as
$F_{5}$ is a forbidden subgraph, and every vertex not on the cycles is of
degree one or two due to the fact that $F_{4}$ and $F_{5}$ are both forbidden
subgraphs. This implies that $G\cong B(0;h_{1},h_{2},h_{3},h_{4})$, where
$h_{i}\geq 0$ for $1\leq i\leq 4$.

Suppose that $d_{G}(u_{1})\geq 5$. Let
$N_{G}(u_{1})\setminus\\{u_{2},u_{3},v_{2},v_{3}\\}=\\{x_{1},\ldots,x_{t}\\}$,
where $t\geq 1$. Since $F_{7}$ is a forbidden subgraph, one has
$d_{G}(x_{1})=\dots=d_{G}(x_{t})=1$. Moreover,
$\max\\{d_{G}(u_{2}),d_{G}(u_{3}),d_{G}(v_{2}),d_{G}(v_{3})\\}\leq 3$ as
$F_{5}$ is a forbidden subgraph.

If $d_{G}(u_{1})\geq 6$, then, since $F_{1}$ is a forbidden subgraph, we have
$d_{G}(u_{2})=d_{G}(u_{3})=d_{G}(v_{2})=d_{G}(v_{3})=2$, which implies that
$G\cong B^{\infty}_{t}$, where $t\geq 2$.

Suppose that $d_{G}(u_{1})=5$. Assume that
$\max\\{d_{G}(u_{2}),d_{G}(u_{3}),d_{G}(v_{2}),d_{G}(v_{3})\\}$
$=d_{G}(u_{2})$. Since $F_{5}$ is a forbidden subgraph, we have
$d_{G}(u_{2})\leq 3$. Then, if $d_{G}(u_{2})=2$, then $G\cong B^{\infty}_{1}$.
Suppose that $d_{G}(u_{2})=3$. Since $F_{3}$ is a forbidden subgraph, we have
$d_{G}(v_{2})=d_{G}(v_{3})=2$. As $F_{6}$ is a forbidden subgraph, we have
$d_{G}(u_{3})=2$. Let $w_{1}$ denote the neighbor of $u_{2}$ not on the
cycles. Since $F_{5}$ is a forbidden subgraph, we have $d_{G}(w_{1})\leq 2$.
Then $G\cong B_{5}$ if $d_{G}(w_{1})=1$. If $d_{G}(w_{1})=2$, then denoting by
$w_{2}$ the neighbor of $w_{1}$ different from $u_{2}$, one has
$d_{G}(w_{2})=1$ as $F_{4}$ is a forbidden subgraph, which implies that
$G\cong B_{4}$. Hence, $G\cong B_{4},B_{5}$, or $B^{\infty}_{1}$.

Combining Cases 1–3, we have $G\cong B_{1},\dots,B_{5}$, or $G\cong
B(s;h_{1},h_{2},h_{3},h_{4})$, where $s\geq 0$ and $h_{i}\geq 0$ for $1\leq
i\leq 4$, or $G\cong B^{\infty}_{t}$ with $t\geq 0$.

Conversely, suppose that $G\cong B_{1},\dots,B_{5}$, or $G\cong
B(s;h_{1},h_{2},h_{3},h_{4})$, where $s\geq 0$ and $h_{i}\geq 0$ for $1\leq
i\leq 4$, or $G\cong B^{\infty}_{t}$ with $t\geq 0$. By a direct calculation,
we have $\lambda_{2}(B_{1})\approx-0.5110<-\frac{1}{2}$,
$\lambda_{2}(B_{4})\approx-0.5023<-\frac{1}{2}$. Since $B_{5}$ is an induced
distance-preserving subgraph of $B_{4}$. By Lemma 2.3, we have
$\lambda_{2}(B_{5})\leq\lambda_{2}(B_{4})<-\frac{1}{2}$. Note that $B_{2}\cong
BGA$ and $B_{3}$ is an induced distance-preserving subgraph of $BGA$. By Lemma
2.1, we have $\lambda_{2}(B_{3})\leq\lambda_{2}(B_{2})<-\frac{1}{2}$. If
$G\cong B(s;h_{1},h_{2},h_{3},h_{4})$, where $s\geq 0$ and $h_{i}\geq 0$ for
$1\leq i\leq 4$, then $G$ is a loose block graph; If $G\cong B^{\infty}_{t}$
with $t\geq 0$, then $G$ is a block star. In either case, we have by Lemma 2.1
that $\lambda_{2}(G)<-\frac{1}{2}$. ∎

Next, we move to give proof of Lemma 4.2.

###### Proof of Lemma 4.2.

By Theorem 3.1, $G$ is chordal, so $\theta(2,2,1)$ is an induced subgraph of
$G$. Let $u_{0}$ and $u_{2}$ be the two vertices of $\theta(2,2,1)$ with
degree three, and $u_{1}$ and $v_{1}$ the two vertices of $\theta(2,2,1)$ with
degree two. Since $F_{9}$ is a forbidden subgraph, one gets
$d_{G}(u_{1})=d_{G}(v_{1})=2$. Since $F_{10}$ is a forbidden subgraph, there
can only be some pendant edges at $x$ or $y$. Assume that $d_{G}(x)\geq
d_{G}(y)$. Then $d_{G}(y)\leq 4$, as otherwise, there would be a forbidden
subgraph $F_{2}$.

If $d_{G}(y)=4$, then, since $F_{1}$ is a forbidden subgraph, we have
$d_{G}(x)\leq 5$, so, $G\cong B_{6},B_{7}$. If $d_{G}(y)=3$, then $G\cong
B^{\theta}_{k}$, where $k\geq 0$.

Conversely, suppose that $G\cong B_{6},B_{7}$, or $G\cong B_{k}^{\theta}$ with
$k\geq 0$. By a direct calculation, we have
$\lambda_{2}(B_{6})\approx-0.5578<-\frac{1}{2}$ and
$\lambda_{2}(B_{7})\approx-0.5119<-\frac{1}{2}$. Assume that
$G=B_{k}^{\theta}$ with $k\geq 0$. Then $|V(G)|=k+4$. If $k=0$, then
$G\cong\theta(2,2,1)$. By a direct calculation, we have
$\lambda_{2}(\theta(2,2,1))\approx-0.5616<-\frac{1}{2}$. Assume that $k\geq
1$. Then $n\geq 5$. Let $V_{1}$ be the set of two vertices of degree $2$ and
$V_{2}$ be the set of vertices of degree one. So we partition $V(G)$ as
$V(G)=\\{w\\}\cup\\{z\\}\cup V_{1}\cup V_{2}$, where $w$ is the vertex with
maximum degree and $z$ is the vertex with degree $3$. Under this partition, we
have

$D(G)+2I_{n}=\left(\begin{array}[]{cccc}2&1&J_{1\times 2}&J_{1\times(n-4)}\\\
1&2&J_{1\times 2}&2J_{1\times(n-4)}\\\ J_{2\times 1}&J_{2\times
1}&2J_{2}&2J_{2\times(n-4)}\\\ J_{(n-4)\times 1}&2J_{(n-4)\times
1}&2J_{(n-4)\times 2}&2J_{n-4}\\\ \end{array}\right).$

It is easily seen that $D(G)+2I_{n}$ is of rank $4$, which implies that $0$ is
an eigenvalue of $D(G)+2I_{n}$ with multiplicity $n-4$. Note that the above
partition for $D(G)+2I_{n}$ is equitable, thus the eigenvalues of its quotient
matrix $Q$ are also the eigenvalues of $D(G)+2I_{n}$, see [3, Lemma 2.3.1],
where

$Q=\left(\begin{array}[]{cccc}2&1&2&n-4\\\ 1&2&2&2(n-4)\\\ 1&1&4&2(n-4)\\\
1&2&4&2(n-4)\\\ \end{array}\right).$

Let $f(\lambda)=\det(\lambda I_{4}-Q)$. By a direct calculation,

$f(\lambda)=\lambda^{4}-2n\lambda^{3}+3(n+1)\lambda^{2}+4(n-6)\lambda-6n+24.$

Note that $f(+\infty)>0$, $f(7)=2404-517n<0$,
$f(\frac{3}{2})=-\frac{3}{16}<0$, $f(0)=24-6n<0$, $f(-\infty)>0$,
$f(\frac{3n-1}{2n})=\frac{1}{16n^{4}}(n^{4}-12n^{3}+70n^{2}-12n+1)$. Let
$g(x)=x^{4}-12x^{3}+70x^{2}-12x+1$. Then
$g^{\prime}(x)=4(x^{3}-9x^{2}+35x-3)$, $g^{\prime\prime}(x)=4(3x^{2}-18x+35)$.
Since $g^{\prime\prime}(x)>0$ for all $x$, $g^{\prime}(x)\geq
g^{\prime}(5)=288>0$ for $x\geq 5$, which implies that $g(x)\geq g(5)=816>0$.
Thus $f(\frac{3n-1}{2n})>0$. It follows that the second largest root
$\lambda^{(2)}$ of $f(\lambda)=0$ satisfies
$\frac{3n-1}{2n}<\lambda^{(2)}<\frac{3}{2}$. By the above argument,
$\lambda^{(2)}$ is the second largest eigenvalues of $D(G)+2I_{n}$, i.e.,
$\lambda_{2}(G)<-\frac{1}{2}$. ∎

From the proof of Lemma 4.2, we have
$-\frac{n+1}{2n}<\lambda_{2}(B_{k}^{\theta})<-\frac{1}{2}$. Thus, the limit of
$\lambda_{2}(B_{k}^{\theta})$ as $n$ approaches $+\infty$ is $-\frac{1}{2}$.

##  5 Split graphs $G$ with $\lambda_{2}(G)<-\frac{1}{2}$

In the following, we view a clique $K$ of cardinality $s$ of a graph $G$ as
$K_{s}$, the subgraph of $G$ induced by $K$.

For nonnegative integer $t$, let $SP^{t}$ be the split graph consisting of a
clique $K_{4}$ and an independent set $I=\\{x_{1},\dots,x_{t},w\\}$ so that
$x_{1},\dots,x_{t}$ have a unique neighbor $u\in V(K_{4})$ and $w$ has exactly
two neighbors $u,v\in V(K_{4})$, see Fig. 8. In particular, $SP^{0}$ is the
split graph with a clique $K_{4}$ and an independent set $I=\\{w\\}$ so that
$w$ has exactly two neighbors in $V(K_{4})$.

Figure 8: Graph $SP^{t}$.

Let $s$ be an integer with $s\geq 2$. Let $K_{s}(x_{1},\dots,x_{r})$ be the
split graph obtained from $K_{s}$ with vertex set $\\{v_{1},\dots,v_{s}\\}$ by
attaching $x_{i}$ pendant edges at $v_{i}$ for $i=1,\dots,r$, where $1\leq
r\leq s$. In particular, $K_{2}(1,1)\cong P_{4}$ and $K_{2}(t)\cong S_{t+2}$.

######  Theorem 5.1.

Let $G$ be a split graph. Then $\lambda_{2}(G)<-\frac{1}{2}$ if and only if

  * •

$G\cong SP_{1},SP_{2}$ as displayed in Fig. 9, or

  * •

$G\cong B_{6},B_{7}$, or

  * •

$G\cong SP^{t}$, where $t\geq 0$, or

  * •

$G\cong B^{\theta}_{k}$, where $k\geq 0$, or

  * •

$G\cong K_{s}(t)$, where $s\geq 2$ and $t\geq 0$, or

  * •

$G\cong K_{s}(a,1)$, where $s\geq 2$ and $a=1,2$, or

  * •

$G\cong K_{s}(\underbrace{1,\dots,1}_{t})$, where $s\geq 3$ and $t\geq 3$.

Figure 9: Graphs $SP_{1},SP_{2}$.

To prove Theorem 5.1, we need two lemmas.

######  Lemma 5.1.

Let $G=SP^{t}$ with $t\geq 0$. Then $\lambda_{2}(G)<-\frac{1}{2}$.

###### Proof.

Note that $SP^{t}$ is a distance-preserving induced subgraph of $SP^{t+1}$.
From Lemma 2.3, we have $\lambda_{2}(SP^{t})\leq\lambda_{2}(SP^{t+1})$, which
implies that $\\{\lambda_{2}(SP^{t}):t=0,1,\dots\\}$ is a non-decreasing
sequence. So it suffices to show that $\lambda_{2}(G)<-\frac{1}{2}$ for large
enough $t$.

Let $G=SP^{t}$. Then $|V(G)|=t+5$. We partition $V(G)$ as $V(G)=V_{1}\cup
V_{2}\cup\\{u\\}\cup\\{v\\}\cup\\{w\\}$, where
$V_{1}=K_{4}\setminus\\{u,v\\}$, $V_{2}=\\{x_{1},\dots,x_{t}\\}$. Under this
partition, we have

$D(G)=\left(\begin{array}[]{ccccc}J_{2}-I_{2}&2J_{2\times t}&J_{2\times
1}&J_{2\times 1}&2J_{2\times 1}\\\ 2J_{t\times 2}&2(J_{t}-I_{t})&J_{t\times
1}&2J_{t\times 1}&2J_{t\times 1}\\\ J_{1\times 2}&J_{1\times t}&0&1&1\\\
J_{1\times 2}&2J_{1\times t}&1&0&1\\\ 2J_{1\times 2}&2J_{1\times
t}&1&1&0\end{array}\right).$

The first two rows of $-I_{t+5}-D(G)$ are equal, implying that $-1$ is a
distance eigenvalue of $G$ with multiplicity at least $1$, and in
$-2I_{t+5}-D(G)$, there are $t$ equal rows, implying that $-2$ is a distance
eigenvalue of $G$ with multiplicity at least $t-1$.

Let $Q$ be the quotient matrix of $D(G)$ with respect to the above partition
on $V(G)$. Then

$Q=\left(\begin{array}[]{ccccc}1&2t&1&1&2\\\ 4&2(t-1)&1&2&2\\\ 2&t&0&1&1\\\
2&2t&1&0&1\\\ 4&2t&1&1&0\end{array}\right).$

Note that the above partition is equitable. Thus the spectrum of $Q$ is
contained in the distance spectrum of $G$, see [3, Lemma 2.3.1]. Let
$f(\lambda)=\det(\lambda I_{5}-Q)$. By a direct calculation,

$f(\lambda)=\lambda^{5}-(2t-1)\lambda^{4}-(15t+17)\lambda^{3}-(33t+49)\lambda^{2}-(23t+44)\lambda-5t-12.$

Note that $f(+\infty)>0$, $f(-0.5)\approx-0.094<0$,
$f(-\frac{t+1}{2t})=\frac{1}{32t^{5}}(t^{5}+19t^{4}-142t^{3}+62t^{2}-3t+1)>0$
for large enough $t$, $f(-1)=-2t<0$, $f(-3)=10t-24>0$ for $t\geq 4$,
$f(-\infty)<0$. It follows that the second largest root $\lambda^{(2)}$ of
$f(\lambda)=0$ satisfies $-\frac{t+1}{2t}<\lambda^{(2)}<-\frac{1}{2}$ for
large enough $t$. By the above argument, $\lambda^{(2)}$ is the second largest
distance eigenvalue of $G$, i.e., $\lambda_{2}(G)<-\frac{1}{2}$. ∎

######  Lemma 5.2.

Let $G=K_{s}(2,1)$, where $s\geq 2$. Then $\lambda_{2}(G)<-\frac{1}{2}$.

###### Proof.

Let $G=K_{s}(2,1)$. Then $|V(G)|=s+3$. Since $K_{s}(2,1)$ is a distance-
preserving induced subgraph of $K_{s+1}(2,1)$, it follows from Lemma 2.3 that
$\lambda_{2}(K_{s}(2,1))\leq\lambda_{2}(K_{s+1}(2,1))$. Then the sequence
$\\{\lambda_{2}(K_{s}(2,1)):s=2,3,\dots\\}$ does not decrease with $s$. So it
suffices to show that $\lambda_{2}(G)<-\frac{1}{2}$ for large enough $s$.

Let $I=\\{w_{1},w_{2},w_{3}\\}$ be the independent set of $K_{s}(2,1)$. We use
$u$ to denote the only common neighbour of $w_{1}$ and $w_{2}$, $v$ denotes
the neighbor of $w_{3}$ in $K_{s}(2,1)$. Then we may partition $V(G)$ as
$V(G)=V_{1}\cup V_{2}\cup\\{u\\}\cup\\{v\\}\cup\\{w_{3}\\}$, where
$V_{1}=K_{s}\setminus\\{u,v\\}$, $V_{2}=\\{w_{1},w_{2}\\}$. Under this
partition, we have

$D(G)=\left(\begin{array}[]{ccccc}J_{s-2}-I_{s-2}&2J_{(s-2)\times
2}&J_{(s-2)\times 1}&J_{(s-2)\times 1}&2J_{(s-2)\times 1}\\\
2J_{2\times(s-2)}&2(J_{2}-I_{2})&J_{2\times 1}&2J_{2\times 1}&3J_{2\times
1}\\\ J_{1\times(s-2)}&J_{1\times 2}&0&1&2\\\ J_{1\times(s-2)}&2J_{1\times
2}&1&0&1\\\ 2J_{1\times(s-2)}&3J_{1\times 2}&2&1&0\end{array}\right).$

It is easy to see that $-1$ is a distance eigenvalue of $G$ with multiplicity
at least $s-3$ and $-2$ is a distance eigenvalue of $G$ with multiplicity at
least $1$.

Let $Q$ be the quotient matrix of $D(G)$ respect the above partition on
$V(G)$. Then

$Q=\left(\begin{array}[]{ccccc}s-3&4&1&1&2\\\ 2(s-2)&2&1&2&3\\\ s-2&2&0&1&2\\\
s-2&4&1&0&1\\\ 2(s-2)&6&2&1&0\end{array}\right).$

As the above partition is equitable, the eigenvalues of $Q$ are also distance
eigenvalues of $G$. Let $f(\lambda)=\det(\lambda I_{5}-Q)$. Then

$f(\lambda)=\lambda^{5}-(s-1)\lambda^{4}-12(s+1)\lambda^{3}-(40s+2)\lambda^{2}-(37s-10)\lambda-10s+4.$

Note that $f(+\infty)>0$, $f(0)=-10s+4<0$,
$f(-\frac{1}{2})=\frac{1-2s}{32}<0$,
$f(-\frac{s+1}{2s})=\frac{1}{32s^{5}}(2s^{6}-89s^{5}+233s^{4}-170s^{3}-44s^{2}+3s+1)>0$
for large enough $s$, $f(-1)=4-2s<0$ for $s\geq 3$, $f(-4)=2(5s-34)>0$ for
$s\geq 7$, $f(-\infty)<0$. It follows that the second largest root
$\lambda^{(2)}$ of $f(\lambda)=0$ satisfies
$-\frac{s+1}{2s}<\lambda^{(2)}<-\frac{1}{2}$ for large enough $s$. Thus,
$\lambda^{(2)}$ is the second largest distance eigenvalue of $G$, i.e.,
$\lambda_{2}(G)<-\frac{1}{2}$. ∎

For a graph $G$ and its subgraph $H$ and a vertex $v$ of $G$ outside $H$, let
$N_{H}(v)=N_{G}(v)\cap V(H)$ and $d_{H}(v)=|N_{H}(v)|$.

We are now ready to prove Theorem 5.1.

###### Proof of Theorem 5.1.

Let $G$ be a split graph of order $n=s+t$. Let $K_{s}$ be the maximum clique
and $I$ be the independent set in $G$ of size $t$. Then $s\geq 2$.

If $I=\emptyset$, then $G\cong K_{s}=K_{s}(0)$ and
$\lambda_{2}(G)=-1<-\frac{1}{2}$.

Suppose that $I\neq\emptyset$.

Suppose first that $s=2$. Let $V(K_{2})=\\{u,v\\}$. Assume that $d_{G}(u)\geq
d_{G}(v)$. Since $F_{2}$ is a forbidden subgraph, we have $d_{G}(v)=1,2$. If
$d_{G}(v)=2$, then $d_{G}(u)\leq 3$ due to $F_{1}$ being forbidden. Thus
$G\cong K_{2}(1,1)$ or $K_{2}(2,1)$. If $d_{G}(v)=1$, then $G\cong K_{2}(t)$.

By a direct calculation, we have
$\lambda_{2}(K_{2}(2,1))\approx-0.5120<-\frac{1}{2}$. From Lemma 2.1, we have
$\lambda_{2}(K_{2}(1,1))<-\frac{1}{2}$ and
$\lambda_{2}(K_{2}(t))<-\frac{1}{2}$.

Suppose next that $s\geq 3$. Let $ID_{G}=\\{z\in I:d_{K_{s}}(z)\geq 2\\}$.

Claim. $|ID_{G}|=0,1$.

Otherwise, there exist $z_{1}$ and $z_{2}$ in $I$ with $d_{K_{s}}(z_{1})\geq
2$ and $d_{K_{s}}(z_{2})\geq 2$. Let $x_{1},x_{2}\in N_{K_{s}}(z_{1})$ and
$y_{1},y_{2}\in N_{K_{s}}(z_{2})$. There are three possibilities.

Suppose that $|\\{x_{1},x_{2}\\}\cap\\{y_{1},y_{2}\\}|=2$, i.e.,
$\\{x_{1},x_{2}\\}=\\{y_{1},y_{2}\\}$. Note that $s\geq 3$. Since $K_{s}$ is a
maximum clique, there exist $x_{3},x_{4}\in V(K_{s})$ such that
$x_{3}z_{1}\notin E(G)$, $x_{4}z_{2}\notin E(G)$. Suppose that $x_{3}\neq
x_{4}$. Then $x_{3}z_{2}\notin E(G)$. Otherwise,
$G[\\{x_{1},x_{2},x_{3},x_{4},z_{2}\\}]\cong K_{5}-e$, and
$\lambda_{2}(K_{5}-e)\approx-0.4495>-\frac{1}{2}$, a contradiction. So either
$x_{3}\neq x_{4}$ and $x_{3}z_{2}\notin E(G)$ or $x_{3}=x_{4}$. In either
case, $G[\\{x_{1},x_{2},x_{3},z_{1},z_{2}\\}]\cong F_{12}$, also a
contradiction.

Suppose that $|\\{x_{1},x_{2}\\}\cap\\{y_{1},y_{2}\\}|=1$. Without loss of
generality, let $y_{1}=x_{1},y_{2}\neq x_{2}$. Then
$G[\\{x_{1},x_{2},y_{2},z_{1},z_{2}\\}]\cong F_{11}$, a contradiction.

Suppose that $|\\{x_{1},x_{2}\\}\cap\\{y_{1},y_{2}\\}|=0$. Then
$G[\\{x_{1},x_{2},y_{1},y_{2},z_{1},z_{2}\\}]\cong F_{13}$, also a
contradiction.

Therefore, the claim follows.

By the above claim, $|ID_{G}|=0,1$.

Case 1. $|ID_{G}|=1$.

Let $ID_{G}=\\{z\\}$.

If $s=3$, then $G$ is a $\theta$-bicyclic graph, so by Lemma 4.2, we have
$\lambda_{2}(G)\leq-\frac{1}{2}$ if and only if $G\cong
B_{6},B_{7},B^{\theta}_{k}$.

Assume that $s\geq 4$. Then $d_{K_{s}}(z)=2$. Otherwise, there is a distance-
preserving subgraph isomorphic to $K_{5}-e$ and
$\lambda_{2}(K_{5}-e)\approx-0.4495>-\frac{1}{2}$, a contradiction. Suppose
that $s\geq 5$. Then there is a distance-preserving subgraph, say $H$, induced
by $V(K_{5})\cup\\{z\\}$. By a direct calculation, we have
$\lambda_{2}(H)\approx-0.4913>-\frac{1}{2}$, a contradiction. It thus follows
that $s=4$. Let $N_{K_{s}}(z)=\\{u,v\\}$ with $d_{G}(u)\geq d_{G}(v)$. Let
$w\in V(K_{4})\setminus\\{u,v\\}$. Since $F_{9}$ is a forbidden graph,
$d_{G}(w)=3$. Since $F_{2}$ is a forbidden subgraph, we have $d_{G}(v)=4,5$.
If $d_{G}(v)=5$, then, since $F_{1}$ is a forbidden subgraph, we have
$d_{G}(u)=5,6$. So $G\cong SP_{1}$ if $d_{G}(u)=6$ and $G\cong SP_{2}$ if
$d_{G}(u)=5$. If $d_{G}(v)=4$, then $G\cong SP^{t}$ with $t\geq 0$.

By a direct calculation, we have
$\lambda_{2}(SP_{1})\approx-0.5106<-\frac{1}{2}$. Since $SP_{2}$ is an induced
distance-preserving subgraph of $SP_{1}$,
$\lambda_{2}(SP_{2})\leq\lambda_{2}(SP_{1})<-\frac{1}{2}$. By Lemma 5.1,
$\lambda_{2}(SP^{t})<-\frac{1}{2}$.

Case 2. $|ID_{G}|=0$.

If there is exactly one vertex in $K_{s}$ with degree not less than $s$, then
$G\cong K_{s}(t)$, which is a block star, and by Lemma 2.1,
$\lambda_{2}(K_{s}(t))<-\frac{1}{2}$.

Suppose that there are exactly two vertices in $K_{s}$, say $u,v$ with
$d_{G}(u)\geq d_{G}(v)\geq s$. Since $F_{2}$ is a forbidden subgraph, we have
$d_{G}(v)\leq s$. So $d_{G}(v)=s$. Furthermore, since $F_{1}$ is a forbidden
subgraph, we have $d_{G}(u)\leq s+1$. Then $G\cong K_{s}(2,1)$ if
$d_{G}(u)=s+1$, and $G\cong K_{s}(1,1)$ if $d_{G}(u)=s$. From Lemma 5.2,
$\lambda_{2}(K_{s}(2,1))<-\frac{1}{2}$. Since $K_{s}(1,1)$ is an induced
distance-preserving subgraph of $K_{s}(2,1)$,
$\lambda_{2}(K_{s}(1,1))\leq\lambda_{2}(K_{s}(2,1))<-\frac{1}{2}$.

If there are $t$ vertices in $K_{s}$ with degree not less than $s$ with $t\geq
3$, then, since $F_{6}$ is a forbidden subgraph, we have $G\cong
K_{s}(\underbrace{1,\dots,1}_{t})$, which is a loose block. From Lemma 2.1, we
have $\lambda_{2}(K_{s}(\underbrace{1,\dots,1}_{t}))<-\frac{1}{2}$. ∎

##  6 Conclusions

In this article, we consider graphs for which the second largest distance
eigenvalue is less than $-\frac{1}{2}$. We show that any such graph must be
chordal and characterize all bicyclic graphs and split graphs with the desired
property. Characterizing graphs with the desired property among graphs with
forbidden subgraphs or minors would be a way to have a fuller understanding
the behavior of the second largest distance eigenvalue.

Acknowledgements

This work was supported by the National Natural Science Foundation of China
(Nos. 12101249 and 12071158).

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