eqsplit

$\begin{split}\BODY\end{split}$ (1)

# Random model of flow decorrelation

Piotr Bożek<EMAIL_ADDRESS>AGH University of Krakow, Faculty of
Physics and Applied Computer Science, al. Mickiewicza 30, 30-059 Cracow,
Poland  Hadi Mehrabpour<EMAIL_ADDRESS>PRISMA+ Cluster of Excellence &
Mainz Institute for Theoretical Physics, Johannes Gutenberg-Universität Mainz,
55099 Mainz, Germany

###### Abstract

The collective flow generated in relativistic heavy-ion collisions fluctuates
from event to event. The fluctuations lead to a decorrelation of flow vectors
measured in separate bins in phase space. These effects have been measured in
experiments and observed in numerical simulations in hydrodynamic models. We
present a simple random model of flow decorrelation in pseudorapidity.
Analytical expressions for the flow factorization breaking coefficients for
flow vectors, flow vector magnitudes, and flow angles are derived. The model
explains the relations between different factorization breaking coefficients
found in experimental data and model simulations. In particular, it is found
that the flow angle decorrelation constitutes about one half of the total flow
vector decorrelation.

ultra-relativistic nuclear collisions, event-by-event fluctuations,forward-
backward harmonic flow correlations

##  I Introduction

One of the methods of investigation of the hot and dense matter created in the
interaction region of relativistic heavy-ion collisions is the analysis of the
collective flow from the spectra of emitted particles Huovinen and Ruuskanen
(2006); Voloshin  _et al._ (2010); Hirano  _et al._ (2010); Heinz (2010);
Heinz and Snellings (2013). Strong pressure gradients in the fireball cause a
rapid expansion. The azimuthal asymmetry of the collective flow can be
quantified using the harmonic flow coefficients, describing the magnitude and
the azimuthal direction of the flow. The dynamical model commonly used to
describe the generation of the collective flow in the rapid expansion of the
source is the relativistic viscous hydrodynamics Schenke  _et al._ (2011).

The collective flow generated reflects the properties of the initial state of
the collision. The initial conditions fluctuate from event to event and the
final flow observables fluctuate as well Aguiar  _et al._ (2002); Takahashi
_et al._ (2009); Alver and Roland (2010); Schenke  _et al._ (2011, 2012). One
of the aspects of such fluctuations is the decorrelation of the flow harmonics
measured in different pseudorapidity bins. Model calculations predict a
deviation from unity of the correlation coefficient between two flow vectors
in separate pseudorapidity or transverse momentum bins Bożek  _et al._ (2011);
Gardim  _et al._ (2013). This correlation coefficient of flow vectors is
called the flow factorization breaking coefficient. It has been measured in
experiments Khachatryan  _et al._ (2015); Aaboud  _et al._ (2018); Aad  _et
al._ (2021); Nie (2019); Acharya  _et al._ (2023) and qualitatively reproduced
in models Gardim  _et al._ (2013); Kozlov  _et al._ (2014); Gardim  _et al._
(2018); Zhao  _et al._ (2017); Bożek (2018); Barbosa  _et al._ (2021); Bożek
_et al._ (2011); Jia and Huo (2014); Pang  _et al._ (2016); Sakai  _et al._
(2022); Cimerman  _et al._ (2021). In this paper we study the decorrelation of
the collective flow for different bins in pseudorapidity.

Correlations of higher moments of flow vectors in different phase-space bins
have been studied as well, both in experiment Aaboud  _et al._ (2018); Acharya
_et al._ (2023) and in models Jia and Huo (2014); Bożek and Broniowski (2018);
Wu  _et al._ (2018); Bożek (2018); Bożek and Samanta (2022). The measurement
of the decorrelation using four-particle correlators allows one to estimate
separately the flow vector magnitude and flow angle decorrelation in different
bins. Experimental and model calculation show that the total flow vector
decorrelation is composed in approximately equal parts from the flow magnitude
and flow angle decorrelation. The second observation is that the decorrelation
of higher powers of flow vectors is stronger than the decorrelation of simple
flow vectors. The factorization breaking coefficient for the second or third
power of the flow vector can be approximated as the second or third power of
the simple flow vector factorization breaking coefficient. Finally, it has
been observed in model calculations that the decorrelation is the strongest
for events (or classes of events) where the overall flow is the smallest.

These effects have been observed in numerical simulations, but no simple
understanding has been given. In this paper, we present a simple model of flow
decorrelation in different phase-space bins. We assume that the flow vector in
a small pseudorapidity bin can be written as a sum of the overall flow
(averaged over the whole event) and of a random vector component. Assuming the
independence of the directions of random component of the flow and of the
average flow, the model can explain qualitatively the effects observed in
model simulations and in the experimental data. We present a number of
analytical results for the factorization coefficients for flow vectors, for
powers of flow vectors, for flow magnitudes and for flow angles. We show that
qualitatively similar relations between different factorization breaking
coefficients are found in simulations using a realistic hydrodynamic model.

We study three- and four-bin measures of the flow decorrelation in
pseudorapidity, used in experimental analyses Khachatryan  _et al._ (2015);
Aaboud  _et al._ (2018). The random model of flow decorrelation can describe
qualitatively the relations between different three- and four-bin measures of
the flow vector, flow magnitude, and flow angle decorrelation observed in
model calculations and in the experimental data. Also in this case a relation
between the decorrelation of the second or third power of flow vectors and the
second or third power of the simple flow vector decorrelation is found.

The flow factorization breaking effect is briefly described in Sec. II. Our
model of the flow decorrelation due to a random component in the flow vector
is introduced in Sec. III. In the following sections applications of the
random flow decorrelation model for the flow decorrelation in pseudorapidity
are presented: the factorization breaking coefficient of the flow in two bins
in pseudorapidity (Sec. IV), the analysis of flow angle decorrelation (Sec.
V), and the calculations for three- and four-bin measures of flow
deccorelation (Sec. VI). The results are summarized in the last section.

##  II Flow correlation

---
Figure 1: Pseudorapidity bins used in definitions of factorization breaking
coefficients of the collective flow (two, three, or all four bins depending on
the case).

The flow in an event can be defined using the harmonic coefficients of the
azimuthal distribution of emitted particles. We use the notation
$V_{n}=v_{n}e^{in\Psi_{n}}$, where $v_{n}$ and $\Psi_{n}$ are the magnitude
and event-plane angle for the $n$th order harmonic flow. The flow vector
$V_{n}$ cannot be reconstructed in each event. Rotationally invariant
combinations of moments of flow vectors can be estimated from the moments of
the corresponding $q_{n}$ vectors in a phase-space region $A$,

$q_{n}(A)=\frac{1}{N}\sum_{k\in A}e^{in\phi_{k}},$ (2)

where the sum runs over all $N$ hadrons in the selected phase-space region
$A$, and $\phi_{k}$ are their azimuthal angles. The event average of the
$q_{n}$ vector moments is an estimator of the corresponding moments of the
flow vectors, e.g.,

$\langle(V_{n})^{m}(V_{n}^{*})^{m}\rangle=\langle
q_{n}^{m}(q_{n}^{\star})^{m}\rangle,$ (3)

where the angular brackets denote an average over the events.

The factorization breaking of the collective flow means that the flow moment
calculated in different regions in phase space ($A$ and $B$),

$\langle V_{n}(A)V_{n}^{*}(B)\rangle=\frac{1}{N_{A}N_{B}}\sum_{k\in A,j\in
B}e^{in(\phi_{k}-\phi_{j})},$ (4)

does not factorize into flow moments calculated Bożek  _et al._ (2011); Gardim
_et al._ (2013) in the same bin,

$\langle V_{n}(A)V_{n}^{*}(A)\rangle=\frac{1}{N_{A}(N_{A}-1)}\sum_{k\neq j\in
A}e^{in(\phi_{k}-\phi_{j})}\ ,$ (5)

which gives

$\langle V_{n}(A)V_{n}^{*}(B)\rangle\neq\sqrt{\langle
V_{n}(A)V_{n}^{*}(A)\rangle\langle V_{n}(B)V_{n}^{*}(B)\rangle}\ .$ (6)

The factorization breaking coefficient is the correlation coefficient of flow
vectors in two phase-space regions

$\rho_{V_{n}}(A,B)=\frac{\langle V_{n}(A)V_{n}^{*}(B)\rangle}{\sqrt{\langle
V_{n}(A)V_{n}^{*}(A)\rangle\langle V_{n}(B)V_{n}^{*}(B)\rangle}}.$ (7)

If the flow dominates the multiparticle correlation we have
$\rho_{V_{n}}(A,B)\leq 1$. The flow correlation coefficient (factorization
breaking coefficient) can be used as a measure of flow decorrelation for two
bins in pseudorapidity Bożek  _et al._ (2011) or in transverse momentum Gardim
_et al._ (2013). The factorization breaking coefficient $r_{n}(p_{1},p_{2})$
in transverse momentum can be measured in experiment Khachatryan  _et al._
(2015); Chatrchyan  _et al._ (2014); Zhou (2014) and calculated in models
Gardim  _et al._ (2013); Kozlov  _et al._ (2014); Gardim  _et al._ (2018);
Zhao  _et al._ (2017); Bożek (2018); Barbosa  _et al._ (2021).

The factorization breaking coefficient in pseudorapidity, Eq. (7), contains a
significant contribution from nonflow effects Bożek  _et al._ (2011). A
modified factorization breaking coefficient has been proposed Khachatryan  _et
al._ (2015), defined as a ratio of two flow vector covariances taken for two
different pairs of bins (Fig. 1),

$\mathcal{R}^{(1)}_{n;V}=\frac{\langle
V_{n}(-\eta)V_{n}^{*}(\eta_{ref})\rangle}{\langle
V_{n}(\eta)V_{n}^{*}(\eta_{ref})\rangle},$ (8)

where $\eta_{ref}$ is the reference pseudorapidity common to the numerator and
the denominator. The bins are placed such that both $|\eta_{ref}-\eta|$ and
$|\eta_{ref}+\eta|$ are large enough to suppress nonflow correlations. The
experimentally measured Khachatryan  _et al._ (2015); Huo (2017); Aaboud  _et
al._ (2018) decorrelation in pseudorapidity using Eq. (8) can be qualitatively
reproduced in hydrodynamic and cascade models Bożek  _et al._ (2015); Pang
_et al._ (2016); Xiao  _et al._ (2016). Besides the simple factorization
breaking coefficient defined in Eq. (7), factorization breaking coefficients
for higher moments of flow vectors can be defined Jia  _et al._ (2017); Aaboud
_et al._ (2018); Bożek (2018); Bożek and Samanta (2022); Acharya  _et al._
(2023). Based on these factorization-breaking coefficients of higher order
flow moments, the flow magnitude and flow angle decorrelation can be estimated
separately Jia and Huo (2014); Bożek (2018). In the following, we discuss a
simple model of flow decorrelation that explains qualitatively the relations
between different factorization breaking coefficients. To illustrate the
effects in model calculations, we use a 3+1-dimensional viscous hydrodynamic
model Schenke  _et al._ (2011) with particle emission through statistical
hadronization Chojnacki  _et al._ (2012) at freeze-out. Details of the three-
dimensional fluctuating initial conditions and hydrodynamic model can be found
in Ref Bożek and Broniowski (2018). In the present work, we use Pb+Pb
collisions at $\sqrt{s_{NN}}=5.02$TeV for two centralities, $0-5$% and
$30-40$%, as a numerical example of the analytical identities discusses in the
random model of flow decorrelation.

##  III Random model of flow decorrelation

The flow in the same event, but in two separate phase-space regions, is
slightly decorrelated. To understand the basic features of this effect, we
study a simple random flow decorrelation model. The model is simple enough so
that, in the limit of small flow decorrelations, many observables, such as the
factorization breaking coefficients and flow correlations, can be estimated
analytically.

In a given event, the flow measured in two specific bins differs. Also, the
flow $V_{i}=V+K_{i}$ defined in a small bin $i$ in phase space differs from
the flow $V$ averaged over the whole acceptance region depicted in Fig. 1. The
flow in a small bin is composed of the average flow in the whole acceptance
region and a random component $K_{i}$. Both $V$ and $K_{i}$ have nontrivial
event-by-event distributions. In the following, we assume that all the small
bins considered are of the same size and that in each bin $i$ the flow
distribution is the same. It is a good approximation for the flow defined in
pseudorapidity bins. Alternatively, one can transfer the definition of the
model to the scaled flow $V_{i}/\langle V_{i}\rangle$, e.g., for the modeling
of flow factorization breaking in transverse momentum.

The combined probability distribution of two components of the flow $V$ and
$K_{i}$, $i=1,\dots,N$ can be very complicated in principle. We make one
simplifying assumption:

$\langle V^{m}(K_{i}^{\star})^{m}\rangle=0\ \ ,\ \textrm{}\ \ m=1,2,\dots\ .$
(9)

We have checked quantitatively the magnitude of the above term in comparison
to the terms that are kept in the expansion. We find that the approximation
works well for the triangular flow and for the elliptic flow in central
collisions ($|\langle VK^{\star}\rangle|<0.05\langle KK^{\star}\rangle$ ). The
approximation is not as good, but still acceptable, for the elliptic flow in
semicentral collisions ($|\langle VK^{\star}\rangle|\simeq 0.13\langle
KK^{\star}\rangle$ ). Note that the above properties are fulfilled under the
assumption of a random relative orientation of flow vectors $V$ and $K_{i}$.
On the other hand, the magnitudes could still be correlated,

$\langle v^{m}|K_{i}|^{l}\rangle\neq 0\ .$ (10)

The local random component of the flow $K_{i}$ fulfills the constraint

$\langle(V+K_{i})(V+K_{i})^{\star}\rangle=Var(V)+Var(K_{i})=Var(V)+C^{2}\ ,$
(11)

where $C^{2}$ is a constant defined as the difference between flow vector
variance in a small bin and flow vector variance in the whole acceptance
region. Nontrivial flow correlation between two different bins is encoded in
the correlation of the random components $K_{i}$ and $K_{j}$ of the flow in
the two bins. Phenomenologically, one expects that as the separation between
the two bins increases the random components of flow vectors $K_{i}$ and
$K_{j}$ become less aligned. This means that the covariance

$\langle K_{i}K_{j}^{\star}\rangle$ (12)

decreases with increasing bin separation. Another correlation between random
components comes from the global constraint

$\sum_{i}K_{i}=0\ .$ (13)

For the study of the covariances between flows in two bins, we define

$A=\frac{K_{i}+K_{j}}{2}\quad\text{and}\quad\Delta=\frac{K_{i}-K_{j}}{2}\ .$
(14)

Note that, due to the properties of $K_{i}$ and $K_{j}$, both $A$ and $\Delta$
depend on the separation between the bins. The distribution of $\Delta$ gets
wider as the bin separation increases (reflecting the increasing decorrelation
of $K_{i}$ and $K_{j}$). On the other hand, the distribution of $A$ gets
narrower,

$\langle AA^{\star}\rangle=C^{2}-\langle\Delta\Delta^{\star}\rangle,$ (15)

due to the constraint from Eq. (11). We have

$\langle A\Delta\rangle=0\ .$ (16)

For bins in pseudorapidity, we specify $V(\eta)=V+K_{i}$ for the flow vector
in a bin of fixed size around the pseudorapidity $\eta$. As indicated above,
the distribution of $A$ and $\Delta$ depends only on the pseudorapidity
separation $|\eta_{1}-\eta_{2}|$ between the two bins. Besides the
mathematical constraint in Eq. (13) or global momentum conservation
constraints Dasgupta  _et al._ (2023), the correlations between the two random
components $K_{i}$ and $K_{j}$ can have a nontrivial physical origin. In the
hydrodynamic model, fluctuations in the initial distribution in space-time
rapidity generate fluctuations in the final flow in pseudorapidity Bożek  _et
al._ (2015). Local fluctuations of the initial flow increase the decorrelation
of the final flow Pang  _et al._ (2016). Finally, dynamical hydrodynamic
fluctuations Kapusta  _et al._ (2012) lead to fluctuations of the final flow
of a finite range in rapidity Sakai  _et al._ (2019). Also thermodynamic
fluctuation and contributions from energy deposition from jets could
contribute to the flow decorrelation. The contribution to the local collective
flow from initial momentum flow, hydrodynamic fluctuations, thermal
fluctuations, or jets is expected to be, in the first approximation,
independent of the average collective flow. Therefore, it could be effectively
described as a random component added to the flow vector as well.

##  IV Factorization breaking coefficient of the collective flow

The factorization breaking coefficient of the collective flow, which is
equivalent to the correlation coefficient for the complex variables
$(V_{n}(\eta))^{m}$ and $(V_{n}(-\eta))^{m}$, can be generalized to any power
of the flow:

$\rho_{V_{n}}^{(m)}(\eta,-\eta)=\frac{\langle(V(\eta))^{m}(V^{\star}(-\eta))^{m}\rangle}{\sqrt{\langle(V(\eta))^{m}(V^{\star}(\eta))^{m}\rangle\langle(V(-\eta))^{m}(V^{\star}(-\eta))^{m}\rangle}}\
.$ (17)

The factorization breaking coefficient of the $m$th power of the flow vector
in Eq. (17) can be estimated in principle from an average of the moments of
the experimentally measured $q$ vectors,

$\rho_{V_{n}}^{(m)}(\eta,-\eta)=\frac{\langle(q(\eta))^{m}(q^{\star}(-\eta))^{m}\rangle}{\sqrt{\langle(q(\eta))^{m}(q^{\star}(\eta))^{m}\rangle\langle(q(-\eta))^{m}(q^{\star}(-\eta))^{m}\rangle}}\
.$ (18)

However, for correlations in pseudorapidity the measure is strongly influenced
by nonflow effects Bożek  _et al._ (2011). In the models results shown here,
the nonflow correlations are reduced by oversampling the final state hadrons
for each hydrodynamic evolution event. We chose the number of oversampling so
that the remaining contribution of nonflow resonance decays to the relative
error is smaller than the statistical errors in the calculation. On the other
hand, such observables can be used as an experimental estimate for higher
moment factorization breaking coefficients in transverse momentum Bożek
(2018); Bożek and Samanta (2022); Acharya  _et al._ (2023), if rapidity gaps
are used to reduce nonflow correlations. Similar relations could be studied
for analogous higher order factorization breaking coefficients in transverse
momentum.

|
---|---
|
Figure 2: A comparison of the factorization breaking coefficient for the second power of flow magnitude (black lines, dots) with the factorization breaking coefficient for the second power of flow vectors (blue lines, diamonds) is shown in the top panels [central and semicentral collisions, panels (a) and (b) respectively]. The red lines with squares represents the flow angle factorization breaking coefficient, Eq. (29). The factorization breaking coefficients for the fourth power of the flow are shown in panels (c) and (d). All results are obtained from the viscous hydrodynamic model for Pb-Pb collisions at $\sqrt{s_{NN}}=5.02$TeV.  |
---|---
|
Figure 3: Same as in Fig. 2 but for the triangular flow.

In the random model of flow decorrelation, the flow in each bin is decomposed
into global and local random components. The factorization breaking
coefficient takes the form

$\displaystyle\rho_{V_{n}}^{(m)}(\eta,-\eta)$ $\displaystyle=$
$\displaystyle\frac{\langle(V_{n}+A_{n}+\Delta_{n})^{m}(V_{n}^{\star}+A_{n}^{\star}-\Delta_{n}^{\star})^{m}\rangle}{\sqrt{\langle(V_{n}+A_{n}+\Delta_{n})^{m}(V_{n}^{\star}+A_{n}^{\star}+\Delta_{n}^{\star})^{m}\rangle\langle(V_{n}+A_{n}-\Delta_{n})^{m}(V_{n}^{\star}+A_{n}^{\star}-\Delta_{n}^{\star})^{m}\rangle}}$
(19) $\displaystyle=$
$\displaystyle\frac{\langle(V_{n}^{{}^{\prime}}+\Delta_{n})^{m}(V_{n}^{{}^{\prime}\star}-\Delta_{n}^{\star})^{m}\rangle}{\langle(V_{n}^{{}^{\prime}}+\Delta_{n})^{m}(V_{n}^{{}^{\prime}\star}+\Delta_{n}^{\star})^{m}\rangle}\
,$

where $V_{n}^{{}^{\prime}}=V_{n}+A_{n}$. If the random component of the flow
is relatively small $\delta_{n}=|\Delta_{n}|\ll|V_{n}|$ (then also
$\delta_{n}=|\Delta_{n}|\ll|V_{n}^{{}^{\prime}}|$), the factorization breaking
coefficient can be expanded to second order in $\Delta$,

$\rho_{V_{n}}^{(m)}(\eta,-\eta)\simeq 1-2m^{2}\frac{\langle
v_{n}^{2m-2}\delta^{2}\rangle}{\langle v_{n}^{2m}\rangle}\ .$ (20)

The above formula shows the generic properties of the factorization-breaking
coefficient. On general grounds, one expects that the lowest order dependence
of $\Delta$ on the bin separation $\Delta\eta$ is linear Bożek  _et al._
(2011); Bożek and Broniowski (2016). This leads to a quadratic dependence of
the factorization-breaking coefficient on the bin separation,

$\rho_{V_{n}}^{(m)}(\eta_{1},\eta_{2})\simeq
1-2m^{2}\kappa(\eta_{1}-\eta_{2})^{2}\ .$ (21)

The flow moment decorrelation, i.e. the deviation of the factorization
breaking coefficient from unity, increases with the rank $m$ of the flow
moment. The formula can be further simplified if the factorization of moments
$v$ and $\delta$ is assumed:

$\displaystyle\rho_{V_{n}}^{(m)}(\eta,-\eta)$ $\displaystyle\simeq$
$\displaystyle 1-2m^{2}\frac{\langle
v_{n}^{2m-2}\rangle\langle\delta^{2}\rangle}{\langle v_{n}^{2m}\rangle}$ (22)
$\displaystyle\simeq$ $\displaystyle
1-2m\frac{\langle\delta^{2}\rangle}{\langle v_{n}^{2}\rangle}\ ,$

where the last equality is obtained for fluctuation-dominated flow. If the
collective flow is dominated by fluctuations in the initial state, as for the
triangular flow or the elliptic flow in central collisions, we have $\langle
v_{n}^{2m}\rangle=m!\langle v_{n}^{2}\rangle^{m}$.

The deviation of the factorization breaking coefficient from $1$ comes from
two effects, the flow angle decorrelation and flow magnitude decorrelation, as
noted in Heinz  _et al._ (2013); Jia  _et al._ (2017). Both effects can be
studied separately in model calculations Bożek (2018). The factorization
breaking coefficient for the flow magnitudes is:

$\rho_{v_{n}}^{(m)}(\eta,-\eta)=\frac{\langle
v_{n}(\eta)^{m}v_{n}(-\eta)^{m}\rangle}{\sqrt{\langle
v_{n}(\eta)^{2m}\rangle\langle v_{n}(-\eta)^{2m}\rangle}}\ .$ (23)

The factorization breaking coefficient for the flow magnitudes could be
estimated experimentally (at least in principle) for even $m=2k$:

$\rho_{v_{n}}^{(m)}(\eta,-\eta)=\frac{\langle(q_{n}(\eta))^{k}(q_{n}^{\star}(\eta))^{k}(q_{n}(-\eta))^{k}(q_{n}^{\star}(-\eta))^{k}\rangle}{\sqrt{\langle(q_{n}(\eta))^{2k}(q_{n}^{\star}(\eta))^{2k}\rangle\langle(q_{n}(\eta))^{2k}(q_{n}^{\star}(\eta))^{2k}\rangle}}\
.$ (24)

Figure 4: Flow factorization breaking coefficients for different moments of
flow vectors calculated in the hydrodynamic model. Panels (a) and (b) are for
central ($0-5$%) and semi-central ($30-40$%) collisions, respectively. The
factorization breaking coefficients $\rho^{(1)}$, $\rho^{(2)}$, $\rho^{(3)}$,
and $\rho^{(4)}$ are denoted with circles, squares, diamonds, and triangles
respectively. The powers of the first order coefficient $[\rho^{(1)}]^{m}$ are
represented with dashed lines of the same color as the corresponding
coefficients $\rho^{(m)}$.

Figure 5: same as in Fig. 4 but for the triangular flow.

The flow magnitude factorization breaking can be calculated using our model
with a random local flow component,

$\displaystyle\rho_{v_{n}}^{(m)}(\eta,-\eta)$ $\displaystyle=$
$\displaystyle\frac{\langle|V_{n}+A_{n}+\Delta_{n}|^{m}|V_{n}+A_{n}-\Delta_{n}|^{m}\rangle}{\sqrt{\langle|V_{n}+A_{n}+\Delta_{n}|^{2m}\rangle\langle|V_{n}+A_{n}-\Delta_{n}|^{2m}\rangle}}$
(25) $\displaystyle=$
$\displaystyle\frac{\langle|V_{n}^{{}^{\prime}}+\Delta_{n}|^{m}|V_{n}^{{}^{\prime}}-\Delta_{n}|^{m}\rangle}{\langle|V_{n}^{{}^{\prime}}+\Delta_{n}|^{2m}\rangle}\
.$

Expansion to the second order in $\delta/v$ gives

$\rho_{v_{n}}^{(m)}(\eta,-\eta)\simeq 1-m^{2}\frac{\langle
v_{n}^{2m-2}\delta^{2}\rangle}{\langle v_{n}^{2m}\rangle}\ .$ (26)

In the random model of flow decorrelation, the decorrelation of the flow
vector magnitudes is approximately one half of the decorrelation of the flow
vectors. The same property can be observed in the viscous hydrodynamic model
results (Figs. 2 and 3).

To estimate the flow angle decorrelation between two bins centered at $\eta$
and $-\eta$ a simple average of the cosine of flow angle difference could be
used:

$\left\langle\cos\left(mn(\Psi_{n}(\eta)-\Psi_{n}(-\eta))\right)\right\rangle=\left\langle\frac{\left(V_{n}(\eta)V_{n}^{\star}(-\eta)\right)^{m}}{\left(|V_{n}(\eta)||V_{n}(-\eta)|\right)^{m}}\right\rangle\
.$ (27)

The flow angle decorrelation defined above cannot be directly measured in the
experiment, unlike the flow vector factorization breaking coefficient in Eq.
(17) (for any $m$) or the flow magnitude factorization breaking coefficient in
Eq. (23) (for even $m$). Note that in the definition of flow factorization
breaking the event average is taken separately in the numerator and the
denominator. In contrast, for the angle decorrelation defined in Eq. (27), the
event average is taken for the whole ratio. The expansion of the flow angle
decorrelation for small $\delta/v$ gives

$\langle\cos\left(mn(\Psi_{n}(\eta)-\Psi_{n}(-\eta))\right)\rangle\simeq
1-m^{2}\left\langle\frac{\delta_{n}^{2}}{v_{n}^{2}}\right\rangle.$ (28)

The simple flow angle decorrelation of order $m$ in Eq. (27) should not be
confused with the angular component of the flow factorization breaking,
$\rho^{(m)}_{V_{n}}(\eta,-\eta)$, Eq. (17). Under the event average, the flow
magnitude to power $m$ is present in the flow factorization breaking
coefficient of order $m$. It has been noticed that the flow angle
decorrelation is strongly anticorrelated with flow magnitude in the event
Bożek and Broniowski (2018). Therefore, the relevant flow angle factorization
breaking coefficient in order $m$ is defined as

$\rho_{\Psi_{n}}^{(m)}(\eta,-\eta)=\frac{\langle
v^{2m}\cos\left(mn(\Psi_{n}(\eta)-\Psi_{n}(-\eta))\right)\rangle}{\langle
v^{2m}\rangle}\ .$ (29)

In experiment, the flow angle factorization breaking coefficient can be
estimated as the ratio of the flow vector and flow vector magnitude
factorization breaking coefficients,

$\rho_{\Psi_{n}}^{(m)}(\eta,-\eta)=\frac{\rho_{V_{n}}^{(m)}(\eta,-\eta)}{\rho_{v_{n}}^{(m)}(\eta,-\eta)}\
,$ (30)

with almost the same results as from the definition in Eq. (29). For small
flow decorrelation, we have

$\rho_{\Psi_{n}}^{(m)}(\eta,-\eta)\simeq 1-m^{2}\frac{\langle
v^{2m-2}\delta^{2}\rangle}{\langle v^{2m}\rangle}.$ (31)

In Figs. 2 and 3, we compare the results for measures of flow decorrelation in
two pseudorapidity bins: the flow factorization breaking coefficient, the flow
magnitude factorization breaking coefficient, and the flow angle factorization
breaking coefficient in the hydrodynamic model. The numerical results follow
the relation obtained in our random flow decorrelation model. In particular,
the flow factorization breaking can be decomposed into the flow magnitude
factorization breaking and the flow angle decorrelation,

$\rho_{V_{n}}^{(m)}(\eta,-\eta)\simeq\rho_{v_{n}}^{(m)}(\eta,-\eta)\rho_{\Psi_{n}}^{(m)}(\eta,-\eta),$
(32)

where, with considerable accuracy, we find:

$[1-\rho_{\Psi_{n}}^{(m)}(\eta,-\eta)]\simeq[1-\rho_{v_{n}}^{(m)}(\eta,-\eta)]\simeq[1-\rho_{V_{n}}^{(m)}(\eta,-\eta)]/2.$
(33)

The flow angle decorrelation is roughly the same as the flow magnitude
decorrelation (compare red and black lines in Figs. 2 and 3). If the above
relation holds, Eq (33) could be used as an experimental estimate of the flow
angle decorrelation also for odd $m$.

Finally, we test the relation given in Eq. (22) between flow factorization
coefficients of different order $m$. When the event-by-event flow follows the
Bessel-Gaussian distribution Mehrabpour (2020); Mehrabpour and Tabatabaee
(2023) (central collisions for the elliptic flow and all centralities for the
triangular flow) and for small deviations of the factorization breaking
coefficients from $1$, one finds:

$\rho_{V_{n}}^{(m)}(\eta,-\eta)\approx\Big{[}\rho_{V_{n}}^{(1)}(\eta,-\eta)\Big{]}^{m}\
.$ (34)

Numerical results from the viscous hydrodynamic model confirm these relations
as illustrated in Figs. 4 and 5. The factorization breaking coefficient of
order $m=2,3$ can be approximated as a power of the first-order factorization
breaking coefficient for the elliptic flow at $0$-$5$% centrality and the
triangular flow at both centralities studied. The elliptic flow in semicentral
events ($30$-$40$%) is not dominated by fluctuations and the relation from Eq.
(22) for the factorization breaking coefficients of different orders is not
true. $\rho^{(4)}$ deviates strongly from $1$ and the approximate relation in
Eq. (34) is broken.

##  V Flow angle decorrelation and the overall flow magnitude

Figure 6: Scatter plot of the scaled elliptic flow in an event $v/\langle
v\rangle$ versus $\cos(n(\Psi_{n}(\eta)-\Psi_{n}(-\eta))$ for central (panel
a)) and semicentral (panel b)) collisions. All the points are from the viscous
hydrodynamic model. The red points represent the expected flow angle
decorrelation as a function of the fixed value of the flow in the event, Eq.
(36).

Figure 7: Same as in Fig. 6 but for the triangular flow.

The flow angle decorrelation is stronger if the overall flow is small. The
largest flow decorrelation occurs for the triangular flow and for the elliptic
flow in central events Bożek  _et al._ (2011). Moreover, for a given
centrality class, in events with a smaller magnitude of the overall flow, the
angle decorrelation is the largest Bożek and Broniowski (2018); Bożek (2018).
In Figs. 6 and 7 are shown the scatter plots of the overall flow magnitude in
the event and of the cosine of the flow decorrelation for a sample of central
and semicentral events. The increase of the flow angle decorrelation with
decreasing overall flow magnitude $v$ can be understood from Eq. (28). Taking

$\langle\cos\left(n(\Psi_{n}(\eta)-\Psi_{n}(-\eta))\right)\rangle\simeq
1-\left\langle\frac{\delta_{n}^{2}}{v_{n}^{2}}\right\rangle,$ (35)

at a fixed value of the flow magnitude, $v_{n}$, one has:

$\langle\cos\left(n(\Psi_{n}(\eta)-\Psi_{n}(-\eta))\right)\rangle|_{v_{n}}\simeq
1-\frac{\langle\delta_{n}^{2}\rangle}{v_{n}^{2}}.$ (36)

We illustrate the relation between the flow angle decorrelation and the
overall flow magnitude for $m=1$. The anticorrelation of the flow angle
decorrelation $\Psi_{n}(\eta)-\Psi_{n}(-\eta)$ with $v_{n}$ [Eq. (36] is
denoted with red points in Figs. 6 and 7. The random model of flow
decorrelation explains the anticorrelation between the overall flow magnitude
$v_{n}$ and the angle correlation
$\langle\cos\left(n(\Psi_{n}(\eta)-\Psi_{n}(-\eta))\right)\rangle$ observed in
numerical simulations of the hydrodynamic model.

##  VI three-bin and four-bin measures of flow decorrelation

Figure 8: The factorization breaking coefficient of the elliptic flow in three
pseudorapidity bins [central and semicentral collisions, panels (a) and (b)
respectively]. The flow vector factorization breaking coefficients, Eq. (44),
are denoted with filled circles, the flow magnitude factorization breaking
coefficients, Eq. (45), are denoted with squares, and the simple flow angle
factorization breaking coefficients, Eq. (39), are denoted with diamonds. The
blue dashed lines indicate an estimate of the flow vector factorization
breaking coefficient, Eq. (38), as a product of the flow magnitude
factorization breaking coefficient and the weighted flow angle factorization
breaking coefficient in Eq. (48).

Figure 9: Same as in Fig. 8 but for the triangular flow.

The two-bin factorization breaking coefficients studied in Sec. IV are very
sensitive to nonflow effects. Alternatively, a three-bin measure of flow
factorization breaking in the longitudinal direction can be used Khachatryan
_et al._ (2015), Eq. (8). This measure can be generalized to higher moments of
the flow Jia  _et al._ (2017); Aaboud  _et al._ (2018),

$\mathcal{R}^{(m)}_{n;V}(\eta)=\frac{\langle V_{n}^{m}(-\eta)V_{n}^{\star
m}(\eta_{ref})\rangle}{\langle V_{n}^{m}(\eta)V_{n}^{\star
m}(\eta_{ref})\rangle}\ .$ (37)

An analogous formula can be written for the factorization breaking coefficient
of the flow magnitude as follows:

$\mathcal{R}^{(m)}_{n;v}(\eta)=\frac{\langle
v_{n}^{m}(-\eta)v_{n}^{m}(\eta_{ref})\rangle}{\langle
v_{n}^{m}(\eta)v_{n}^{m}(\eta_{ref})\rangle}\ .$ (38)

A third measure can be defined to estimate the simple flow angle
decorrelation:

$\mathcal{R}^{s(m)}_{n;\Psi}(\eta)=\frac{\langle\cos\left(mn(\Psi_{n}(-\eta)-\Psi_{n}(\eta_{ref}))\right)\rangle}{\langle\cos\left(mn(\Psi_{n}(\eta)-\Psi_{n}(\eta_{ref}))\right)\rangle}\
.$ (39)

Using the local random component (K) for the flow in a given bin
$V(\eta)=V+K(\eta)$ and defining

$A_{1}=\frac{K(\eta)+K(\eta_{ref})}{2}\quad\text{and}\quad\Delta_{1}=\frac{K(\eta)-K(\eta_{ref})}{2}\
,$ (40)

and

$A_{2}=\frac{K(-\eta)+K(\eta_{ref})}{2}\quad\text{and}\quad\Delta_{2}=\frac{K(-\eta)-K(\eta_{ref})}{2}\
,$ (41)

one can write:

$\mathcal{R}^{(m)}_{n;V}(\eta)=\frac{\langle(V+A_{2}+\Delta_{2})^{m}(V+A_{2}-\Delta_{2})^{*m}\rangle}{\langle(V+A_{1}+\Delta_{1})^{m}(V+A_{1}-\Delta_{1})^{*m}\rangle}\
,$ (42)
$\mathcal{R}^{(m)}_{n;v}(\eta)=\frac{\langle|V+A_{2}+\Delta_{2}|^{m}|V+A_{2}-\Delta_{2}|^{m}\rangle}{\langle|V+A_{1}+\Delta_{1}|^{m}|V+A_{1}-\Delta_{1}|^{m}\rangle}\
.$ (43)

The expansion of the factorization breaking coefficients to second order in
$\delta^{2}$, taking into account that
$\langle|A|^{2}\rangle=C^{2}-\delta^{2}$, takes the form

$\mathcal{R}^{(m)}_{n;V}(\eta)=1-2m^{2}\frac{\langle
v^{2m-2}\delta_{2}^{2}\rangle-\langle v^{2m-2}\delta_{1}^{2}\rangle}{\langle
v^{2m}\rangle},$ (44)

for the flow vector decorrelation and

$\mathcal{R}^{(m)}_{n;v}(\eta)=1-m^{2}\frac{\langle
v^{2m-2}\delta_{2}^{2}\rangle-\langle v^{2m-2}\delta_{1}^{2}\rangle}{\langle
v^{2m}\rangle},$ (45)

for the flow magnitude decorrelation. We find that the deviation of the flow
factorization breaking coefficient $R_{n;V}^{(m)}$ from $1$ is twice as large
as the deviation of the flow magnitude coefficient $R_{n;v}^{(m)}$ from $1$:

$[1-R_{n;v}^{(m)}(\eta)]\simeq[1-R_{n;V}^{(m)}(\eta)]/2\ .$ (46)

The above relation is approximately fulfilled in numerical simulations and the
experimental data Bożek and Broniowski (2018). Moreover, we can find the
general formula for the simple flow angle decorrelation in Eq. (39) as
follows:

$\mathcal{R}^{s(m)}_{n;\Psi}=1-m^{2}\langle\frac{\delta_{2}^{2}}{v^{2}}\rangle+m^{2}\langle\frac{\delta_{1}^{2}}{v^{2}}\rangle\
.$ (47)

This is not the flow angle decorrelation that is measured in experiment. The
simple flow angle decorrelation is as strong or stronger than the full flow
vector decorrelation (Figs. 8 and 9). Similarly to the two-bin correlator, the
correct flow angle decorrelation weighted with the power of the flow magnitude
can be defined as

$\mathcal{R}^{(m)}_{n;\Psi}(\eta)=\frac{\langle
v_{n}^{2m}\cos\left(mn(\Psi_{n}(-\eta)-\Psi_{n}(\eta_{ref}))\right)\rangle}{\langle
v_{n}^{2m}\cos\left(mn(\Psi_{n}(\eta)-\Psi_{n}(\eta_{ref}))\right)\rangle}\ .$
(48)

Expanding to the order $\delta_{j}^{2}$ we find

$\mathcal{R}^{(m)}_{n;\Psi}\approx 1-m^{2}\frac{\langle
v^{m-2}\delta_{2}^{2}\rangle}{\langle v^{2m}\rangle}+m^{2}\frac{\langle
v^{2m-2}\delta_{1}^{2}\rangle}{\langle v^{2m}\rangle}\ .$ (49)

Comparing the above results with the results for the flow vector
decorrelation, Eq. (44), and flow magnitude decorrelation, Eq. (45), one
observes

$\mathcal{R}^{(m)}_{V}\approx\mathcal{R}^{(m)}_{v}\mathcal{R}^{(m)}_{\Psi}\ .$
(50)

We expect that the above factorization of the flow vector coefficient works
well for both elliptic and triangular flows and all centralities. This
factorization is observed in numerical calculations Bożek and Broniowski
(2018) (Figs. 8 and 9). It means that the relation
$\mathcal{R}^{(m)}_{V}/\mathcal{R}^{(m)}_{v}$ can be used as an estimate of
the weighted flow angle decorrelation $\mathcal{R}^{(m)}_{n;\Psi}$, Eq. (48).
On the other hand, the simple angle decorrelation, Eq. (39), is significantly
larger.

Figure 10: Comparison of three-bin (red lines) and four-bin (black lines) flow
vector factorization breaking coefficients for the elliptic flow (panel a))
and for the triangular flow (panel b)), for $30-40$% centrality. Red dash
lines show the decomposition of the flow vector factorization breaking
coefficient $\mathcal{R}_{V}^{(2)}$ into the flow magnitude and flow angle
decorrelation using Eq. (53).

Figure 11: The factorization breaking coefficients for the first (black line
and dots), second (red line and squares), and third (blue line and diamonds)
powers of the flow vector $\mathcal{R}^{(m)}_{V}$ compared with the second and
third powers $\Big{[}\mathcal{R}^{(1)}_{V}\Big{]}^{m}$ (dashed lines). Results
for the elliptic flow and the triangular flow are shown in panels (a) and (b)
respectively, for the centrality range $30-40$%.

In the experiment, due to nonflow correlations, only the three-bin flow vector
correlator $\mathcal{R}^{(m)}_{n;V}(\eta)$ can be measured. There is another
practical way to estimate flow angle decorrelation at different
pseudorapidities using a four-bin correlator Aaboud  _et al._ (2018),

$\mathcal{R}_{n;VV}=\frac{\langle
V_{n}(-\eta_{ref})V_{n}(-\eta)V_{n}^{*}(\eta)V_{n}^{*}(\eta_{ref})\rangle}{\langle
V_{n}(-\eta_{ref})V_{n}^{*}(-\eta)V_{n}(\eta)V_{n}^{*}(\eta_{ref})\rangle}.$
(51)

In the random model of flow decorrelation, the above four-bin correlator takes
the form (in the lowest order in $\delta$)

$\mathcal{R}_{n;VV}\approx 1-4\frac{\langle
v^{2}\delta_{2}^{2}\rangle}{\langle v^{4}\rangle}+4\frac{\langle
v^{2}\delta_{1}^{2}\rangle}{\langle v^{4}\rangle}\ .$ (52)

The four-bin correlator is an estimate of the weighted flow angle
factorization breaking coefficient, Eq. (48). The flow vector factorization
breaking coefficient $\mathcal{R}_{V}$ approximately factorizes into the flow
magnitude and flow angle factorization breaking coefficient,

$\mathcal{R}^{(2)}_{V}\approx\mathcal{R}_{n;VV}\mathcal{R}^{(2)}_{v}\ .$ (53)

Numerical results from the viscous hydrodynamic model for the correlators
$\mathcal{R}_{n;VV}$, $\mathcal{R}^{(2)}_{V}$, and the above factorization are
shown in Fig. 10. There is a good consistency between the results of the
hydrodynamic simulations and the factorization given in Eq. (53) (red dashed
line) in second- and third-order harmonics for both central (not shown) and
semicentral collisions.

Finally, we test in the hydrodynamic model the scaling of the factorization
breaking coefficients for different moments of the harmonic flow vectors:

$\mathcal{R}^{(m)}_{V}\approx\left[\mathcal{R}^{(1)}_{V}\right]^{m}\ .$ (54)

Figure 11 present a comparison of the factorization breaking coefficients for
the second and third moments of the flow vector $\mathcal{R}^{(m)}_{V}$ (for
$m=2-3$) (red and blue lines respectively) with the respective powers of the
factorization breaking coefficients of the flow vectors
$\Big{[}\mathcal{R}^{(1)}_{V}\Big{]}^{m}$ for $m=2-3$ (red and blue dashed
lines). The above relation is expected for fluctuation dominated flow
(triangular flow) from Eq. (44) derived in the random model of flow
decorrelation.

##  VII Conclusions

We analyze the decorrelation of the flow vectors in separate rapidity bins. A
simple random model of flow decorrelation is able to reproduce qualitatively
the scaling relations between factorization breaking coefficients. Numerical
simulations in the hydrodynamic model show that the flow vector magnitude and
flow vector angle decorrelations are approximately equal and sum up in the
total flow vector decorrelation. The same relation can be obtained in a random
model of flow decorrelation, where the flow in a small rapidity bin is written
as a sum of the average flow in the event and of a random vector component.
Assuming that the random component direction is independent of the average
flow and its magnitude is much smaller than the average flow, analytical
expressions for the factorization breaking coefficients for flow vectors, flow
magnitudes, and flow angles are given, with similar relations between them as
in the full hydrodynamic simulation.

The factorization breaking coefficients for higher powers of the flow vectors
shows a large deviation from $1$, reflecting the stronger decorrelation for
the second or third power of the flow vectors than for the flow vectors only.
In the random model of flow decorrelation this property comes from the general
analytical expressions for the correlations of different moments of flow
vectors. In the case when the flow is fluctuation dominated (triangular flow
or elliptic flow in central collisions) the factorization breaking
coefficients of different powers $m$ of the flow are related

$\rho^{(m)}(-\eta,\eta)\simeq\left[\rho^{(1)}(-\eta,\eta)\right]^{m}.$ (55)

The above relation is found analytically in the random decorrelation model, as
well as in numerical simulations in the hydrodynamic model.

The flow angle decorrelation is larger if the overall flow is small. This can
be observed on an event-by-event basis in the hydrodynamic simulations and is
encoded in the analytical expressions obtained in the random model of flow
decorrelation. As a consequence the flow decorrelation is larger for the
triangular flow and for the elliptic flow in central collisions than for the
elliptic flow in semicentral collisions.

Analytical expression of the factorization breaking coefficients are given for
the three- and four-bin measures of flow decorrelation in pseudorapidity. Such
measures are used in experimental analyses in order to reduce nonflow effects.
The random model of flow decorrelation reproduces qualitatively the relation
observed in experimental data and in hydrodynamic simulations:

  * •

the flow angle decorrelation is approximately one-half of the flow vector
decorrelation,

  * •

the decorrelation of the second or third powers of the flow is given as the
second or third powers of the flow vector decorrelation, when the overall flow
is fluctuation dominated.

The proposed random model of flow decorrelation in pseudorapidity is
surprisingly simple, yet it reproduces qualitatively a number of
phenomenological relations observed in experimental data or in realistic
hydrodynamic simulations. The model can serve as a way to understand the
fluctuations inherent in models of heavy-ion collisions. Moreover, any
deviations from these simple scalings in the experimental data could suggest
the existence of specific correlations between the average flow and the random
flow decorrelations, which could be interesting in the study of realistic
initial conditions for the hydrodynamic simulations of heavy-ion collisions.
The possible existence of a similar simple description of the decorrelation of
the flow in different transverse momentum bins is open and left for future
studies.

## Acknowledgments

P.B. acknowledges support from the National Science Centre Grant
2018/29/B/ST2/00244. H.M. thanks CERN-TH group for support. H.M. is funded by
the Cluster of Excellence Precision Physics, Fundamental Interactions, and
Structure of Matter (PRISMA+ EXC 2118/1) funded by the German Research
Foundation (DFG) within the German Excellence Strategy (Project ID 39083149).

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[table]capposition=top

# On the sum of the two largest signless Laplacian eigenvalues

Zi-Ming Zhou , Chang-Xiang He
College of Science,
University of Shanghai for Science and Technology, Shanghai, P. R. China.
Corresponding author; Email addresses: changxiang-he@163.com(C.-X. He)

###### Abstract

Let $G$ be a simple connected graph and let $S_{k}(G)$ be the sum of the first
$k$ largest signless Laplacian eigenvalues of $G$. It was conjectured by
Ashraf, Omidi and Tayfeh-Rezaibe in 2013 that $S_{k}(G)\leq
e(G)+\binom{k+1}{2}$ holds for $1\leq k\leq n-1$. They gave a proof for the
conjecture when $k=2$, but applied an incorrect key lemma. Therefore, the
conjecture is still open when $k=2$. In this paper, we prove that
$S_{2}(G)<e(G)+3$ is true for any graphs which also confirm the conjecture
when $k=2$.

Keywords: Graph; Signless Laplacian matrix; Sum of signless Laplacian
eigenvalue

##  1 Introduction

Let $G=(V(G),E(G))$ be a simple connected graph. The number of its vertices
and edges are denoted by $n(G)$ (or $n$ for short) and $e(G)$, respectively.
Let $N(v)$ denote the set of neighbors of vertex $v$ and $d(v)=|N(v)|$ denote
the degree of the vertex $v$ in $G$. For nonempty set $X\subset V(G)$, let
$N(X)$ denote the set of vertices outside $X$ and each vertex of $N(X)$ has at
least one neighbor in $X$. Let $e(X,Y)$ be the number of edges whose one
endpoint in $X$ and one endpoint in $Y$. For some vertex $v\notin X$, let
$N_{X}(v)$ denote the neighbors of vertex $v$ in $X$. Suppose that
$V^{\prime}$, $E^{\prime}$ are nonempty subsets of $V(G)$ and $E(G)$,
respectively. The subgraph of $G$ whose vertex set is $V^{\prime}$ and whose
edge set is the set of those edges of $G$ that have both ends in $V^{\prime}$
is called the induced subgraph of $G$, denoted by $G[V^{\prime}]$. The
subgraph of $G$ whose vertex set is the set of ends of edges in $E^{\prime}$
and whose edge set is $E^{\prime}$ is called the edge-induced subgraph of $G$,
denoted by $G[E^{\prime}]$. Two distinct edges of $G$ are called independent
if they have no common endpoint. A set of pairwise independent edges in $G$ is
called a matching. The maximum size of a matching in $G$ is known as the
matching number of $G$, denoted by $m(G)$. For two graphs $G_{1}$ and $G_{2}$,
the union of $G_{1}$ and $G_{2}$, denoted by $G_{1}\cup G_{2}$, is the graph
with vertex set $V(G_{1})\cup V(G_{2})$ and edge set $E(G_{1})\cup E(G_{2})$.
If $H$ is a subgraph of $G$, let $G-E(H)$ denote the graph obtained by
removing the edges in $H$ from $G$.

Let $A(G)$ be the adjacency matrix of $G$ and
$D(G)=diag(d(v_{1}),d(v_{2}),\ldots,d(v_{n}))$ be the diagonal matrix of
vertex degrees of $G$. The Laplacian matrix and the signless Laplacian matrix
of G are defined as $L(G)=D(G)-A(G)$ and $Q(G)=D(G)+A(G)$, respectively. For a
graph $G$, let $M=M(G)$ be a graph matrix associated with $G$. The
$M$-polynomial is defined as $P_{M}(G,x)=$ det$(xI-M)$, where $I$ is the
identity matrix. The $M$-eigenvalues are the roots of the $M$-polynomial.
Usually, the graph matrices considered are $A(G),L(G)$ and $Q(G)$. For a
square matrix $M$, the characteristic polynomial of $M$ is simply denoted by
$P(M,x)=$ det$(xI-M)$.

It is well known that both $L(G)$ and $Q(G)$ are positive semidefinite and
hence their eigenvalues are nonnegative real numbers. The eigenvalues of
$L(G)$ and $Q(G)$ are called the Laplacian eigenvalues and the signless
Laplacian eigenvalues of $G$, respectively, and are denoted by
$\mu_{1}(G)\geq\mu_{2}(G)\geq\cdots\geq\mu_{n}(G)$ (or
$\mu_{1}\geq\mu_{2}\geq\cdots\geq\mu_{n}$ for short) and $q_{1}(G)\geq
q_{2}(G)\geq\cdots\geq q_{n}(G)$ (or $q_{1}\geq q_{2}\geq\cdots\geq q_{n}$ for
short), respectively. Let $S_{k}(G)=\sum_{i=1}^{k}q_{i}(G)$ be the sum of the
first $k$ largest signless Laplacian eigenvalues of $G$.

In [1], Brouwer proposed the following conjecture for Laplacian eigenvalues.

######  Conjecture 1.

_For any graph $G$ with $n$ vertices and for any $k\in\\{1,2,\ldots,n\\}$,_

$\sum_{i=1}^{k}\mu_{i}(G)\leq e(G)+\binom{k+1}{2}.$

Haemers et al.[2] showed that Conjecture 1 is true for all trees and for
$k=2$, that is $\mu_{1}(G)+\mu_{2}(G)\leq e(G)+3$ for any graph $G$.

In analogy to Brouwer’s conjecture, Ashraf et al.[3] proposed the following
conjecture on signless Laplacian eigenvalues.

######  Conjecture 2.

_For any graph $G$ with $n$ vertices and for any $k\in\\{1,2,\ldots,n\\}$,_

$S_{k}(G)\leq e(G)+\binom{k+1}{2}.$

Clearly, Conjecture 2 is true for $k=n-1$ and $k=n$, respectively. And the
correctness of Conjecture 2 for trees follows from Brouwer’s conjecture since
$L(G)$ and $Q(G)$ are similar if $G$ is bipartite[4]. Ashraf et al.[3] proved
that Conjecture 2 is true for regular graphs, they also proved that Conjecture
2 is true for $k=2$, that is $S_{2}(G)\leq e(G)+3$ for any graph $G$, but the
key lemma ([3], Lemma 13) they used is incorrect which has a counterexample
in[5]. Zheng[5] proved that Conjecture 2 is true for all connected triangle-
free graphs when $k=2$. Yang and You[6] proved that Conjecture 2 is true for
unicyclic graphs and bicyclic graphs for all $k$ and tricyclic graphs when
$k\neq 3$.

Therefore, Conjecture 2 is still open when $k=2$. In this paper, we prove that
$S_{2}(G)<e(G)+3$ is true for any graphs which also confirm the conjecture 2
when $k=2$. Our main result is as follows.

######  Theorem 1.1.

For any graph $G$ with $n$ vertices,

$S_{2}(G)<e(G)+3.$

By using computer computations, we check Theorem1.1 holds for all graphs with
at most 10 vertices.

The organization of the remaining of the paper is as follows. In section 2, we
give some lemmas which are useful in the proof of Theorem 1.1. In section 3,
we give the proof of Theorem 1.1.

##  2 Preliminaries

######  Lemma 2.1.

For any graph $G$ with at most 10 vertices, $S_{2}(G)<e(G)+3$.

######  Lemma 2.2 ([4]).

Let $n$ be a positive integer.

  * (1)

The signless Laplacian eigenvalues of $K_{n}$ are $2n-2$ and $n-2$ with
multiplicities 1 and $n-1$, respectively.

  * (2)

The signless Laplacian eigenvalues of $K_{1,n-1}$ are $n$, 1 and 0 with
multiplicities 1, $n-2$, and 1, respectively.

The following result which can be found in matrix theory which plays an
important role in our proofs. For a symmetric matrix $M$ of order $n$, let
$\lambda_{1}(M)\geq\cdots\geq\lambda_{n}(M)$ be its eigenvalues.

######  Lemma 2.3 ([7]).

Let $A$ and $B$ be two real symmetric matrices of size $n$. Then for any
$1\leq k\leq n$,

$\sum^{k}_{i=1}\lambda_{i}(A+B)\leq\sum^{k}_{i=1}\lambda_{i}(A)+\sum^{k}_{i=1}\lambda_{i}(B).$

An immediate result of Lemma 2.3 is the following corollary which will be used
frequently.

######  Corollary 2.1.

Let $G$ be a graph of order $n$ such that there exist edge disjoint subgraphs
of $G$, say $G_{1},\cdots,G_{r}$, with $E(G)=\bigcup^{r}_{i=1}E(G_{i})$. Then
$S_{k}(G)\leq\sum^{r}_{i=1}S_{k}(G_{i})$ for any integer $k$ with $1\leq k\leq
n$.

The following result is also straightforward.

######  Lemma 2.4.

For some $k$, if Theorem1.1 holds for $G$ and $H$, then it does for $G\cup H$.

The next lemma is the key to our approach. It gives a sufficient condition for
the truth of Theorem1.1, that holds for almost all graphs.

######  Lemma 2.5.

If $G$ is a graph with a nonempty subgraph $H$ for which $S_{2}(H)\leq e(H)$,
then $S_{2}(G)<e(G)+3$.

###### Proof.

Assume that $G$ is a counterexample with a minimum possible number of edges.
By Corollary 2.1, we have $e(G)+3\leq S_{2}(G)\leq S_{2}(H)+S_{2}(G-H)$. This
implies that $S_{2}(G-H)\geq e(G-H)+3$, since $e(G)=e(H)+e(G-H)$ and
$S_{2}(H)\leq e(H)$, which contradicts the minimality of $e(G)$. ∎

In [8], Li and Guo propose the full Brouwer’s Laplacian spectrum conjecture
and prove the conjecture holds for $k=2$ with the unique extremal graph (non-
bipartite) which indicate $\mu_{1}(G)+\mu_{2}(G)<e(G)+3$ holds for bipartite
graphs. Using the fact that signless Laplacian matrix and Laplacian matrix are
similar for a bipartite graph we have the following.

######  Lemma 2.6.

For any bipartite graph $G$, $S_{2}(G)<e(G)+3$.

######  Lemma 2.7 ([4]).

Let $G$ be a graph with $n$ vertices and let $G^{\prime}$ be a graph obtained
from $G$ by inserting a new edge into $G$. Then the signless Laplacian
eigenvalues of $G$ and $G^{\prime}$ interlace, that is,

$q_{1}(G^{\prime})\geq q_{1}(G)\geq\cdots\geq q_{n}(G^{\prime})\geq q_{n}(G).$

For the sum of $k$ largest signless Laplacian eigenvalues of $G$ and
$G^{\prime}$, we can discuss similarly as what Haemers et al. did in [2] for
the sum of $k$ largest Laplacian eigenvalues of $G$. We improve it as follows.

######  Lemma 2.8.

Let $G$ be a graph with $n$ vertices. Suppose that there exist two non-
adjacent vertices $u,v\in V(G)$ such that $q_{k}(G)\geq d(u)+d(v)+2$ for some
integer $k$, $1\leq k\leq n$. If $G^{\prime}$ is the graph obtained from $G$
by inserting an edge $e=uv$ into $G$, then $S_{k}(G^{\prime})<S_{k}(G)+1$.

###### Proof.

For $i=1,\ldots,n$, define $\sigma_{i}=q_{i}(G^{\prime})-q_{i}(G)$. By Lemma
2.7, $\sigma_{i}\geq 0$, for any $i$. Let $d_{1}\geq\cdots\geq d_{n}$ and
$d_{1}^{\prime}\geq\cdots\geq d_{n}^{\prime}$ be vertex degrees of $G$ and
$G^{\prime}$, respectively. Recall that for any graph $H$, considering the
trace of the matrix $Q(H)^{2}$, we have

$\sum^{|V(H)|}_{i=1}q^{2}_{i}(H)=\sum_{v\in V(H)}d^{2}(v)+2e(H).$

Applying this fact, we have

$\displaystyle\sum^{n}_{i=1}q^{2}_{i}(G^{\prime})$
$\displaystyle=\sum^{n}_{i=1}d_{i}^{\prime 2}+2e(G^{\prime})$
$\displaystyle=\sum^{n}_{i=1}d_{i}^{2}+2e(G)+2d(u)+2d(v)+4$
$\displaystyle=\sum^{n}_{i=1}q^{2}_{i}(G)+2(d(u)+d(v)+2).$

This yields that

$\displaystyle 2q_{k}(G^{\prime})\sum^{k}_{i=1}\sigma_{i}$
$\displaystyle\leq\sum^{k}_{i=1}2\sigma_{i}q_{i}(G)$
$\displaystyle<\sum^{n}_{i=1}q^{2}_{i}(G^{\prime})-\sum^{n}_{i=1}q^{2}_{i}(G)$
$\displaystyle=2(d(u)+d(v)+2).$

Since $q_{k}(G^{\prime})\geq d(u)+d(v)+2$,
$S_{k}(G^{\prime})-S_{k}(G)=\sum^{k}_{i=1}\sigma_{i}<1$, and the result
follows. ∎

In particular, if $S_{k}(G)<e(G)+\binom{k+1}{2}$, then
$S_{k}(G^{\prime})<e(G^{\prime})+\binom{k+1}{2}$.

$H$-join operation of graphs: Let $H$ be a graph with $V(H)=\\{1,2,...,k\\}$,
and $G_{i}$ be disjoint $r_{i}$-regular graphs of order $n_{i}(i=1,2,...,k)$.
Then the graph $\bigvee_{H}\\{G_{1},G_{2},...,G_{k}\\}$ is formed by taking
the graphs $G_{1},G_{2},...,G_{k}$ and joining every vertex of $G_{i}$ to
every vertex of $G_{j}$ whenever $i$ is adjacent to $j$ in $H$.

Define $N_{i}=\sum_{j\in N_{H}(i)}n_{j}$ for each $i\in V(H)$ and

$\displaystyle(Q^{\pi})_{ij}=\begin{cases}2r_{i}+N_{i},&i=j,\\\ n_{j},&ij\in
E(H),\\\ 0,&otherwise.\end{cases}$

######  Lemma 2.9 ([9]).

Let $H$ be a graph with $V(H)=\\{1,2,...,k\\}$, and $G_{i}$ be a
$r_{i}$-regular graph of order $n_{i}(i=1,2,...,k)$. If
$G=\bigvee_{H}\\{G_{1},G_{2},...,G_{k}\\}$, then

$\displaystyle
P_{Q}(G,x)=P(Q^{\pi},x)\prod_{i=1}^{k}\frac{P_{Q}(G_{i},x-N_{i})}{x-2r_{i}-N_{i}}.$

Lexicographically ordered $k$-sets: Fix a number $k$ between 1 and $n$,
inclusive. Consider all possible sets of $k$ distinct numbers between 1 and
$n$. Order the elements of each set by using a lexicographic or dictionary
order. Denote these sets by $S_{1},S_{2},\ldots,S_{\tbinom{n}{k}}$ so that
$S_{1}<S_{2}<\ldots<S_{\tbinom{n}{k}}$.

######  Example 1.

Let $n=4$ and $k=2$. Then $S_{1}=\\{1,2\\},\ S_{2}=\\{1,3\\},\
S_{3}=\\{1,4\\},\ S_{4}=\\{2,3\\},\ S_{5}=\\{2,4\\},\ S_{6}=\\{3,4\\}$.

Compound matrix: Let $A_{i,j}$ be the $k\times k$ matrix formed by the
intersection, in $A$, of the rows whose numbers are in the set $S_{i}$ and the
columns whose numbers are in the set $S_{j}$. Let $A$ be a complex matrix of
order $n$, define $C_{k}(A)$ as its $k$-th compound matrix, which is the
matrix of size $\tbinom{n}{k}\times\tbinom{n}{k}$ whose $(i,j)$-th entry is
the determinant of the matrix $A_{i,j}$.

Additive compound matrix: The matrix $C_{k}(I+tA)$ is an
$\tbinom{n}{k}\times\tbinom{n}{k}$ matrix, each entry of which is a polynomial
in $t$ of degree at most $k$. The $k$-th additive compound matrix is defined
as

$\displaystyle\Delta_{k}(A)=\frac{d}{dt}C_{k}(I_{n}+tA)\big{|}_{t=0}.$

For additive compound matrix, the following lemma is a well-known result.

######  Lemma 2.10 ([10]).

If $\lambda_{1},\ \lambda_{2},\ldots,\lambda_{n}$ are all eigenvalues of $A$,
then the eigenvalues of $\Delta_{k}(A)$ are the $\tbinom{n}{k}$ distinct sums
of the $\lambda_{i}$ taken $k$ at a time.

######  Corollary 2.2.

If $q_{1},\ q_{2},\ldots,q_{n}$ are signless eigenvalues of $G$, and
$P(Q^{\pi},x)$ contains the largest two roots of $P_{Q}(G,x)$, then the
largest eigenvalue of $\Delta_{2}(Q^{\pi})$ equals $S_{2}(G)$.

######  Lemma 2.11 ([11]).

Let $G$ be a connected graph. Then $q_{1}(G)\leq max\\{d(v)+m(v)\mid v\in
V(G)\\}$, with equality holding if and only if $G$ is either semiregular
bipartite or regular, where $m(v)$ is the average of the degrees of the
vertices of $G$ adjacent to the vertex $v$.

##  3 Proof of Theorem 1.1

By Lemma 2.1, 2.4 and 2.6, we may assume that $G$ is a non-bipartite connected
graph with at least 11 vertices. Noting that for $H=4K_{2}$ or $H=3K_{1,2}$,
one has $S_{2}(H)=e(H)$, using Lemma 2.5, we may assume $G$ contains neither
$H=4K_{2}$ nor $H=3K_{1,2}$ as a subgraph. It is sufficient to consider only
graphs $G$ whose matching number $m(G)$ is at most 3. In the following, we
prove Theorem 1.1 for $m(G)=1,2$ and 3, respectively.

######  Lemma 3.1.

Let $G$ be a graph with $m(G)=1$. Then $S_{2}(G)<e(G)+3$.

###### Proof.

Let $n=|V(G)|$. Since $m(G)=1$, it is easily checked that either
$G=K_{1,k-1}\cup(n-k)K_{1}$ for some $k$, $1\leq k\leq n$ or
$G=K_{3}\cup(n-3)K_{1}$. By Lemma 2.2, the assertion holds. ∎

A pendant neighbor of $G$ is a vertex adjacent to a pendant vertex. Let $p(G)$
be the number of pendant vertices of $G$ and $q(G)$ be the number of pendant
neighbors.

######  Lemma 3.2.

Let $G$ be a graph with $m(G)=2$. Then $S_{2}(G)<e(G)+3.$

###### Proof.

We may assume that $G$ is a connected graph with at least 11 vertices. First
suppose that $G$ has no $K_{3}$ as a subgraph, since $m(G)=2$, $G$ has no odd
cycle as a subgraph. It is obvious that $G$ is a bipartite graph, the
assertion holds by Lemma 2.6.

Next suppose that $G$ has a subgraph $H=K_{3}$ with $V(H)=\\{u,v,w\\}$.

If every edge of $G$ has at least one endpoint in $V(H)$, since $m(G)=2$, we
only need to consider $q(G)=0,1$ and 2, respectively. Then $G$ is isomorphic
to $G_{1}\ (1\leq t_{1}\leq n-2,\ 0\leq t_{2},\ t_{3}\leq n-3)$ or $G_{2}\
(7\leq t=n-4)$ in Fig. 1.

If there exists an edge $e=ab$ has no endpoint in $V(H)$, let
$M=V(G)-\\{a,b,u,v,w\\}$, since $m(G)=2$, there are no edges between $V(H)$
and $M$, and all vertices in $M$ are adjacent to one of the endpoints of $e$,
say $a$. Hence there are no edges between $b$ and $V(H)$, we only need to
consider $|N(a)\cap V(H)|=1$ and 3, respectively (isomorphic to $G_{2}$ when
$|N(a)\cap V(H)|=2$). Then $G$ is isomorphic to $G_{3}\ (7\leq t=n-4)$ or
$G_{4}\ (7\leq t=n-4)$ in Fig. 1.

Figure 1: Possible forms of graphs $G$ with $m(G)=2$

When $G\cong G_{1}$, the graph $G$ has an obvious partition
$G=\bigvee_{H}\\{G_{1},G_{2},...,G_{k}\\}(k=5)$ where the corresponding
quotient matrix is

$\displaystyle Q^{\pi}=\left(\begin{array}[]{ccccc}2&0&0&1&1\\\ 0&1&0&1&0\\\
0&0&1&0&1\\\ t_{1}&t_{2}&0&t_{1}+t_{2}+1&1\\\
t_{1}&0&t_{3}&1&t_{1}+t_{3}+1\end{array}\right).$

By Lemma 2.9 and direct calculation, the characteristic polynomial of $Q(G)$
is

$\displaystyle P_{Q}(G,x)=P(Q^{\pi},x)(x-2)^{t_{1}-1}(x-1)^{t_{2}+t_{3}-2}.$

The characteristic polynomial of 2-th additive compound matrix
$\Delta_{2}(Q^{\pi})$ is defined as $P(\Delta_{2}(Q^{\pi}),x)$. By using
computer computations, we checked all coefficients of
$P(\Delta_{2}(Q^{\pi}),x+e(G)+3)$ are positive, therefore its roots are all
negative, which indicates the largest root of $P(\Delta_{2}(Q^{\pi}),x)$ (the
largest eigenvalue of $\Delta_{2}(Q^{\pi})$) less than $e(G)+3$. Since
$q_{2}>2$ by Lemma 2.7, $P(Q^{\pi},x)$ contains the largest two roots of
$P_{Q}(G,x)$. By Corollary 2.2, we have $S_{2}(G)<e(G)+3$.

When $G\cong G_{2}$, the graph $G$ has an obvious partition
$G=\bigvee_{H}\\{G_{1},G_{2},...,G_{k}\\}(k=4)$ where the corresponding
quotient matrix is

$\displaystyle Q^{\pi}=\left(\begin{array}[]{cccc}1&1&0&0\\\ t&t+2&2&0\\\
0&1&4&1\\\ 0&0&2&2\end{array}\right).$

By Lemma 2.9 and direct calculation, the characteristic polynomial of $Q(G)$
is

$\displaystyle P_{Q}(G,x)=P(Q^{\pi},x)(x-2)(x-1)^{t-1}.$

By simple calculations we obtain

$\displaystyle P(Q^{\pi},t+3.25)$
$\displaystyle=0.25t^{3}+0.94t^{2}-6.33t-13.89>0,$ $\displaystyle
P(Q^{\pi},4.75)$ $\displaystyle=-0.3t-19.98<0,$ $\displaystyle P(Q^{\pi},2)$
$\displaystyle=4t>0,$ $\displaystyle P(Q^{\pi},1)$ $\displaystyle=-t<0,$
$\displaystyle P(Q^{\pi},0)$ $\displaystyle=8>0,$

which indicates $q_{1}(G)<t+3.25$ and $q_{2}(G)<4.75$, we have
$S_{2}(G)<t+8=e(G)+3$.

When $G\cong G_{3}$, we use the quotient matrix approach, when $G\cong G_{4}$,
we use the 2-th additive compound matrix approach, details are deferred in the
Appendix. This completes the proof. ∎

######  Lemma 3.3.

Let $G$ be a graph with $m(G)=3$. Then $S_{2}(G)<e(G)+3.$

###### Proof.

Without loss of generality, we may suppose that $e_{1}=a_{1}b_{1}$,
$e_{2}=a_{2}b_{2}$ and $e_{3}=a_{3}b_{3}$ are three independent edges in $G$.

We first suppose that $G$ contains $K_{3}+2K_{2}$ as a subgraph and
$V(K_{3})=\\{a_{1},b_{1},u\\}$. Let $V_{1}=V(K_{3}+2K_{2})$,
$M^{\prime}=V(G)-V_{1}$. Then $|M^{\prime}|\geq 4$. Since $m(G)=3$, so
$M^{\prime}$ is an independent set and $e(V(K_{3}),M^{\prime})=0$. Since $G$
has neither $3K_{1,2}$ nor $4K_{2}$ as a subgraph and $|M^{\prime}|\geq 4$,
there exists exactly one vertex $a_{3}$(w.l.o.g.) which is adjacent to
vertices of $M^{\prime}$ and $d(b_{3})=1$.

Let $V_{2}=\\{a_{1},b_{1},u,a_{2},b_{2}\\}$, $E_{1}=\\{a_{3}w|w\in V_{2}\\}$.
Then $G[E_{1}]$ is a star with center $a_{3}$, and $G-E_{1}$ is a disjoint
union of a star $K_{1,|M^{\prime}|+1}$ with center $a_{3}$ and a graph
$G[V_{2}]$ containing $K_{3}+K_{2}$.

If $G[V_{2}]$ is connected, by Lemma 2.11, we have
$q_{1}(G[V_{2}])<e(G[V_{2}])$. Since $n(K_{1,|M^{\prime}|+1})\geq 6$, so
$q_{2}(G[V_{2}])\leq q_{1}(K_{1,|M^{\prime}|+1})$. Hence
$S_{2}(G-E_{1})=q_{1}(G[V_{2}])+q_{1}(K_{1,|M^{\prime}|+1})<e(G[V_{2}])+e(K_{1,|M^{\prime}|+1})+1=e(G-E_{1})+1$.
Thus by Corollary 2.1, $S_{2}(G)\leq
S_{2}(G[E_{1}])+S_{2}(G-E_{1})<e(G[E_{1}])+2+e(G-E_{1})+1=e(G)+3$.

If $G[V_{2}]$ is disconnected, then $G[V_{2}]=K_{3}+K_{2}$. Let
$t=d(a_{2})+d(b_{2})$. We have $t=3,4$. It is not hard to see that
$m(G-e_{2})=2$ and $G-e_{2}$ contains disjoint union of a star
$K_{1,|M^{\prime}|+1}$ with center $a_{3}$ and $K_{3}$. Therefore, by Lemma
2.7, $q_{2}(G-e_{2})\geq t$. Using Lemmas 3.2 and 2.8, we find that
$S_{2}(G)<S_{2}(G-e_{2})+1<(e(G-e_{2})+3)+1=e(G)+3$, as required.

Next suppose that $G$ has no subgraph $K_{3}+2K_{2}$. Since $m(G)=3$,
$M=V(G)-V(\\{e_{1},e_{2},e_{3}\\})$ is an independent set and $|M|\geq 5$.
Since $G$ has no $4K_{2}$ and $K_{3}+2K_{2}$ as subgraph, either
$N_{M}(a_{i})=\emptyset$ or $N_{M}(b_{i})=\emptyset$ for $i=1,2,3$. Without
loss of generality, we may assume that $N(M)\subseteq\\{a_{1},a_{2},a_{3}\\}$.
We consider the following three cases.

Figure 2: Possible forms of graphs containing a $P_{5}$

Case 1: $|N(M)|=3$.

We have $|N(M)|=\\{a_{1},a_{2},a_{3}\\}$. Since G has no $3K_{1,2}$, the
bipartite subgraph $G-\\{b_{1},b_{2},b_{3}\\}$ has no perfect matching. By
Hall’s Theorem, there exists a subset of $\\{a_{1},a_{2},a_{3}\\}$ with 2
elements, say $\\{a_{2},a_{3}\\}$, such that $|N(\\{a_{2},a_{3}\\})|\cap
M|=1$. This means that there exists exactly one vertex $y\in M$ which is
adjacent to both $a_{2}$ and $a_{3}$. If $d(b_{1})\geq 2$, then we clearly
find a subgraph isomorphic to $3S_{3}$ in $G$, a contradiction. Therefore,
$d(b1)=1$. Let $V_{1}=\\{a_{2},b_{2},a_{3},b_{3},y\\}$, $E_{1}=\\{a_{1}w|w\in
V_{1}\\}$. Then $G[E_{1}]$ is a star with center $a_{1}$, and $G-E_{1}$ is a
disjoint union of a star $K_{1,|M|}$ with center $a_{1}$ and a graph
$G[V_{1}]$ containing $P_{5}$. Since G has no $K_{3}+2K_{2}$, two edges
$a_{2}a_{3}$ and $b_{2}b_{3}$ can not exist at the same time, therefore
$G[V_{1}]$ is isomorphic to graphs in Fig. 2.

By direct calculation, we have $q_{1}(G[V_{1}])<e(G[V_{1}])$. Since
$n(K_{1,|M|})\geq 6$, so $q_{2}(G[V_{1}])\leq q_{1}(K_{1,|M|})$. Hence
$S_{2}(G-E_{1})=q_{1}(G[V_{1}])+q_{1}(K_{1,|M|})<e(G[V_{1}])+e(K_{1,|M|})+1=e(G-E_{1})+1$.
Thus by Corollary 2.1, $S_{2}(G)\leq
S_{2}(G[E_{1}])+S_{2}(G-E_{1})<e(G[E_{1}])+2+e(G-E_{1})+1=e(G)+3$.

Case 2: $|N(M)|=2$.

Without loss of generality, suppose that $|N(M)|=\\{a_{1},a_{2}\\}$. Since
$m(G)=3$, $b_{1}$ is not adjacent to $b_{2}$. If $b_{1}$ is adjacent to
$a_{3}$ or $b_{3}$, then changing the role of $e_{1}$, $e_{2}$, $e_{3}$ by
three independent edges $\\{a_{1},z\\}$, $e_{2}$, $e_{3}$ for some vertex
$z\in M\cap N(a_{1})$, we have Case 1. Therefore, we may assume that $b_{1}$,
and similarly $b_{2}$, is adjacent to none of the vertices $a_{3}$ and
$b_{3}$. Hence we only need to consider $|N(\\{a_{3},b_{3}\\})|=1$ and 2,
respectively. Consider the existence of the edge $a_{1}a_{2}$, possible forms
of graph $G$ are in Fig. 3.

When $G\cong G_{i}$, $i=5,6,8,11$, let $t=d(a_{3})+d(b_{3})$, we have $t=3,4$.
It is not hard to see that $m(G-e_{3})=2$ and $G-e_{3}$ contains two disjoint
stars $S_{t}$ with centers $a_{1}$ and $a_{2}$. Therefore, by Lemma 2.7,
$q_{2}(G-e_{3})\geq t$. Using Lemmas 3.2 and 2.8, we find that
$S_{2}(G)<S_{2}(G-e_{3})+1<(e(G-e_{3})+3)+1=e(G)+3$, as required.

When $G\cong G_{i}$, $i=7,9,10,12,13$, we use the 2-th additive compound
matrix approach, details are deferred in the Appendix.

Figure 3: Possible forms of graphs $G$ with $m(G)=3$ and $|N(M)|=2$

Case 3: $|N(M)|=1$.

Without loss of generality, suppose that $|N(M)|=\\{a_{1}\\}$. If $d(b1)\geq
2$, then we clearly find three independent edges $e^{\prime}_{1}$,
$e^{\prime}_{2}$, $e^{\prime}_{3}$ in $G$ such that the set
$M^{\prime}=V(G)-V(\\{e^{\prime}_{1},e^{\prime}_{2},e^{\prime}_{3}\\}$ is an
independent set and $|N(M^{\prime})|\geq 2$ which is dealt with as the
previous cases. Therefore, $d(b_{1})=1$. Let
$V_{1}=\\{a_{2},b_{2},a_{3},b_{3}\\}$, $E_{1}=\\{a_{1}w|w\in V_{1}\\}$. Then
$G[E_{1}]$ is a star with center $a_{1}$, and $G-E_{1}$ is a disjoint union of
a star $K_{1,|M|+1}$ with center $a_{1}$ and a graph $G[V_{1}]$ containing
$2K_{2}$. Therefore $G[V_{1}]$ is isomorphic to graphs in Fig. 4.

Figure 4: Possible forms of graphs containing $2K_{2}$

By direct calculation, we have $q_{1}(G[V_{1}])<e(G[V_{1}])+1$, hence
$S_{2}(G-E_{1})=q_{1}(G[V_{1}])+q_{1}(K_{1,|M|+1})<e(G[V_{1}])+1+e(K_{1,|M|})+1=e(G-E_{1})+2$.
If $|N(a_{1})|\cap V_{1}|=1$, which means $|E_{1}|=1$, by Corollary 2.1,
$S_{2}(G)\leq
S_{2}(G[E_{1}])+S_{2}(G-E_{1})<e(G[E_{1}])+1+e(G-E_{1})+2=e(G)+3$. Hence we
only need to consider $|N(a_{1})|\cap V_{1}|=2$, 3 and 4, respectively.
Possible forms of graph $G$ are in Fig. 5.

Figure 5: Possible forms of graphs $G$ with $m(G)=3$ and $|N(M)|=1$

When $G\cong G_{i}$, $i=14,18,33$, it is obvious that $G$ is a bipartite
graph, the assertion holds by Lemma 2.6.

When $G\cong G_{i}$, $i=17,24,25$, let $t=d(a_{3})+d(b_{3})$ (pick $a_{2}$ and
$b_{2}$ when $G\cong G_{25}$), we have $t=3,4$. It is not hard to see that
$m(G-e_{3})=2$ and $G-e_{3}$ contains two disjoint stars $S_{t}$ with centers
$a_{1}$ and $b_{2}$ (contains disjoint union of a star $S_{t}$ with center
$a_{1}$ and a $K_{3}$ when $G\cong G_{25}$). Therefore, by Lemma 2.7,
$q_{2}(G-e_{3})\geq t$. Using Lemmas 3.2 and 2.8, we find that
$S_{2}(G)<S_{2}(G-e_{3})+1<(e(G-e_{3})+3)+1=e(G)+3$, as required.

When $G$ is isomorphic to other graphs in Fig. 5, we use the quotient matrix
approach, details are deferred in the Appendix. This completes the proof. ∎

Proof of Theorem 1.1: The result follows immediately from Lemmas 2.5, 3.1,
3.2and 3.3.

Remark. Ashraf et al.[3] proved that Conjecture 2 is asymptotically tight for
any $k$ for the graph $K_{k}\vee\overline{K_{t}}$, the join of $K_{k}$ and the
empty graph $\overline{K_{t}}$. Oliveira et al.[12] proved that inequality
$q_{1}(G)+q_{2}(G)\leq e(G)+3$ is tighter for the graph $S^{+}_{n}$ (the star
graph with an additional edge) among all firefly graphs and also tighter to
$S^{+}_{n}$ than to the graphs $K_{k}\vee\overline{K_{t}}$, they also
conjectured that $S^{+}_{n}$ minimizes $f(G)=e(G)-q_{1}(G)-q_{2}(G)$ among all
graphs $G$ on $n$ vertices.

## Appendix

Table 1: 2-th additive compound matrix approach No. | Corresponding quotient matrix $Q^{\pi}$ | Characteristic polynomial $P_{Q}(G,x)$
---|---|---
$G_{4}$ | $\left(\begin{array}[]{ccc}1&1&0\\\ t&t+3&3\\\ 0&1&5\end{array}\right)$ | $P(Q^{\pi},x)(x-2)^{2}(x-1)^{t-1}$
$G_{7}$ | $\left(\begin{array}[]{cccccc}2&0&0&1&1&0\\\ 0&1&0&1&0&0\\\ 0&0&1&0&1&0\\\ t_{1}&t_{2}&0&t_{1}+t_{2}+3&1&2\\\ t_{1}&0&t_{3}&1&t_{1}+t_{3}+1&0\\\ 0&0&0&1&0&3\end{array}\right)$ | $P(Q^{\pi},x)(x-2)^{t_{1}-1}(x-1)^{t_{2}+t_{3}-1}$
$G_{9}$ | $\left(\begin{array}[]{ccccccc}2&0&0&1&1&0&0\\\ 0&1&0&1&0&0&0\\\ 0&0&1&0&1&0&0\\\ t_{1}&t_{2}&0&t_{1}+t_{2}+2&0&1&1\\\ t_{1}&0&t_{3}&0&t_{1}+t_{3}+1&0&1\\\ 0&0&0&1&0&2&1\\\ 0&0&0&1&1&1&3\end{array}\right)$ | $P(Q^{\pi},x)(x-2)^{t_{1}-1}(x-1)^{t_{2}+t_{3}-2}$
$G_{10}$ | $\left(\begin{array}[]{cccccc}2&0&0&1&1&0\\\ 0&1&0&1&0&0\\\ 0&0&1&0&1&0\\\ t_{1}&t_{2}&0&t_{1}+t_{2}+2&0&2\\\ t_{1}&0&t_{3}&0&t_{1}+t_{3}+2&2\\\ 0&0&0&1&1&4\end{array}\right)$ | $P(Q^{\pi},x)(x-2)^{t_{1}}(x-1)^{t_{2}+t_{3}-2}$
$G_{12}$ | $\left(\begin{array}[]{ccccccc}2&0&0&1&1&0&0\\\ 0&1&0&1&0&0&0\\\ 0&0&1&0&1&0&0\\\ t_{1}&t_{2}&0&t_{1}+t_{2}+3&1&1&1\\\ t_{1}&0&t_{3}&1&t_{1}+t_{3}+2&0&1\\\ 0&0&0&1&0&2&1\\\ 0&0&0&1&1&1&3\end{array}\right)$ | $P(Q^{\pi},x)(x-2)^{t_{1}-1}(x-1)^{t_{2}+t_{3}-2}$
$G_{13}$ | $\left(\begin{array}[]{cccccc}2&0&0&1&1&0\\\ 0&1&0&1&0&0\\\ 0&0&1&0&1&0\\\ t_{1}&t_{2}&0&t_{1}+t_{2}+3&1&2\\\ t_{1}&0&t_{3}&1&t_{1}+t_{3}+3&2\\\ 0&0&0&1&1&4\end{array}\right)$ | $P(Q^{\pi},x)(x-2)^{t_{1}}(x-1)^{t_{2}+t_{3}-2}$
Table 2: Quotient matrix approach No. | Corresponding quotient matrix $Q^{\pi}$ |  Characteristic polynomial $P_{Q}(G,x)$
---|---|---
$G_{3}$ | $\left(\begin{array}[]{cccc}1&1&0&0\\\ t&t+1&1&0\\\ 0&1&3&2\\\ 0&0&1&3\end{array}\right)$ |  $[x^{4}-(t+8)x^{3}+(6t+19)x^{2}-(7t+16)x+4](x-1)^{t}$
$G_{15}$ | $\left(\begin{array}[]{ccccc}1&1&0&0&0\\\ t&t+3&2&0&1\\\ 0&1&3&0&0\\\ 0&0&0&1&1\\\ 0&1&0&1&2\end{array}\right)$ |  $[x^{5}-(t+10)+x^{4}+(6t+34)x^{3}-(10t+48)x^{2}+(3t+27)x-4](x-1)^{t}$
$G_{16}$ | $\left(\begin{array}[]{cccc}1&1&0&0\\\ t&t+4&2&2\\\ 0&1&3&0\\\ 0&1&0&3\end{array}\right)$ |  $[x^{3}-(t+8)x^{2}+(3t+15)x-8](x-3)(x-1)^{t+1}$
$G_{19}$ | $\left(\begin{array}[]{cccccc}1&1&0&0&0&0\\\ t&t+2&1&0&0&1\\\ 0&1&2&1&0&0\\\ 0&0&1&2&1&0\\\ 0&0&0&1&2&1\\\ 0&1&0&0&1&2\end{array}\right)$ |  $[x^{4}-(t+8)x^{3}+(5t+20)x^{2}-(5t+17)x+4](x^{2}-3x+1)(x-1)^{t-1}$
$G_{20}$ | $\left(\begin{array}[]{cccccc}1&1&0&0&0&0\\\ t&t+2&0&1&1&0\\\ 0&0&1&1&0&0\\\ 0&1&1&3&1&0\\\ 0&1&0&1&3&1\\\ 0&0&0&0&1&1\end{array}\right)$ |  $[x^{4}-(t+8)x^{3}+(5t+18)x^{2}-(3t+15)x+4](x^{2}-3x+1)(x-1)^{t-1}$
$G_{21}$ | $\left(\begin{array}[]{cccccc}1&1&0&0&0&0\\\ t&t+3&1&1&1&0\\\ 0&1&2&1&0&0\\\ 0&1&1&3&1&0\\\ 0&1&0&1&3&1\\\ 0&0&0&0&1&1\end{array}\right)$ |  $[x^{6}-(t+13)x^{5}+(9t+62)x^{4}-(26t+140)x^{3}+(27t+158)x^{2}-(8t+84)x+16](x-1)^{t-1}$
$G_{22}$ | $\left(\begin{array}[]{cccccc}1&1&0&0&0&0\\\ t&t+3&1&1&0&1\\\ 0&1&2&1&0&0\\\ 0&1&1&3&1&0\\\ 0&0&0&1&2&1\\\ 0&1&0&0&1&2\end{array}\right)$ |  $[x^{6}-(t+13)x^{5}+(9t+63)x^{4}-(27t+145)x^{3}+(31t+165)x^{2}-(11t+87)x+16](x-1)^{t-1}$
$G_{23}$ | $\left(\begin{array}[]{cccccc}1&1&0&0&0&0\\\ t&t+4&1&1&1&1\\\ 0&1&2&1&0&0\\\ 0&1&1&3&1&0\\\ 0&1&0&1&3&1\\\ 0&1&0&0&1&2\end{array}\right)$ |  $[x^{4}-(t+11)x^{3}+(6t+37)x^{2}-(7t+47)x+20](x-3)(x-1)^{t}$
$G_{26}$ | $\left(\begin{array}[]{cccccc}1&1&0&0&0&0\\\ t&t+2&1&0&0&1\\\ 0&1&2&1&0&0\\\ 0&0&1&3&1&1\\\ 0&0&0&1&2&1\\\ 0&1&0&1&1&3\end{array}\right)$ |  $[x^{5}-(t+12)x^{4}+(9t+51)x^{3}-(24t+94)x^{2}+(19t+71)x-16](x-1)^{t}$
$G_{27}$ | $\left(\begin{array}[]{cccccc}1&1&0&0&0&0\\\ t&t+2&0&1&0&1\\\ 0&0&1&1&0&0\\\ 0&1&1&4&1&1\\\ 0&0&0&1&2&1\\\ 0&1&0&1&1&3\end{array}\right)$ |  $[x^{6}-(t+13)x^{5}+(10t+61)x^{4}-(31t+135)x^{3}+(35t+150)x^{2}-(12t+80)x+16](x-1)^{t-1}$
$G_{28}$ | $\left(\begin{array}[]{cccccc}1&1&0&0&0&0\\\ t&t+3&1&1&0&1\\\ 0&1&2&1&0&0\\\ 0&1&1&4&1&1\\\ 0&0&0&1&2&1\\\ 0&1&0&1&1&3\end{array}\right)$ |  $[x^{6}-(t+15)x^{5}+(11t+84)x^{4}-(40t+228)x^{3}+(58t+319)x^{2}-(29t+221)x+60](x-1)^{t-1}$
$G_{29}$ | $\left(\begin{array}[]{ccccc}1&1&0&0&0\\\ t&t+3&1&0&2\\\ 0&1&2&1&0\\\ 0&0&1&3&2\\\ 0&1&0&1&4\end{array}\right)$ |  $[x^{5}-(t+13)x^{4}+(9t+59)x^{3}-(23t+115)x^{2}+(16t+92)x-24](x-2)(x-1)^{t-1}$
$G_{30}$ | $\left(\begin{array}[]{ccccc}1&1&0&0&0\\\ t&t+3&0&1&2\\\ 0&0&1&1&0\\\ 0&1&1&4&2\\\ 0&1&0&1&4\end{array}\right)$ |  $[x^{5}-(t+13)x^{4}+(9t+57)x^{3}-(21t+107)x^{2}+(10t+86)x-24](x-2)(x-1)^{t-1}$
$G_{31}$ | $\left(\begin{array}[]{ccccc}1&1&0&0&0\\\ t&t+4&1&1&2\\\ 0&1&2&1&0\\\ 0&1&1&4&2\\\ 0&1&0&1&4\end{array}\right)$ |  $[x^{4}-(t+12)x^{3}+(7t+43)x^{2}-(8t+56)x+24](x-3)(x-2)(x-1)^{t-1}$
$G_{32}$ | $\left(\begin{array}[]{cccccc}1&1&0&0&0&0\\\ t&t+2&1&1&0&0\\\ 0&1&3&1&0&1\\\ 0&1&1&3&1&0\\\ 0&0&0&1&2&1\\\ 0&0&1&0&1&2\end{array}\right)$ |  $[x^{4}-(t+10)x^{3}+(7t+32)x^{2}-(11t+39)x+16](x^{2}-3x+1)(x-1)^{t-1}$
$G_{34}$ | $\left(\begin{array}[]{ccccc}1&1&0&0&0\\\ t&t+3&1&2&0\\\ 0&1&3&2&0\\\ 0&1&1&3&1\\\ 0&0&0&2&2\end{array}\right)$ |  $[x^{5}-(t+12)x^{4}+(8t+49)x^{3}-(17t+86)x^{2}+(8t+64)x-16](x-3)(x-1)^{t-1}$
$G_{35}$ | $\left(\begin{array}[]{ccc}1&1&0\\\ t&t+4&4\\\ 0&1&5\end{array}\right)$ |  $[x^{3}-(t+10)x^{2}+(5t+25)x-16](x-3)^{2}(x-1)^{t}$
$G_{36}$ | $\left(\begin{array}[]{cccccc}1&1&0&0&0&0\\\ t&t+2&1&1&0&0\\\ 0&1&4&1&1&1\\\ 0&1&1&3&1&0\\\ 0&0&1&1&3&1\\\ 0&0&1&0&1&2\end{array}\right)$ |  $[x^{6}-(t+15)x^{5}+(12t+84)x^{4}-(48t+228)x^{3}+(77t+319)x^{2}-(41t+221)x+60](x-1)^{t-1}$
$G_{37}$ | $\left(\begin{array}[]{cccc}1&1&0&0\\\ t&t+2&0&2\\\ 0&0&4&2\\\ 0&1&2&3\end{array}\right)$ |  $[x^{4}-(t+10)x^{3}+(7t+29)x^{2}-(8t+28)x+8](x-3)(x-2)(x-1)^{t-1}$
$G_{38}$ | $\left(\begin{array}[]{cccc}1&1&0&0\\\ t&t+2&2&0\\\ 0&1&5&2\\\ 0&0&2&2\end{array}\right)$ |  $[x^{3}-(t+9)x^{2}+(6t+18)x-8](x-3)(x-2)(x-1)^{t}$
$G_{39}$ | $\left(\begin{array}[]{ccccc}1&1&0&0&0\\\ t&t+3&1&2&0\\\ 0&1&4&2&1\\\ 0&1&1&3&1\\\ 0&0&1&2&3\end{array}\right)$ |  $[x^{5}-(t+14)x^{4}+(10t+68)x^{3}-(28t+146)x^{2}+(23t+139)x-48](x-3)(x-1)^{t-1}$
$G_{40}$ | $\left(\begin{array}[]{ccccc}1&1&0&0&0\\\ t&t+3&2&1&0\\\ 0&1&5&1&1\\\ 0&1&2&3&0\\\ 0&0&2&0&2\end{array}\right)$ |  $[x^{5}-(t+14)x^{4}+(10t+67)x^{3}-(27t+142)x^{2}+(20t+136)x-48](x-3)(x-1)^{t-1}$
$G_{41}$ | $\left(\begin{array}[]{cccc}1&1&0&0\\\ t&t+4&2&2\\\ 0&1&5&2\\\ 0&1&2&3\end{array}\right)$ |  $[x^{4}-(t+13)x^{3}+(8t+51)x^{2}-(11t+75)x+36](x-3)^{2}(x-1)^{t-1}$
$G_{42}$ | $\left(\begin{array}[]{cccc}1&1&0&0\\\ t&t+2&2&0\\\ 0&1&5&2\\\ 0&0&2&4\end{array}\right)$ |  $[x^{4}-(t+12)x^{3}+(9t+43)x^{2}-(16t+56)x+24](x-3)(x-2)(x-1)^{t-1}$
$G_{43}$ | $\left(\begin{array}[]{ccccc}1&1&0&0&0\\\ t&t+3&2&1&0\\\ 0&1&5&1&1\\\ 0&1&2&4&1\\\ 0&0&2&1&3\end{array}\right)$ |  $[x^{4}-(t+13)x^{3}+(9t+51)x^{2}-(15t+75)x+36](x-3)^{2}(x-1)^{t-1}$
$G_{44}$ | $\left(\begin{array}[]{ccc}1&1&0\\\ t&t+4&4\\\ 0&1&7\end{array}\right)$ |  $[x^{3}-(t+12)x^{2}+(7t+35)x-24](x-3)^{3}(x-1)^{t-1}$

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