# Classifying optical (out)bursts in cataclysmic variables: the distinct
observational characteristics of dwarf novae, micronovae, stellar flares and
magnetic gating

Krystian Iłkiewicz Astronomical Observatory, University of Warsaw, Al.
Ujazdowskie 4, 00-478 Warszawa, Poland  Centre for Extragalactic Astronomy,
Department of Physics, Durham University, DH1 3LE, UK  Simone Scaringi Centre
for Extragalactic Astronomy, Department of Physics, Durham University, DH1
3LE, UK  Martina Veresvarska Centre for Extragalactic Astronomy, Department of
Physics, Durham University, DH1 3LE, UK  Domitilla De Martino
INAF–Osservatorio Astronomico di Capodimonte, salita Moiariello 16, I-80131,
Napoli, Italy  Colin Littlefield Department of Physics, University of Notre
Dame, Notre Dame, IN 46556, USA  Christian Knigge Department of Physics and
Astronomy. University of Southampton, Southampton SO17 1BJ, UK  John A. Paice
Centre for Extragalactic Astronomy, Department of Physics, Durham University,
DH1 3LE, UK  Anwesha Sahu Department of Physics, University of Warwick, Gibbet
Hill Road, Coventry CV4 7AL, UK

###### Abstract

Cataclysmic variables can experience short optical brightenings, which are
commonly attributed to phenomena such as dwarf novae outbursts, micronovae,
donor flares or magnetic gating bursts. Since these events exhibit similar
observational characteristics, their identification has often been ambiguous.
In particular, magnetic gating bursts and micronovae have been suggested as
alternative interpretations of the same phenomena. Here we show that the
timescales and energies separate the optical brightenings into separate
clusters consistent with their different classifications. This suggest that
micronovae and magnetic gating bursts are in fact separate phenomena. Based on
our finding we develop diagnostic diagrams that can distinguish between these
bursts/flares based on their properties. We demonstrate the effectiveness of
this approach on observations of a newly identified intermediate polar, CTCV
J0333-4451, which we classify as a magnetic gating system. CTCV J0333-4451 is
the third high spin-to-orbital period ratio intermediate polar with magnetic
gating, suggesting that these bursts are common among these rare systems.

Cataclysmic variable stars — Optical bursts — Dwarf novae — Flare stars — Time
domain astronomy — DQ Herculis stars

##  1 Introduction

Cataclysmic variables (CVs) are binary stars in which an accreting white dwarf
accretes material typically from a main sequence mass donor. CVs can exhibit
various forms of short optical bursts. Among the best-studied bursts in CVs
are dwarf novae. Dwarf nova eruptions are caused by a thermal-viscous
instability in the accretion disc (Lasota, 2001). Based on the properties of
the bursts, dwarf novae are divided into several subtypes (e.g. Warner, 2003).
However, their duration and amplitude depend mainly on the size of the
accretion disc. In particular, their duration can range from a few days (e.g.
Cannizzo et al., 2012) to years (e.g. Iłkiewicz et al., 2023).

In CVs with a magnetic white dwarf the accretion disc is truncated. This makes
dwarf novae outbursts less likely to occur in magnetic systems compared to
non-magnetic CVs (Hameury & Lasota, 2017). In these magnetic systems, two
other kinds of bursts seem more likely: micronovae and magnetic gating bursts.
Micronovae occur on timescales of hours, show energies 10-6 times smaller
compared to classical novae, and have outburst shapes similar to Type I X-ray
bursts in accreting neutron stars (Scaringi et al., 2022b). The proposed
mechanism behind micronovae are localized thermonuclear runaways in
magnetically confined accretion streams (Scaringi et al., 2022a). Another
explanation for micronovae could be magnetic reconnection events in the
magnetic disc (Schaefer et al., 2022). On the other hand, in the magnetic
gating model, accretion is halted by the white dwarf magnetic field until
enough pressure builds up in the accretion disc allowing for a short burst of
accretion (D’Angelo & Spruit, 2010, 2012; Scaringi et al., 2017). While
magnetic gating is a widely accepted phenomenon, the reality of micronovae is
still under question. In particular, magnetic gating bursts have been proposed
as an alternative interpretation of the claimed micronovae (Hameury et al.,
2022).

Among the most rare short brightenings observed in CVs are stellar flares
originating from the mass donor (Ramsay et al., 2021). They are expected to
have similar properties to flares and superflares from single stars. However,
they are rarely observed in CVs. This is likely because the mass donor is
tidally locked, making it spin rapidly, and rapidly rotating main sequence
stars are unlikely to show flaring activity (Ramsay et al., 2020).

Here we explore observational properties of short optical bursts in CVs. Based
on the studied systems we discover that the short bursts observed in CVs fall
into separate clusters based on their burst energies and timescales. In
particular, we show that magnetic gating systems and micronovae display
distinct sets of characteristics, implying different physical mechanisms are
at play in these bursts. Based on our findings, we advocate the use of
diagnostic diagrams to identify the nature of a given burst. We employ the
newly proposed method to classify bursts in the newly identified intermediate
polar CTCV J0333-4451 (hereafter J0333). We show that J0333 falls in the
cluster of magnetic gating systems.

##  2 Observations and Literature Data

We employed observations of J0333 made by Transiting Exoplanet Survey
Satellite (Ricker et al., 2015, TESS). The observations were carried out
during TESS sectors 30 and 31 (23 September – 18 November 2020) with a 120s
cadence. The data was processed with the Science Processing Operations Center
(SPOC) pipeline (Jenkins et al., 2016).

In order to flux-calibrate TESS observations we followed a method employed by
Scaringi et al. (2022b). Namely, we found nearly simultaneous observations of
TESS and ASAS-SN. Then, we fitted a linear relationship between the TESS SAP
flux in electrons per second and ASAS-SN flux in $g$ band. We then used this
relationship to scale the TESS data to fluxes in Jy. The J0333 light curve is
presented in Fig. 1. We employed the same flux calibration method with TESS
data of J0333 and observations of other objects.

Figure 1: Sectors 30 (top) and 31 (bottom) of the $TESS$ observations of J0333
with flux scale calibrated to the ASAS-SN $g$ filter.The top inset figure show
a zoom on an asymmetrical burst, while the bottom inset figure shows a
symmetrical burst.

Here we consider properties of short bursts observed in CVs, i.e. bursts with
duration of order of days or shorter. Due to the short timescales of the
bursts, we limited the comparison to bursts observed with $Kepler$ and $TESS$,
where the cadence is higher and more consistent compared to ground based
observations. Our analysis is intended for CVs with short orbital periods
($<$10 hours) and will have limited application to CVs with evolved donors.
This is because evolved stars can experience stellar flares that have energies
similar to the energies of micronovae (see e.g. fig. 9 of Tu et al., 2021).
However, the only CV with short bursts and a long orbital period observed by
either $TESS$ or $Kepler$ is V2487 Oph, with the distance to the object being
too uncertain for a meaningful analysis (Schaefer et al., 2022). In order to
estimate the energies of bursts in each system we assumed distances based on a
Gaia DR3 parallax (Bailer-Jones et al., 2021). We also exclude classical novae
from our analysis, even though they can occur on short timescales (e.g.
Sokolovsky et al., 2023). This is because classical novae have luminosities
several orders of magnitude larger compared to other bursts in CVs. We assume
that interstellar reddening to all of the systems is negligible. This is
consistent with extinction A${}_{g}<$0.4 mag estimated with a 3D reddening map
for all objects for which data was available (Green et al., 2019).

In order to measure the properties of short bursts in CVs we reanalyzed
already published TESS observations of V1025 Cen (Littlefield et al., 2022)
and TW Pic (Scaringi et al., 2022c). Moreover, in order to measure individual
bursts in MV Lyr we flux calibrated the $Kepler$ light curve to the $V$ band
using calibration of Scaringi et al. (2017). In the case of TV Col, EI UMa and
ASASSN-19bh, we corrected the published burst peak luminosities by subtracting
the quiescent flux of the systems (Scaringi et al., 2022b). The measurements
of burst in CP Pup are directly taken from Veresvarska et al. (2023). We
supplemented our sample with dwarf novae outbursts reported by Otulakowska-
Hypka et al. (2016) that had a duration of seven days or less and were
observed by TESS. We note that the constraints on the duration of outbursts
exclude most intermediate polars with suspected dwarf nova type outbursts
(e.g. GK Per, V455 And). However, we measured a suspected dwarf nova outbursts
in one intermediate polar, FS Aur.

Table 1: Properties of short optical bursts observed in CVs together with the
reference to the source of their identification. The luminosities and energies
are in $TESS$ or $Kepler$ bands.

Object | Outburst type | Peak optical | Total optical | Burst | Frequency | Reference
---|---|---|---|---|---|---
luminosity [erg/s] | energy [erg] | duration [days] | [day-1]
J0333 | Magnetic gating | (0.9$\pm$0.5)$\times 10^{32}$ | (13.2$\pm$9.8)$\times 10^{35}$ | 0.58$\pm$0.18 | 0.13 | 1
TW Pic | Magnetic gating | (1.5$\pm$0.3)$\times 10^{32}$ | (1.9$\pm$1.2)$\times 10^{35}$ | 0.05$\pm$0.03 | 16.11 | 2
MV Lyr | Magnetic gating | (3.0$\pm$0.7)$\times 10^{32}$ | (8.7$\pm$4.3)$\times 10^{35}$ | 0.12$\pm$0.03 | 6.39 | 3
V1025 Cen | Magnetic gating | (0.9$\pm$0.1)$\times 10^{32}$ | (4.6$\pm$3.5)$\times 10^{35}$ | 0.36$\pm$0.06 | 0.35 | 4
TV Col | Micronova | (0.8$\pm$0.2)$\times 10^{34}$ | (1.2$\pm$0.5)$\times 10^{38}$ | 0.52$\pm$0.13 | 0.05 | 5
EI UMa | Micronova | (1.9$\pm$0.2)$\times 10^{34}$ | (2.6$\pm$0.3)$\times 10^{38}$ | 0.36$\pm$0.07 | 0.04 | 5
ASASSN-19bh | Micronova | 3.4$\times 10^{34}$ | 11.6$\times 10^{38}$ | 6.96 | 0.04 | 5
CP Pup | Micronova | (0.32$\pm$0.06)$\times 10^{34}$ | 0.6$\times 10^{38}$ | 0.8$\pm$0.2 | 0.017 | 6
MQ Dra | Donor flare | 2.6$\times 10^{30}$ | 2.2$\times 10^{33}$ | 0.035 | 0.012 | 7
V1504 Cyg | Dwarf nova | (3.6$\pm$1.3)$\times 10^{32}$ | (4.4$\pm$0.3)$\times 10^{37}$ | 4.3$\pm$0.2 | 0.09 | 8
IX Dra | Dwarf nova | (2.2$\pm$0.4)$\times 10^{32}$ | (2.5$\pm$0.6)$\times 10^{37}$ | 3.9$\pm$0.2 | 0.14 | 8
WX Hyi | Dwarf nova | (1.5$\pm$0.2)$\times 10^{32}$ | (1.3$\pm$0.3)$\times 10^{37}$ | 2.8$\pm$0.4 | 0.18 | 8
SS UMi | Dwarf nova | (8.8$\pm$0.3)$\times 10^{31}$ | (8.9$\pm$1.7)$\times 10^{36}$ | 3.8$\pm$0.2 | 0.09 | 8
FS Aur | Dwarf nova (IP) | (1.3$\pm$0.1)$\times 10^{32}$ | (2.0$\pm$0.1)$\times 10^{37}$ | 5.7$\pm$0.1 | 0.04 | 8
YZ Cnc | Dwarf nova | (3.7$\pm$0.8)$\times 10^{32}$ | (6.8$\pm$1.9)$\times 10^{37}$ | 5.8$\pm$1.2 | 0.09 | 8
V485 Cen | Dwarf nova | (4.7$\pm$0.1)$\times 10^{31}$ | (4.5$\pm$0.2)$\times 10^{36}$ | 3.8$\pm$0.5 | 0.11 | 8
VW Hyi | Dwarf nova | (2.4$\pm$0.3)$\times 10^{32}$ | (3.2$\pm$0.6)$\times 10^{37}$ | 6.2$\pm$0.9 | 0.02 | 8
X Leo | Dwarf nova | (8.0$\pm$0.1)$\times 10^{32}$ | (1.8$\pm$0.1)$\times 10^{38}$ | 6.6$\pm$0.3 | 0.09 | 8
BI Ori | Dwarf nova | (3.7$\pm$0.2)$\times 10^{32}$ | (8.5$\pm$0.1)$\times 10^{37}$ | 7.0$\pm$0.3 | 0.07 | 8
AT Cnc | Dwarf nova | (6.9$\pm$0.3)$\times 10^{32}$ | (2.1$\pm$0.3)$\times 10^{38}$ | 7.7$\pm$0.3 | 0.05 | 8

References: (1) This work; (2) Scaringi et al. (2022c); (3) Scaringi et al.
(2017); (4) Littlefield et al. (2022); (5) Scaringi et al. (2022b); (6)
Veresvarska et al. (2023); (7) Ramsay et al. (2021); (8) Otulakowska-Hypka et
al. (2016)

We compared the bursts in CVs to stellar flares from main sequence stars as
they can occur on similar timescales. The only CV with flares originating from
the donor observed by $TESS$ is MQ Dra (Ramsay et al., 2021). MQ Dra was below
the detection limit of ASAS-SN during the $TESS$ observations. Hence, we
employed the Zwicky Transient Facility (ZTF; Masci et al. 2019) observations
in $g$ filter to flux calibrate $TESS$ data. Since main sequence stars can
experience superflares that are more energetic than the normal flare in MQ
Dra, we expanded the comparison to superflares in single main sequence stars
(Tu et al., 2020). The superflares properties have been transformed to $g$
filter from bolometric values using corrections from Chen et al. (2019).

In order to estimate the frequency of bursts we follow eq. 9 of Tu et al.
(2020), i.e. the frequency of bursts is equal to the number of bursts divided
by the continuous monitoring period. In the case of bursts that were occurring
only during a low state of the system, we measured bursts frequency only
during the time interval when the bursts were present, rather than the entire
monitoring window (i.e. during a low state of MV Lyr and TW Pic, see e.g.
Scaringi et al. 2022c).

The start of a bursts was assumed to be at a time when the flux of the object
visibly rose above the quiescent or noise level. The burst end was assumed to
be at a time when the brightness returned to the pre-burst level. We note that
in some of the objects the measurements of start and end point of a bursts
might have been affected by the changes in brightness due to orbital
variability. Therefore, the measured duration of bursts might have a
systematic error of up to a few percent.

The collected sample of bursts in CVs is presented in Table 2. The final
reported values are the mean of measurements for all of the observed bursts
with the reported range corresponding to the largest deviation from the mean
of an individual outburst. We note that none of the measured energies or
luminosities have bolometric correction. However, they are measured in a
consistent fashion, i.e. they are measured in the $TESS$ or $Kepler$ bands and
calibrated to either $g$ or $V$ bands.

##  3 Results

As a result of the comparison between the bursts we discovered that the bursts
properties appear to fall into separate clusters which seem to not be
connected. This suggests different physical mechanism behind bursts in each
cluster. In fact, these clusters appear to be consistent with the
classification suggested in the literature, being micronovae, magnetic gating
bursts, dwarf novae, and donor flares (Fig. 2). We note that Hameury et al.
(2022) questioned the micronova interpretation and suggested that micronovae
can be interpreted as magnetic gating bursts instead. However, it is
immediately clear that systems classified as micronovae have energies orders
of magnitude higher compared to bursts that were identified as magnetic gating
(Table 2). This confirms that magnetic gating and micronovae are two different
phenomena (Scaringi et al., 2022b). The previous confusion between micronovae
and magnetic gating is likely due to the fact that energies of the bursts were
not included in the analysis of Hameury et al. (2022).

Figure 2: Properties of short bursts in CVs with their identification from
literature. Superflares from single main sequence stars are plotted for
comparison. Effect of extinction of A${}_{g}=0.5$ mag on the measured
properties is marked with a black line.

We propose that the four panels in Fig. 2 can be used as diagnostic diagrams
that can differentiate between the burst types. A distinction between magnetic
gating bursts and stellar flares based solely on energies is not possible.
Instead, one has to rely on a relationship between flares/bursts frequency and
their average duration (Fig. 2). In particular, there is an apparent anti-
correlation between magnetic gating bursts average duration and their
frequency. The anti-correlation between the frequency of magnetic gating
bursts and their duration may be expected, as lower burst frequency implies a
higher mass that was halted in the accretion disc between the bursts. We note
that the micronovae measured frequency is likely significantly overestimated,
since they likely remain dormant for significantly longer periods of time
compared to their continuous monitoring time. The exception is CP Pup, where
the recurrence time of $\sim$60 days was estimated (Veresvarska et al., 2023).
However, micronovae can be distinguished from other classes of bursts using
other properties.

###  3.1 Bursts in J0333

J0333 was classified as a CV by Augusteijn et al. (2010). J0333 showed a blue
continuum and emission lines of He I, He II, Fe II, and Balmer series. Based
on radial velocity study Augusteijn et al. (2010) estimated an orbital period
of the system to be 0.06 days. However, the orbital period determination was
hindered by insufficient sampling of the spectroscopic observations. Moreover,
the authors reported strong, short-term variations in the photometric
observations of J0333. While Augusteijn et al. (2010) classified J0333 as a
dwarf nova there are no recorded outbursts of the system in the literature.

J0333 experienced six bursts with varying amplitude and duration during the
TESS monitoring period (Fig. 1). Majority of bursts appear symmetrical, while
some of the larger bursts show a fast rise and a slow decline. Moreover, the
first bursts observed displayed a secondary brightening during decline.

After masking the bursts we performed a timing analysis of J0333 $TESS$ data.
A Lomb-Scargle periodogram (Lomb, 1976; Scargle, 1982) revealed a variability
at frequency of 14.89686(6) d-1 (97 min), which we identify as the orbital
period (Fig. 3). This orbital period is consistent with the radial velocity
study done by Augusteijn et al. (2010). Moreover, we discover variability at a
frequency of 37.6422(1) d-1 (38 min). We associate this variability with the
white dwarf spin period, suggesting an intermediate polar nature of J0333. The
source was detected in X-rays by $Swift-XRT$ and catalogued as 2SXPS
J033320.3-445139 (Evans et al., 2020). The $Swift-XRT$ signal-to-noise ratio
was insufficient for a detailed timing analysis although hints of a
variability at the 38 min spin period is found. In addition, the X-ray
spectrum is rather hard and consistent with an optically thin plasma at a
temperature of $\sim$7 keV (Section B), further corroborating the intermediate
polar identification. However, we note that the J0333 X-ray luminosity of
$\sim 10^{32}$ erg/s suggests an unexpectedly high mass transfer rate for a
system with a short orbital period.

Figure 3: Lomb-Scargle periodogram of J0333 $TESS$ observations after the
bursts have been masked. Detected periodic variability corresponds to the
orbital period ($\Omega$) and the white dwarf spin period ($\omega$).

The properties of J0333 bursts observed by $TESS$ are presented in Table 2.
The energies of bursts are similar to both magnetic gating bursts and stellar
flares. However, the relatively long duration of bursts identifies J0333 as a
magnetic gating system (Fig. 2).

##  4 Discussion and Conclusions

We showed that short bursts in CVs display distinct observational properties
that divide them into separate groups consistent with their literature
classification. Based on that we proposed diagnostic diagrams that can
distinguish between the short optical bursts types in CVs. The main conclusion
from these diagrams regards intermediate polars. Namely, short dwarf nova
outbursts in intermediate polars were speculated to be less likely to occur
compared to non-magnetic systems (Hameury & Lasota, 2017). However, the
properties of bursts in an intermediate polar FS Aur (Neustroev et al., 2013)
are only consistent with dwarf nova nature of the system (Fig. 2). Moreover,
micronovae and magnetic gating intermediate polars are separated on the
diagnostic diagrams, contrary to the recent suggestion in the literature
(Hameury et al., 2022). We note that this does not confirm the physical
mechanism suggested for these bursts in the literature, but simply it implies
a different physical mechanism are at play in micronovae and magnetic gating
systems.

While all micronovae are separated from other classes of short bursts, there
is a clear divide between them. Namely, the burst in ASASSN-19bh has a
duration and total energy one order of magnitude larger compared to other
micronovae. Together with the different shapes of bursts this mimics the two
types of Type-I X-ray bursts, as was noted by Scaringi et al. (2022b).
However, the comparison to Type-I X-ray bursts is limited due to the fact that
the nuclear reactions expected in Type-I X-ray bursts and micronovae differ.
Moreover, when a larger sample of systems is discovered, it will be possible
to improve the populations in the diagnostic diagrams allowing to confirm or
disprove the segregation of different micronovae. Nevertheless, the shared
relationship between the total energy released and peak flux during outburst
of the currently known sample seem to suggest that they are indeed two classes
of the same phenomenon (Fig. 2). If we consider the short duration micronovae
alone, it seems that there is an apparent relationship between the burst
duration and peak luminosity. While only three such systems are known, a
possibility of using short duration micronovae as distance indicators should
be investigated when a larger sample of objects will be discovered.

Bursts in intermediate polars have been proposed to be connected to the
appearance of superhumps (Mukai & Pretorius, 2023). However, since we have
shown that micronovae and magnetic gating bursts seem to be two separate
phenomena, it seems that superhumps are only connected to the occurrence of
short duration micronovae and do not appear in magnetic gating systems.

We identified J0333 as an intermediate polar with an orbital period of 97 min
and a white dwarf spin period of 38 min. Moreover, we derived diagnostic
diagrams that identified short bursts discovered in J0333 as a signature of
magnetic gating. J0333 has a relatively high spin-to-orbital period ratio of
0.39. The orbital period of J0333 and its spin-to-orbital period ratio is very
close to what is observed in EX Hya, V598 Peg, V1025 Cen and DW Cnc (see fig.
5 of Littlefield et al., 2023). Interestingly, EX Hya, V1025 Cen and DW Cnc
variability has been interpreted as magnetic gating bursts (Mhlahlo et al.,
2007; Duffy et al., 2022; Littlefield et al., 2022). This suggests that
magnetic gating in intermediate polars with a high spin-to-orbital period
ratio may be common.

This work was supported by Polish National Science Center grant Sonatina
2021/40/C/ST9/00186. MV acknowledges the support of the Science and Technology
Facilities Council (STFC) studentship ST/W507428/1. DdM acknowledges financial
support from INAF AstroFund-2022 grant ”FANS”. AS acknowledges the Warwick
Astrophysics PhD prize scholarship made possible thanks to a generous
philanthropic donation. This paper includes data collected by the TESS
mission. Funding for the TESS mission is provided by the NASA’s Science
Mission Directorate.

##  Appendix A Long term variability of J03333

Variability of J0333 on long timescales was analysed using data from the
Catalina Real-time Transient Survey (Drake et al., 2009, CRTS), All-Sky
Automated Survey for Supernovae (Shappee et al., 2014; Kochanek et al., 2017,
ASAS-SN) and Gaia mission (Gaia Collaboration et al., 2016; Babusiaux et al.,
2023). The long-term variability of J0333 is presented in Figure 4. Similar
bursts to what is observed in TESS are present during most of the monitoring
period in the $g$ band (Fig. 4). Presence of the bursts before that time is
inconclusive due to lower cadence of the data, but at least one burst seemed
to be observed in the $V$ band one day after the $Swift$ pointing. Both the
X-ray flux and UV flux in the $UVM2$ band seemed constant during the $Swift$
pointing, suggesting that the burst was over by the time of $Swift$
observation.

Figure 4: A light curve of J0333 in $V$, Gaia $G$ and $g$ filters. The time of
TESS monitoring is marked with a grey area. Time of the Swift pointing is
marked with a blue line. The Swift pointing was one day after an apparent
brightening in the $V$ band.

##  Appendix B The X-ray emission of J0333

J0333 was observed by $Swift-XRT$ on Oct. 19, 2008 for $\sim$21 h accumulating
a total of 10 ks (ObsID: 00037302001). Is was found at a count rate of
0.05$cts\,s^{-1}$ in the 0.3-10 keV range. The sparse X-ray coverage due to
the spacecraft orbit and the low S/N did not allow a period search but folding
the data at the 38 min period a variability with amplitude of
19${}^{+10}_{-9}\%$ was found. The X-ray spectrum averaged over the whole
observation is equally fitted (C-statistic) with an absorbed power law with
index 1.84${}^{+0.23}_{-0.22}$ ($\chi_{red}^{2}$=1.14) and an optically thin
plasma apec with temperature of 6.6${}^{+5.5}_{-1.9}$ keV adopting solar
abundances ($\chi_{red}^{2}$=1.2) models. The more physical apec model fit
gives an hydrogen column density of $\rm N_{H}=3.8^{+4.4}_{-0.2}\times
10^{20}\,cm^{-2}$ consistent with the small distance of J0333 and the total
column density towards the source (HI4PI Collaboration et al., 2016), while
the power law fit gives a much higher value $\rm N_{H}=1.2\times
10^{21}\,cm^{-2}$. The unabsorbed X-ray flux for the apec fit was $\rm
F_{0.3-10}=1.9\times 10^{-12}\,erg\,cm^{-2}\,s^{-1}$ and the bolometric flux
over a dummy range of 0.1-100 keV resulted in $\rm\sim 2.2\times
10^{-12}\,erg\,cm^{-2}\,s^{-1}$. At the Gaia DR3 distance of 578 pc the X-ray
luminosity was found to be $\rm\sim 8.8\times 10^{31}\,erg\,s^{-1}$. The lack
of better quality data and higher energy coverage prevents to infer a possible
temperature gradient and hence a possible estimate of the white dwarf mass.

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u_{i,t}}{\partial\alpha}=-\big{(}g_{i,t}-1\big{)}\Bigg{(}u_{i,t}+\tilde{m}_{i,t-1}-\bigg{(}\mu_{0}+C_{m}\sum_{s=1}^{t-1}\psi_{s}\bigg{)}_{i}\Bigg{)}\frac{\partial\varphi_{i,t}}{\partial\alpha}-\frac{\partial(C_{k}\psi_{t})_{i}}{\partial\alpha},$
(175)

where we use the fact that $\ln(|G_{t}|)=\sum_{i=1}^{n}\ln(|g_{i,t}|)$.
Therefore, in vector form, equations (173) and (175) can be written by

$\frac{\partial}{\partial\alpha}\ln(|G_{t}|)=(g_{t}-i_{n})^{\prime}\frac{\partial\varphi_{t}}{\partial\alpha}$
(176)

and

$\frac{\partial
u_{t}}{\partial\alpha}=-\text{diag}\\{\delta_{t}\\}\frac{\partial\varphi_{t}}{\partial\alpha}-\frac{\partial(C_{k}\psi_{t})}{\partial\alpha}=-\big{[}L_{1}\delta_{t}:\dots:L_{n}\delta_{t}\big{]}\frac{\partial\varphi_{t}}{\partial\alpha}-\frac{\partial(C_{k}\psi_{t})}{\partial\alpha},$
(177)

respectively. Let us denote vectorizations of the matrices $C_{k}$ and $C_{m}$
by $c_{k}:=\text{vec}(C_{k})$ and $c_{m}:=\text{vec}(C_{m})$, respectively.
Consequently, since $\frac{\partial(C_{k}\psi_{t})}{\partial
c_{k}^{\prime}}=\frac{\partial((\psi_{t}^{\prime}\otimes
I_{n})c_{k})}{\partial c_{k}^{\prime}}=(\psi_{t}^{\prime}\otimes I_{n})$ and
$\frac{\partial(C_{m}\psi_{s})}{\partial
c_{m}^{\prime}}=(\psi_{s}^{\prime}\otimes I_{n})$, partial derivatives of the
white noise process $u_{t}$ and $\ln(|G_{t}|)$ with respect to the parameters
$\mu_{0}$, $c_{k}$, and $c_{m}$ are given by the equations (165)–(167). Also,
as $\frac{\partial(C_{m}\psi_{t})}{\partial
c_{m}^{\prime}}=\psi_{t}^{\prime}\otimes I_{n}$, one obtains equation (168).
That completes the proof. ∎

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# Sharp bounds on the zeroth-order general Randić index of trees in terms of
domination number 111This work was supported by the National Natural Science
Foundation of China [61773020] and Postgraduate Scientific Research Innovation
Project of Hunan Province [CX20200033]. Corresponding author: Jianping Li
(lijianping65@nudt.edu.cn).

Chang Liu, Jianping Li
College of Liberal Arts and Sciences, National University of Defense
Technology,
Changsha, China, 410073.

###### Abstract

The zeroth-order general Randić index of graph $G=(V_{G},E_{G})$, denoted by
${}^{0}R_{\alpha}(G)$, is the sum of items $(d_{v})^{\alpha}$ over all
vertices $v\in V_{G}$, where $\alpha$ is a pertinently chosen real number. In
this paper, we obtain the sharp upper and lower bounds on ${}^{0}R_{\alpha}$
of trees with a domination number $\gamma$, in intervals
$\alpha\in(-\infty,0)\cup(1,\infty)$ and $\alpha\in(0,1)$, respectively. The
corresponding extremal graphs of these bounds are also characterized.
AMS Subject Classification: 05C50; 05C35; 05C69
Keywords: The zeroth-order general Randić index; Extremal trees; Domination
number

##  1 Introduction

Let $G$ be a graph with vertex vertex $V_{G}$ and edge set $E_{G}$. The
general Randić index is defined as

$R_{\alpha}=R_{\alpha}(G)=\sum_{uv\in E_{G}}(d_{u}d_{v})^{\alpha},$

where $d_{v}$ denotes the degree of a vertex $v\in V(G)$, and $\alpha$ is an
arbitrary real number. It’s widely known that $R_{-\frac{1}{2}}$, i.e., the
Randić index in original sense, was introduced by the chemist Milan Randić
[18] under the name connectivity index or branching index in 1975, which has a
good correlation with a variety of physico-chemical properties of alkanes,
such as enthalpy of formation, boiling point, parameters in the Antoine
equation, surface area and solubility in water, etc. In the past 30 to 40
years, the Randić index has been widely utilized in physics, chemistry,
biology, and complex networks [5, 19], and many interesting mathematical
properties have been obtained [4, 11, 13]. In 1998, Bollobás and Erdös [1]
generalized this index by replacing $-\frac{1}{2}$ with a real number
$\alpha$, and called it the general Randić index, denoted by
$R_{\alpha}=R_{\alpha}(G)$.

Moreover, there are also many variants of Randić index [6, 9, 20]. In [8],
Kier and Hall proposed the zeroth-order Randić index, denoted by ${}^{0}R$.
The explicit formula of ${}^{0}R$ is

${}^{0}R=^{0}R(G)=\sum_{v\in V_{G}}(d_{v})^{-\frac{1}{2}}.$

In some bibliographies, ${}^{0}R$ is also called the modified first Zagreb
index (${}^{m}M_{1}$). Pavlović [16] determined the extremal $(n,m)$-graphs of
${}^{0}R$ with maximum value. Almost at the same time, Lang et al. considered
similar problems in [12] for the first Zagreb index ($M_{1}$), which is
defined as

$M_{1}=M_{1}(G)=\sum_{v\in V_{G}}(d_{v})^{2}.$

In 2005, Li and Zheng [14] constructed the zeroth-order general Randić index,
written ${}^{0}R_{\alpha}$, is the sum of items $(d_{v})^{\alpha}$ over all
vertices $v\in V_{G}$, where $\alpha$ is an pertinently chosen real number.
Note that ${}^{0}R_{-\frac{1}{2}}={{}^{0}R}={{}^{m}M_{1}}$, and
${}^{0}R_{2}=M_{1}$ in the mathematical sense. For the zeroth-order general
Randić index of trees, Li and Zhao [15] determined the first three maximum and
minimum values with exponent
$\alpha_{0},-\alpha_{0},\dfrac{1}{\alpha_{0}},\dfrac{1}{\alpha_{0}}$, where
$\alpha_{0}\geq 2$ is an integer. In 2007, Hu et al. [7] investigated
connected $(n,m)$-graphs with extremal values of ${}^{0}R_{\alpha}$. Two years
later, in [17], Pavlović et al. corrected some errors in the work of Hu et al.

Recently, the relationships between Randić-type indices and domination number
has attracted much attention of many researchers. In 2016, Borovićanin and
Furtula [2] gave the precise upper and lower bounds on the first Zagreb index
($M_{1}$) of trees in terms of domination number and characterized the
corresponding extremal trees. Later, Bermudo et al. [3] and Liu et al. [10]
answered the same question regarding the Randić index ($R_{-\frac{1}{2}}$) and
the modified first Zagreb index (${}^{m}M_{1}$), respectively. Motivate by [2,
3, 10], in this paper, we intend to establish connections between the zeroth-
order general Randić index of trees and domination number.

For convenience, we first introduce some graph-theoretic terminology and
notions. The number of vertices and edges of graph $G$ are called the order
and size of $G$, respectively. For each $v\in V_{G}$, the set of neighbours of
this vertex is denoted by $N(v)=\left\\{u\in V_{G}|~{}uv\in E_{G}\right\\}$. A
vertex $v$ for which $d_{v}=1$ is called a pendent vertex or a leaf vertex.
The maximum vertex degree in $G$ is denote by $\Delta(G)$. The diameter of a
tree is the longest path between two pendent vertices. The dominating set of
graph $G$ is a vertex subset in $V_{G}$ such that every vertex in
$V_{G}\setminus D$ is adjacent at least one vertex in $D$. A subset $D$ is
called minimum dominating set of $G$ if $D$ contains least vertices among all
dominating sets. Domination number $\gamma$ is defined as $\gamma=|D|$.

Based on the above consideration, the structure of this paper is arranged as
below. In Section 2, we prove a fundamental lemma and simplify the
mathematical formula of several bounds on ${}^{0}R_{\alpha}$. Then in Section
3 and 4, sharp upper and lower bounds on ${}^{0}R_{\alpha}$ of trees with a
given domination number for $\alpha\in(-\infty,0)\cup(1,\infty)$ and
$\alpha\in(0,1)$ are obtained, respectively. Furthermore, the corresponding
extremal trees are characterized.

##  2 Preliminaries

Now, we present a basic lemma, and then show the simplified mathematical
formula of bounds on ${}^{0}R_{\alpha}$.

######  Lemma 2.1.

Let the function
$f(x_{1},x_{2},\cdots,x_{k})=x_{1}^{\alpha}+x_{2}^{\alpha}+\cdots+x_{k}^{\alpha}$,
where $\left\\{x_{1},x_{2},\cdots,x_{k}\right\\}$ are all positive integers,
and $\sum_{i=1}^{k}x_{i}=N$ be a fixed positive integer. If there exist
$x_{i}$ and $x_{j}$ such that $x_{i}-x_{j}\geq 2$,
$i,j\in\left\\{1,2,\cdots,k\right\\}$ and $i\neq j$, then

  * (i)

$f(x_{1},\cdots,x_{i},\cdots,x_{j},\cdots,x_{k})<f(x_{1},\cdots,x_{i}-1,\cdots,x_{j}+1,\cdots,x_{k})$,
for $\alpha\in(0,1)$,

  * (ii)

$f(x_{1},\cdots,x_{i},\cdots,x_{j},\cdots,x_{k})>f(x_{1},\cdots,x_{i}-1,\cdots,x_{j}+1,\cdots,x_{k})$,
for $\alpha\in(-\infty,0)\cup(1,\infty)$.

###### Proof.

From the definition of function $f(x_{1},x_{2},\cdots,x_{k})$, we consider the
following function $g(x_{i},x_{j},\alpha)$,

$\displaystyle g(x_{i},x_{j},\alpha)$
$\displaystyle=f(x_{1},\cdots,x_{i},\cdots,x_{j},\cdots,x_{k})-f(x_{1},\cdots,x_{i}-1,\cdots,x_{j}+1,\cdots,x_{k})$
$\displaystyle=x_{i}^{\alpha}+x_{j}^{\alpha}-(x_{i}-1)^{\alpha}-(x_{j}+1)^{\alpha}.$

Note that $x_{i}-x_{j}\geq 2$, we let $x_{i}=x_{j}+r$, where $r\geq 2$ is a
integer. Hence,
$g(x_{i},x_{j},\alpha)=g(x_{j},r,\alpha)=(x_{j}+r)^{\alpha}+x_{j}^{\alpha}-(x_{j}+r-1)^{\alpha}-(x_{j}+1)^{\alpha}$.
Suppose $r$ is a continuous variable, then we get $\frac{\partial
g(x_{j},r,\alpha)}{\partial
r}=\alpha(x_{j}+r)^{\alpha-1}-\alpha(x_{j}+r-1)^{\alpha-1}$ is positive if
$\alpha\in(-\infty,0)\cup(1,\infty)$, and negative if $\alpha\in(0,1)$. By the
Lagrange mean-value theorem, we have

  * (i)

$\displaystyle~{}~{}~{}~{}x_{i}^{\alpha}+x_{j}^{\alpha}-(x_{i}-1)^{\alpha}-(x_{j}+1)^{\alpha}$
$\displaystyle\leq\left(x_{j}+2\right)^{\alpha}+x_{j}^{\alpha}-2(x_{j}+1)^{\alpha}$
$\displaystyle=\alpha\xi_{1}^{\alpha-1}-\alpha\xi_{2}^{\alpha-1}$
$\displaystyle=\alpha(\alpha-1)(\xi_{1}-\xi_{2})\eta^{\alpha-2}<0,$

for $\alpha\in(0,1)$, where $x_{j}<\xi_{2}<x_{j}+1<\xi_{1}<x_{i}+2$, and
$\xi_{2}<\eta<\xi_{1}$.

  * (ii)

$\displaystyle~{}~{}~{}~{}x_{i}^{\alpha}+x_{j}^{\alpha}-(x_{i}-1)^{\alpha}-(x_{j}+1)^{\alpha}$
$\displaystyle\geq\left(x_{j}+2\right)^{\alpha}+x_{j}^{\alpha}-2(x_{j}+1)^{\alpha}$
$\displaystyle=\alpha\xi_{1}^{\alpha-1}-\alpha\xi_{2}^{\alpha-1}$
$\displaystyle=\alpha(\alpha-1)(\xi_{1}-\xi_{2})\eta^{\alpha-2}>0,$

for $\alpha\in(-\infty,0)\cup(1,\infty)$, where
$x_{j}<\xi_{2}<x_{j}+1<\xi_{1}<x_{i}+2$, and $\xi_{2}<\eta<\xi_{1}$.

This completes the proof. ∎

By repeating Lemma 2.1, finally, we can obtain the following corollary.

######  Corollary 2.2.

Assume the function $f(x_{1},x_{2},\cdots,x_{k})$ is defined as above. Then,

  * (i)

for $\alpha\in(0,1)$, $f(x_{1},x_{2},\cdots,x_{k})$ attains its maximum value
if the difference between any two integer in
$\left\\{x_{1},x_{2},\cdots,x_{k}\right\\}$ at most one, and

  * (ii)

for $\alpha\in(-\infty,0)\cup(1,\infty)$, $f(x_{1},x_{2},\cdots,x_{k})$
attains its minimum value if the difference between any two integer in
$\left\\{x_{1},x_{2},\cdots,x_{k}\right\\}$ at most one.

Note that $D$ is a minimum dominating set in a tree $T$ with order $n$ and
domination number $\gamma$, and $\overline{D}=V(T)\backslash D$. Let
$E_{1}=\left\\{uv\in E_{T}|~{}u\in D,~{}v\in\overline{D}\right\\}$,
$E_{2}=\left\\{uv\in E_{T}|~{}u\in D,~{}v\in D\right\\}$, $E_{3}=\left\\{uv\in
E_{T}|~{}u\in\overline{D},~{}v\in\overline{D}\right\\}$ be three subsets of
$E_{T}$, and $l_{1}=|E_{1}|$, $l_{2}=|E_{2}|$, $l_{3}=|E_{3}|$. It’s obvious
that

$\begin{cases}l_{1}+l_{2}+l_{3}=|E_{T}|=n-1,\\\ \sum\limits_{v\in
D}d_{v}=l_{1}+2l_{2},\\\
\sum\limits_{v^{\prime}\in\overline{D}}d_{v^{\prime}}=l_{1}+2l_{3}.\end{cases}$
(2.1)

Now the zeroth-order general Randić index ${}^{0}R_{\alpha}(T)$ can be given
by

$^{0}R_{\alpha}(T)=\sum\limits_{v\in
D}(d_{v})^{\alpha}+\sum\limits_{v^{\prime}\in\overline{D}}(d_{v^{\prime}})^{\alpha}.$
(2.2)

Since each $v\in\overline{D}$ is adjacent to at least one vertex of $D$, one
can see that $l_{1}\geq n-\gamma$. Then by calculation, we obtain
$l_{2}+l_{3}\leq\gamma-1$, implying

$|l_{2}-l_{3}|\leq\gamma-1.$ (2.3)

If $\alpha\in(0,1)$, then by Corollary 2.2, we can see the sum (2.2)
necessarily attains maximum when degrees
$d_{v}\in\left\\{\lfloor\frac{l_{1}+2l_{2}}{\gamma}\rfloor,\lceil\frac{l_{1}+2l_{3}}{\gamma}\rceil\right\\}$
for any vertex $v\in D$ and degrees
$d_{v^{\prime}}\in\left\\{\lfloor\frac{l_{1}+2l_{2}}{n-\gamma}\rfloor,\lceil\frac{l_{1}+2l_{3}}{n-\gamma}\rceil\right\\}$
for any vertex $v^{\prime}\in\overline{D}$. Therefore, we let
$l_{1}+2l_{2}=q\gamma+t$ ($0\leq t\leq\gamma-1$) and
$l_{1}+2l_{3}=q^{\prime}(n-\gamma)+t^{\prime}$ $(0\leq t^{\prime}\leq
n-\gamma-1)$, where $q=\lfloor\frac{l_{1}+2l_{2}}{\gamma}\rfloor$,
$t=l_{1}+2l_{2}-\gamma\lfloor\frac{l_{1}+2l_{2}}{\gamma}\rfloor$,
$q^{\prime}=\lfloor\frac{l_{1}+2l_{3}}{n-\gamma}\rfloor$ and
$t^{\prime}=l_{1}+2l_{3}-\gamma\lfloor\frac{l_{1}+2l_{3}}{n-\gamma}\rfloor$.
Bearing in mind previous discussion, one can check that the formula shown in
(2.2) will attain its maximum if $D$ contains $t$ vertices with degree $q+1$
and $\gamma-t$ vertices with degree $q$, and $\overline{D}$ contains
$t^{\prime}$ vertices with degree $q^{\prime}+1$ and $n-\gamma-t^{\prime}$
vertices with degree $q^{\prime}$. Thus, we have

$\displaystyle\sum\limits_{v\in D}(d_{v})^{\alpha}$ $\displaystyle\leq
t(q+1)^{\alpha}+(\gamma-t)q^{\alpha}$
$\displaystyle=\left(n-1+l_{2}-l_{3}-\gamma\lfloor\frac{n-1+l_{2}-l_{3}}{\gamma}\rfloor\right)\left[\left(\lfloor\frac{n-1+l_{2}-l_{3}}{\gamma}\rfloor+1\right)^{\alpha}-\left(\lfloor\frac{n-1+l_{2}-l_{3}}{\gamma}\rfloor\right)^{\alpha}\right]$
$\displaystyle~{}~{}~{}~{}+\gamma\left(\lfloor\frac{n-1+l_{2}-l_{3}}{\gamma}\rfloor\right)^{\alpha},$

and

$\displaystyle\sum\limits_{v^{\prime}\in\overline{D}}(d_{v^{\prime}})^{\alpha}$
$\displaystyle\leq
t^{\prime}(q^{\prime}+1)^{\alpha}+(\gamma-t^{\prime})(q^{\prime})^{\alpha}$
$\displaystyle=\left[n-1+l_{2}-l_{3}-(n-\gamma)\lfloor\frac{n-1+l_{3}-l_{2}}{n-\gamma}\rfloor\right]\left[\left(\lfloor\frac{n-1+l_{3}-l_{2}}{n-\gamma}\rfloor+1\right)^{\alpha}-\left(\lfloor\frac{n-1+l_{3}-l_{2}}{n-\gamma}\rfloor\right)^{\alpha}\right]$
$\displaystyle~{}~{}~{}~{}+(n-\gamma)\left(\lfloor\frac{n-1+l_{3}-l_{2}}{n-\gamma}\rfloor\right)^{\alpha},$

which implies that

$\begin{split}{}^{0}R_{\alpha}&\leq\left(n-1+l_{2}-l_{3}-\gamma\lfloor\frac{n-1+l_{2}-l_{3}}{\gamma}\rfloor\right)\left[\left(\lfloor\frac{n-1+l_{2}-l_{3}}{\gamma}\rfloor+1\right)^{\alpha}-\left(\lfloor\frac{n-1+l_{2}-l_{3}}{\gamma}\rfloor\right)^{\alpha}\right]\\\
&~{}~{}~{}+\left[n-1+l_{2}-l_{3}-(n-\gamma)\lfloor\frac{n-1+l_{3}-l_{2}}{n-\gamma}\rfloor\right]\left[\left(\lfloor\frac{n-1+l_{3}-l_{2}}{n-\gamma}\rfloor+1\right)^{\alpha}-\left(\lfloor\frac{n-1+l_{3}-l_{2}}{n-\gamma}\rfloor\right)^{\alpha}\right]\\\
&~{}~{}~{}+\gamma\left(\lfloor\frac{n-1+l_{2}-l_{3}}{\gamma}\rfloor\right)^{\alpha}+(n-\gamma)\left(\lfloor\frac{n-1+l_{3}-l_{2}}{n-\gamma}\rfloor\right)^{\alpha}.\end{split}$
(2.4)

For fixed $n$ and $\gamma$, the right-hand side of the inequality (2.4) can be
viewed as the function $h(l_{2}-l_{3})$, i.e. ${}^{0}R_{\alpha}\leq
h(l_{2}-l_{3})$ for $\alpha\in(0,1)$.

Analogously, if $\alpha\in(-\infty,0)\cup(1,\infty)$, we can derive the
inequality ${}^{0}R_{\alpha}\geq h(l_{2}-l_{3})$. So far, we have obtained a
simplified formula of bounds on ${}^{0}R_{\alpha}$.

##  3 Bounds for the ${}^{0}R_{\alpha\in(0,1)}$ of trees in terms of
domination number

In this section, several upper and lower bounds for the zeroth-order general
Randić index ${}^{0}R_{\alpha\in(0,1)}$ of trees are determined. To
characterize extremal $n$-vertex trees of ${}^{0}R_{2}$ with a given
domination number $\gamma$, Borovićanin and Furtula [2] defined three trees
family, denoted by $\mathcal{F}_{1}(n,\gamma)$, $\mathcal{F}_{2}(n,\gamma)$
and $\mathcal{F}_{3}(n,\gamma)$ in here, which will be shown below.

######  Definition 3.1.

  * (i)

$\mathcal{F}_{1}(n,\gamma)$ is a graph family contains some $n$-vertex trees
with domination number $\gamma$ which consists of the stars of orders
$\lfloor\frac{n-\gamma}{\gamma}\rfloor$ and
$\lceil\frac{n-\gamma}{\gamma}\rceil$ with exactly $\gamma-1$ pairs of
adjacent pendent vertices in neighbouring stars.

  * (ii)

$\mathcal{F}_{2}(n,\gamma)$ is a graph family contains some $n$-vertex trees
$T$ with domination number $\gamma$ such that each vertex in $V_{T}$ has at
most one pendent neighbour and $T$ satisfies: (1) there exists a minimum
dominating set $D$ of $T$ has $3\gamma-n-2$ vertices with degree $3$ and
$2(n-2\gamma)$ vertices with degree $2$, while $\overline{D}$ has
$n-2\gamma+2$ vertices with degree $2$ and $3\gamma-n$ pendent vertices, or
(2) there exists a minimum dominating set $D$ of $T$ has $n-2\gamma$ vertices
with degree $2$ and $3\gamma-n$ pendent vertices, while $\overline{D}$ has
$2(n-2\gamma+1)$ vertices with degree $2$, $3\gamma-n-2$ with degree $3$, and
each vertex in $\overline{D}$ has only one neighbour in domination set $D$.

  * (iii)

$\mathcal{F}_{3}(n,\gamma)$ is a set of trees with order $n$ and domination
number $\gamma$, which are obtained from the star $S_{n-\gamma+1}$ by
attaching a pendant edge to its $\gamma-1$ pendent vertices.

Figure 1: Tow non-isomorphic trees in the graph family
$\mathcal{F}_{1}(n,\gamma)$.

Next, we will give two theorems to prove that graph family
$\mathcal{F}_{i}(n,\gamma)$ ($i=1,2,3$) are also extremal trees of
${}^{0}R_{\alpha}$ with a given domination number $\gamma$, for
$\alpha\in(0,1)$.

######  Theorem 3.2.

Let $T$ be an $n$-vertex tree with domination number $\gamma$,
$\alpha\in(0,1)$, then

  * (i)

${}^{0}R_{\alpha}\leq\left[\left(\lfloor\frac{n-1}{\gamma}\rfloor\right)^{\alpha}-\left(\lfloor\frac{n-1}{\gamma}\rfloor-1\right)^{\alpha}\right]\left(n-\gamma\lfloor\frac{n-1}{\gamma}\rfloor\right)+\gamma\left(\lfloor\frac{n-1}{\gamma}\rfloor-1\right)^{\alpha}+2(2^{\alpha}-1)(\gamma-1)+(n-\gamma),$

for $1\leq\gamma\leq\frac{n}{3}$, with equality holding if and only if
$T\in\mathcal{F}_{1}(n,\gamma)$.

  * (ii)

${}^{0}R_{\alpha}\leq\begin{cases}(n-2)\cdot 2^{\alpha}+2,&\mbox{for
}\gamma=\lceil\frac{n}{3}\rceil,\\\ (-3^{\alpha}+3\cdot
2^{\alpha}-1)n+3(3^{\alpha}-2\cdot
2^{\alpha}+1)\gamma+2(2^{\alpha}-3^{\alpha}),&\mbox{for
}\frac{n+3}{3}\leq\gamma\leq\frac{n}{2},\end{cases}$

with equality holding if and only if $T\in\mathcal{F}_{2}(n,\gamma)$.

###### Proof.

(i). For path $P_{3}$, the theorem holds. Suppose $n\geq 3$, then by
$1\leq\gamma\leq\frac{n}{3}$, we get $n-\gamma\geq\frac{2n}{3}$, i.e.
$\frac{\gamma-1}{n-\gamma}\leq\frac{n-3}{2n}<\frac{1}{2}$. Combining (2.3),
yields

$1=\frac{n-1-\gamma+1}{n-\gamma}\leq\frac{n-1+l_{3}-l_{2}}{n-\gamma}\leq\frac{n-1+\gamma-1}{n-\gamma}=1+2\frac{\gamma-1}{n-\gamma}<2,$

implying

$q^{\prime}=\lfloor\frac{n-1+l_{3}-l_{2}}{n-\gamma}\rfloor=1.$

Then by
$\frac{n-1+l_{2}-l_{3}}{\gamma}\geq\frac{n-1-\gamma+1}{\gamma}=\frac{n-\gamma}{\gamma}\geq\frac{2n}{n}=2$,
we can see $q=\lfloor\frac{n-1+l_{2}-l_{3}}{\gamma}\rfloor\geq 2$. Now, the
function $h(l_{2}-l_{3})$ can be formula that

$\begin{split}h(l_{2}-l_{3})&=\left[(q+1)^{\alpha}-q^{\alpha}+1-2^{\alpha}\right](l_{2}-l_{3})+(n-\gamma
q-1)\left[\left(q+1\right)^{\alpha}-q^{\alpha}\right]+\gamma q^{\alpha}\\\
&~{}~{}~{}~{}+(\gamma-1)(2^{\alpha}-1)+(n-\gamma).\end{split}$ (3.1)

There are two possible cases to be consider.

Case 1. $0\leq l_{2}-l_{3}\leq\gamma-1$. In such a case,
$\frac{n-1}{\gamma}\leq\frac{n-1+l_{2}-l_{3}}{\gamma}\leq\frac{n-1+\gamma-1}{\gamma}<\frac{n-1}{\gamma}+1$,
implying

$\displaystyle q=\lfloor\frac{n-1+l_{2}-l_{3}}{\gamma}\rfloor$
$\displaystyle=\lfloor\frac{n-1}{\gamma}\rfloor,~{}~{}~{}~{}~{}~{}~{}\mbox{for
}0\leq l_{2}-l_{3}\leq\gamma\lfloor\frac{n-1}{\gamma}\rfloor+\gamma-n,$
$\displaystyle q=\lfloor\frac{n-1+l_{2}-l_{3}}{\gamma}\rfloor$
$\displaystyle=\lfloor\frac{n-1}{\gamma}\rfloor+1,~{}~{}\mbox{for
}\gamma\lfloor\frac{n-1}{\gamma}\rfloor+\gamma-n+1\leq
l_{2}-l_{3}\leq\gamma-1.$

Due to $\alpha\in(0,1)$, easily, one can check that $h(l_{2}-l_{3})$ always
decrease in interval
$\left[0,\gamma\lfloor\frac{n-1}{\gamma}\rfloor+\gamma-n\right]$ and
$\left[\gamma\lfloor\frac{n-1}{\gamma}\rfloor+\gamma-n+1,\gamma-1\right]$.
Thus, $h(l_{2}-l_{3})$ will attain its maximum if $l_{2}-l_{3}=0$ or
$l_{2}-l_{3}=\gamma\lfloor\frac{n-1}{\gamma}\rfloor+\gamma-n+1$. We consider
the following difference

$h\left(\gamma\lfloor\frac{n-1}{\gamma}\rfloor+\gamma-n+1\right)-h(0)=\left[\gamma\lfloor\frac{n-1}{\gamma}\rfloor+\gamma-(n-1)\right]\left[\left(\lfloor\frac{n-1}{\gamma}\rfloor+1\right)^{\alpha}-\left(\lfloor\frac{n-1}{\gamma}\rfloor\right)^{\alpha}+1-2^{\alpha}\right].$
(3.2)

See $\frac{n-1}{\gamma}\geq\frac{n-1}{n/3}\geq 2+\frac{n-3}{n}$, we have
$\lfloor\frac{n-1}{\gamma}\rfloor\geq 2$. Hence,

$\left[\left(\lfloor\frac{n-1}{\gamma}\rfloor+1\right)^{\alpha}-\left(\lfloor\frac{n-1}{\gamma}\rfloor\right)^{\alpha}+1-2^{\alpha}\right]\leq
3^{\alpha}+1-2^{\alpha}<0,$ (3.3)

for any $\alpha\in(0,1)$. Combining (3.2) and (3.3), yields
$h\left(\gamma\lfloor\frac{n-1}{\gamma}\rfloor+\gamma-n+1\right)<h(0)$. Then
the function $h(0)$ becomes

$h(0)=\left(n-\gamma\lfloor\frac{n-1}{\gamma}\rfloor-1\right)\left[\left(\lfloor\frac{n-1}{\gamma}\rfloor+1\right)^{\alpha}-\left(\lfloor\frac{n-1}{\gamma}\rfloor\right)^{\alpha}\right]+\gamma\left(\lfloor\frac{n-1}{\gamma}\rfloor\right)^{\alpha}+(n-\gamma)+(\gamma-1)(2^{\alpha}-1).$
(3.4)

Case 2. $1-\gamma\leq l_{2}-l_{3}\leq 0$. Note that
$\frac{n-\gamma-1}{\gamma}\leq\frac{n-\gamma}{\gamma}\leq\frac{n-1+l_{2}-l_{3}}{\gamma}\leq\frac{n-1}{\gamma}$.
From (2.3), we get

$\displaystyle q=\lfloor\frac{n-1+l_{2}-l_{3}}{\gamma}\rfloor$
$\displaystyle=\lfloor\frac{n-1}{\gamma}\rfloor,~{}~{}~{}~{}~{}~{}~{}\mbox{for
}\gamma\lfloor\frac{n-1}{\gamma}\rfloor-n+1\leq l_{2}-l_{3}\leq 0,$ (3.5)
$\displaystyle q=\lfloor\frac{n-1+l_{2}-l_{3}}{\gamma}\rfloor$
$\displaystyle=\lfloor\frac{n-1}{\gamma}\rfloor-1,~{}~{}\mbox{for
}1-\gamma\leq l_{2}-l_{3}\leq\gamma\lfloor\frac{n-1}{\gamma}\rfloor-n.$ (3.6)

Analogously, we conclude that $h(l_{2}-l_{3})$ will attain its maximum if
$l_{2}-l_{3}=\gamma\lfloor\frac{n-1}{\gamma}\rfloor-n+1$ or
$l_{2}-l_{3}=1-\gamma$. Consider the difference
$h\left(\gamma\lfloor\frac{n-1}{\gamma}\rfloor-n+1\right)-h(1-\gamma)$ which
is given by

$h\left(\gamma\lfloor\frac{n-1}{\gamma}\rfloor-n+1\right)-h(1-\gamma)=-\left(n-\gamma\lfloor\frac{n-1}{\gamma}\rfloor-\gamma\right)\left[\left(\lfloor\frac{n-1}{\gamma}\rfloor\right)^{\alpha}-\left(\lfloor\frac{n-1}{\gamma}\rfloor-1\right)^{\alpha}+1-2^{\alpha}\right].$
(3.7)

Since $n-\gamma-\gamma\lfloor\frac{n-1}{\gamma}\rfloor\leq 0$, for $\gamma\geq
2$ and
$\left(\lfloor\frac{n-1}{\gamma}\rfloor\right)^{\alpha}-\left(\lfloor\frac{n-1}{\gamma}\rfloor-1\right)^{\alpha}+1-2^{\alpha}\leq
0$, for $\alpha\in(0,1)$ and $\lfloor\frac{n-1}{\gamma}\rfloor\geq 2$, we have
$h\left(\gamma\lfloor\frac{n-1}{\gamma}\rfloor-n+1\right)\leq h(1-\gamma)$. In
addition, if $n=\gamma\lfloor\frac{n-1}{\gamma}\rfloor+\gamma$, then only the
inequality (3.6) holds. Note that
$\left(\lfloor\frac{n-1}{\gamma}\rfloor\right)^{\alpha}-\left(\lfloor\frac{n-1}{\gamma}\rfloor-1\right)^{\alpha}=2^{\alpha}-1$
if and only if $\lfloor\frac{n-1}{\gamma}\rfloor=2$, implying, $2\gamma+1\leq
n<3\gamma+1$. Hence, $n=3\gamma$. Then we can conclude that the extremal tree
$T$ in such a case consist of $\gamma$ $3$-vertex paths $P_{3}$ with exactly
$\gamma-1$ pairs of adjacent pendent vertices in neighbouring paths.
Obviously, $T\in\mathcal{F}_{1}(n,\gamma)$.

Consequently, in order to find the feasible maximum value of $h(l_{2}-l_{3})$,
we just need to calculate the value of $h(0)-h(1-\gamma)$,

$\begin{split}h(0)-h(1-\gamma)&=\left[\left(\lfloor\frac{n-1}{\gamma}\rfloor+1\right)^{\alpha}-\left(\lfloor\frac{n-1}{\gamma}\rfloor\right)^{\alpha}\right]\left(n-\gamma\lfloor\frac{n-1}{\gamma}\rfloor-1\right)+\gamma\left(\lfloor\frac{n-1}{\gamma}\rfloor\right)^{\alpha}\\\
&~{}~{}~{}~{}-\left[\left(\lfloor\frac{n-1}{\gamma}\rfloor\right)^{\alpha}-\left(\lfloor\frac{n-1}{\gamma}\rfloor-1\right)^{\alpha}\right]\left(n-\gamma\lfloor\frac{n-1}{\gamma}\rfloor\right)-\gamma\left(\lfloor\frac{n-1}{\gamma}\rfloor-1\right)^{\alpha}\\\
&~{}~{}~{}~{}+(n-\gamma)+(\gamma-1)(2^{\alpha}-1)-2(2^{\alpha}-1)(\gamma-1)-(n-\gamma).\end{split}$

Note that
$0<\left(\lfloor\frac{n-1}{\gamma}\rfloor+1\right)^{\alpha}-\left(\lfloor\frac{n-1}{\gamma}\rfloor\right)^{\alpha}<\left(\lfloor\frac{n-1}{\gamma}\rfloor\right)^{\alpha}-\left(\lfloor\frac{n-1}{\gamma}\rfloor-1\right)^{\alpha}$
for $\alpha\in(0,1)$. Then we have

$\displaystyle h(0)-h(1-\gamma)$
$\displaystyle=\left(n-\gamma\lfloor\frac{n-1}{\gamma}\rfloor-1\right)\left[\left(\lfloor\frac{n-1}{\gamma}\rfloor+1\right)^{\alpha}+\left(\lfloor\frac{n-1}{\gamma}\rfloor-1\right)^{\alpha}-2\left(\lfloor\frac{n-1}{\gamma}\rfloor\right)^{\alpha}\right]$
$\displaystyle~{}~{}~{}~{}+(\gamma-1)\left[\left(\lfloor\frac{n-1}{\gamma}\rfloor\right)^{\alpha}-\left(\lfloor\frac{n-1}{\gamma}\rfloor-1\right)^{\alpha}+2^{\alpha}-1\right]<0.$

The inequality is strict. Thus,

${}^{0}R_{\alpha}\leq\left[\left(\lfloor\frac{n-1}{\gamma}\rfloor\right)^{\alpha}-\left(\lfloor\frac{n-1}{\gamma}\rfloor-1\right)^{\alpha}\right]\left(n-\gamma\lfloor\frac{n-1}{\gamma}\rfloor\right)+\gamma\left(\lfloor\frac{n-1}{\gamma}\rfloor-1\right)^{\alpha}+2(2^{\alpha}-1)(\gamma-1)+(n-\gamma),$

for $\alpha\in(0,1)$ and $1\leq\gamma\leq\frac{n}{3}$. Equality holds if and
only if $l_{2}-l_{3}=1-\gamma$. Combining (2.1) and (2.2), yields
$l_{1}=n-\gamma$, $l_{2}=0$ and $l_{3}=\gamma-1$. One can easily check that
the corresponding extremal trees in such a case all belong to
$\mathcal{F}_{1}(n,\gamma)$.

Figure 2: Tow non-isomorphic trees in the graph family
$\mathcal{F}_{2}(n,\gamma)$.

(ii). For $\gamma=\lceil\frac{n}{3}\rceil$, the path $P_{n}$ is the unique
tree such that ${}^{0}R_{\alpha}$ with $\alpha\in(0,1)$ attains maximum. Then
we suppose $\gamma\geq\frac{n+3}{3}$. Now, it holds $2\gamma\leq n\leq
3\gamma-3$, implying $n\geq 6$ and $\gamma\geq 3$. Due to

$1=\frac{n-\gamma}{n-\gamma}\leq\frac{n-1+l_{3}-l_{2}}{n-\gamma}\leq\frac{n-1+\gamma-1}{n-\gamma}=1+2\frac{\gamma-1}{n-\gamma}<3,$

which implies that
$q^{\prime}=\lfloor\frac{n-1+l_{3}-l_{2}}{n-\gamma}\rfloor=2$ or
$q^{\prime}=\lfloor\frac{n-1+l_{3}-l_{2}}{n-\gamma}\rfloor=1$. Therefore, we
consider the following two cases.

Case 1. $q^{\prime}=\lfloor\frac{n-1+l_{3}-l_{2}}{n-\gamma}\rfloor=1$. In this
case, one can see $l_{3}-l_{2}<n-2\gamma+1$, implying $l_{2}-l_{3}\geq
2\gamma-n$. Note that $\gamma\leq\frac{n}{2}$, thus, $2\gamma-n\leq 0$. For
convince, we will divide the discussion into $2\gamma-n\leq-1$ and
$2\gamma-n=0$.

Case 1.1. $2\gamma-n\leq-1$. Obviously,
$2\leq\frac{n-1}{\gamma}\leq\frac{n-1}{n/3+1}<3$, then we have
$\lfloor\frac{n-1}{\gamma}\rfloor=2$. Assume that $2\gamma-n\leq
l_{2}-l_{3}\leq 0$. Similarly, one can see

$\displaystyle q=\lfloor\frac{n-1+l_{2}-l_{3}}{\gamma}\rfloor$
$\displaystyle=\lfloor\frac{n-1}{\gamma}\rfloor=2,~{}~{}~{}~{}~{}~{}~{}\mbox{for
}2\gamma-n+1\leq l_{2}-l_{3}\leq 0,$ (3.8) $\displaystyle
q=\lfloor\frac{n-1+l_{2}-l_{3}}{\gamma}\rfloor$
$\displaystyle=\lfloor\frac{n-1}{\gamma}\rfloor-1=1,~{}~{}\mbox{for
}l_{2}-l_{3}=2\gamma-n.$ (3.9)

Due to $q^{\prime}=\lfloor\frac{n-1+l_{3}-l_{2}}{n-\gamma}\rfloor=1$ and
$\gamma\geq\frac{n+3}{3}$, then the only relation (3.8) holds. Hence,

$\begin{split}h(l_{2}-l_{3})&=(3^{\alpha}-2\cdot
2^{\alpha}+1)(l_{2}-l_{3})+(3^{\alpha}-2^{\alpha}+1)n\\\ &~{}~{}~{}~{}+(4\cdot
2^{\alpha}-2\cdot 3^{\alpha}-2)\gamma-(3^{\alpha}-1),(2\gamma-n+1\leq
l_{2}-l_{3}\leq 0).\end{split}$ (3.10)

Now, we suppose $0\leq l_{2}-l_{3}\leq\gamma-1$. Analogously, one can see

$\displaystyle q=\lfloor\frac{n-1+l_{2}-l_{3}}{\gamma}\rfloor$
$\displaystyle=\lfloor\frac{n-1}{\gamma}\rfloor=2,~{}~{}~{}~{}~{}~{}~{}\mbox{for
}0\leq l_{2}-l_{3}\leq 3\gamma-n,$ $\displaystyle
q=\lfloor\frac{n-1+l_{2}-l_{3}}{\gamma}\rfloor$
$\displaystyle=\lfloor\frac{n-1}{\gamma}\rfloor+1=3,~{}~{}\mbox{for
}3\gamma-n+1\leq l_{2}-l_{3}\leq\gamma-1,$

implying

$\begin{split}h(l_{2}-l_{3})&=(3^{\alpha}-2\cdot
2^{\alpha}+1)(l_{2}-l_{3})+(3^{\alpha}-2^{\alpha}+1)n\\\ &~{}~{}~{}~{}+(4\cdot
2^{\alpha}-2\cdot 3^{\alpha}-2)\gamma-(3^{\alpha}-1),~{}~{}(0\leq
l_{2}-l_{3}\leq 3\gamma-n)\end{split}$ (3.11)

and

$\begin{split}h(l_{2}-l_{3})&=(4^{\alpha}-3^{\alpha}-2^{\alpha}+1)(l_{2}-l_{3})+(4\cdot
3^{\alpha}-3\cdot 4^{\alpha}+2^{\alpha}-2)\gamma\\\
&~{}~{}~{}~{}+(4^{\alpha}-3^{\alpha}+1)n-(4^{\alpha}-3^{\alpha}+2^{\alpha}-1),~{}~{}(3\gamma-n+1\leq
l_{2}-l_{3}\leq\gamma-1)\end{split}$

Then by (3.10) and (3.11), we obtain

$\begin{split}h(l_{2}-l_{3})&=(3^{\alpha}-2\cdot
2^{\alpha}+1)(l_{2}-l_{3})+(3^{\alpha}-2^{\alpha}+1)n\\\ &~{}~{}~{}~{}+(4\cdot
2^{\alpha}-2\cdot 3^{\alpha}-2)\gamma-(3^{\alpha}-1),(2\gamma-n+1\leq
l_{2}-l_{3}\leq 3\gamma-n).\end{split}$ (3.12)

See $h(3\gamma-n+1)-h(3\gamma-n)=(3^{\alpha}-2\cdot 2^{\alpha}+1)<0$ for any
$\alpha\in(0,1)$, we just need consider the relation (3.12).

Note that ${}^{0}R_{\alpha\in(0,1)}(T)$ attains its maximum if and only if $T$
is a path (See [13] Theorem 4.2), we conclude that an extremal $T$, whose
${}^{0}R_{\alpha\in(0,1)}$ is maximum, only consist of vertices with degree
$1$, $2$ and $3$. To determine a sharp upper bound on ${}^{0}R_{\alpha}$, we
must investigate further to find a feasible value of $l_{2}-l_{3}$. For a
minimum dominating set $D$ of tree $T$, the number of vertices with degree $2$
and $3$ are denoted by $s_{2}$ and $s_{3}$, respectively. Also, for the set
$\overline{D}$, the number of vertices with degree $1$ and $2$are denoted by
$s^{\prime}_{1}$ and $s^{\prime}_{2}$, respectively. It’s holds

$\begin{cases}|V_{T}|=s_{2}+s_{3}+s^{\prime}_{1}+s^{\prime}_{2},\\\
s_{2}+s_{3}=\gamma,\\\ s^{\prime}_{1}+s^{\prime}_{2}=n-\gamma.\end{cases}$
(3.13)

Combining $\sum_{v\in
V_{T}}d_{v}=2(n-1)=2(s_{2}+s_{3}+s^{\prime}_{1}+s^{\prime}_{2}-1)=s^{\prime}_{1}+2(s_{2}+s^{\prime}_{2})+3s_{3}$,
yields $s_{3}=s^{\prime}_{1}-2$ and $s_{2}-s^{\prime}_{2}=2\gamma-n+2$. From
(3.13), we get

$\begin{cases}n-1+l_{2}-l_{3}=2s_{2}+3s^{\prime}_{1}-6,\\\
n-1+l_{3}-l_{2}=s^{\prime}_{1}+2s^{\prime}_{2}.\end{cases}$ (3.14)

Based on (2.4) and system (3.14), the function $h(l_{2}-l_{3})$ becomes

$h(s^{\prime}_{1})=(3^{\alpha}-2\cdot
2^{\alpha}+1)s^{\prime}_{1}+2^{\alpha}\cdot
n+2(2^{\alpha}-3^{\alpha}),~{}~{}\mbox{for }2\leq s^{\prime}_{1}\leq\gamma+1.$

Case 1.2. $2\gamma-n=0$, i.e., $\gamma=\frac{n}{2}$ if $n$ is even.

Then, $1\leq\frac{n-1}{\gamma}=1+\frac{\gamma-1}{\gamma}<2$, which implies
$\lfloor\frac{n-1}{\gamma}\rfloor=1$. One can easily check that the following
relations hold.

$\displaystyle q=\lfloor\frac{n-1+l_{2}-l_{3}}{\gamma}\rfloor$
$\displaystyle=\lfloor\frac{n-1}{\gamma}\rfloor=1,~{}~{}~{}~{}~{}~{}~{}\mbox{for
}l_{2}-l_{3}=0,$ $\displaystyle
q=\lfloor\frac{n-1+l_{2}-l_{3}}{\gamma}\rfloor$
$\displaystyle=\lfloor\frac{n-1}{\gamma}\rfloor+1=2,~{}~{}\mbox{for }1\leq
l_{2}-l_{3}\leq\frac{n-2}{2}.$

By the analogous derivation, the function $h(l_{2}-l_{3})$ can be given by

$h(s^{\prime}_{1})=(3^{\alpha}-2\cdot
2^{\alpha}+1)s^{\prime}_{1}+2^{\alpha}\cdot
n+2(2^{\alpha}-3^{\alpha}),~{}~{}\mbox{for }2\leq
s^{\prime}_{1}\leq\frac{n}{2}.$

In [2], Borovićanin and Furtula have proved that $s^{\prime}_{1}\geq
3\gamma-n$ for trees, and $s^{\prime}_{1}>3\gamma-n$ always holds if there
exists a vertex in $V_{T}$ has two pendent neighbours. It’s obvious that
${}^{0}R_{\alpha}$ of trees attains its maximum value if
$s^{\prime}_{1}=3\gamma-n$, i.e., $l_{2}-l_{3}=5\gamma-2n+1$. In such a case,
one can check that corresponding extremal trees all belong to
$\mathcal{F}_{2}(n,\gamma)$. Now, the function $h(l_{2}-l_{3})$ becomes

$\displaystyle h(3\gamma-n)$ $\displaystyle=(3^{\alpha}-2\cdot
2^{\alpha}+1)(3\gamma-n)+2^{\alpha}\cdot n+2(2^{\alpha}-3^{\alpha})$
$\displaystyle=(-3^{\alpha}+3\cdot 2^{\alpha}-1)n+3(3^{\alpha}-2\cdot
2^{\alpha}+1)\gamma+2(2^{\alpha}-3^{\alpha}).$

Case 2. $q^{\prime}=\lfloor\frac{n-1+l_{3}-l_{2}}{n-\gamma}\rfloor=2$. Due to
$2\leq\frac{n-1+l_{3}-l_{2}}{n-\gamma}<3$, we have $l_{2}-l_{3}\leq
2\gamma-n+1<0$. It holds

$1\leq\frac{n-\gamma}{\gamma}\leq\frac{n-1+l_{2}-l_{3}}{\gamma}\leq\frac{2(\gamma-1)}{\gamma}<2,$

implying $q=\lfloor\frac{n-1+l_{2}-l_{3}}{\gamma}\rfloor=1$. If
$l_{3}-l_{2}=n-2\gamma+1$, we have $\frac{n-1+l_{3}-l_{2}}{n-\gamma}=2$,
implying all vertices in $\overline{D}$ has degrees 2, where $D$ is an
arbitrary minimum dominating set. Consequently, all vertices in $D$ have
degree 1 and 2, i.e., $T\cong P_{n}$, a contradiction, since
$\gamma\geq\frac{n+3}{3}$.

Next, we assume $l_{3}-l_{2}\geq n-2\gamma+2$, then by (2.4), we get

$\begin{split}h(l_{2}-l_{3})&=(-3^{\alpha}+2\cdot
2^{\alpha}-1)(l_{2}-l_{3})+(3\cdot 2^{\alpha}-3^{\alpha}-1)n\\\
&~{}~{}~{}~{}+(2\cdot 3^{\alpha}-4\cdot
2^{\alpha}+2)\gamma+(1-3^{\alpha}),~{}~{}(1-\gamma\leq l_{2}-l_{3}\leq
2\gamma-n-2).\end{split}$

Analogously, we have to find the minimum realizable value of $l_{2}-l_{3}$.
For an arbitrary minimum dominating set $D$ of $T$, the number of vertices
with degree 1, and 2 are denoted by $s_{1}$ and $s_{2}$, respectively, and for
the set $\overline{D}$, the number of vertices with degree 2, and 3 are
denoted by $s^{\prime}_{2}$ and $s^{\prime}_{3}$, respectively. Apparently, it
holds $s_{2}-s^{\prime}_{2}=2\gamma-n-2$ and $l_{2}-l_{3}=2\gamma-n-s_{1}+1$.
Hence, the function $h(l_{2}-l_{3})$ can be given by

$h(s_{1})=(3^{\alpha}-2\cdot
2^{\alpha}+1)s_{1}+2^{\alpha}n+2(2^{\alpha}-3^{\alpha}),\mbox{ for }3\leq
s_{1}\leq 3\gamma-n.$

Based on previous discussions, we can determined the only possible value of
$s_{1}$, that is $3\gamma-n$, implying $l_{2}-l_{3}=1-\gamma$, such that there
exists a corresponding extremal trees with order $n$ and domination number
$\gamma$, where $\frac{n+3}{3}\leq\gamma\leq\frac{n}{2}$, satisfying its all
vertices in an arbitrary minimum dominating set $D$ have degrees 1 and 2,
while all vertices in $\overline{D}$ have degrees 2 and 3. Then, the function
$h(l_{2}-l_{3})$ can be written as

$\displaystyle h(3\gamma-n)$ $\displaystyle=(3^{\alpha}-2\cdot
2^{\alpha}+1)(3\gamma-n)+2^{\alpha}\cdot n+2(2^{\alpha}-3^{\alpha})$
$\displaystyle=(-3^{\alpha}+3\cdot 2^{\alpha}-1)n+3(3^{\alpha}-2\cdot
2^{\alpha}+1)\gamma+2(2^{\alpha}-3^{\alpha}).$

At this time, we have $l_{1}=n-\gamma$, $l_{2}=0$, and $l_{3}=\gamma-1$. Based
on previous considerations, one can check that the corresponding extremal
trees all belong to $\mathcal{F}_{2}(n,\gamma)$.

This complete the proof. ∎

Figure 3: The tree family $\mathcal{F}_{3}(n,\gamma)$.

######  Theorem 3.3.

Let $T$ be an $n$-vertex tree with domination number $\gamma$,
$\alpha\in(0,1)$, then

$^{0}R_{\alpha}\geq(n-\gamma)^{\alpha}+(n-\gamma)+(\gamma-1)\cdot 2^{\alpha},$
(3.15)

with equality holding if and only if $T\in\mathcal{F}_{3}(n,\gamma)$.

###### Proof.

Assume first that $\Delta(T)=2$, implying $T$ is a path and
$\gamma(T)=\lceil\frac{n}{3}\rceil$. It holds $T_{2}\in\mathcal{F}_{3}(2,1)$
($\cong P_{2}$), $T_{3}\in\mathcal{F}_{3}(3,1)$ ($\cong P_{3}$) and
$T_{4}\in\mathcal{F}_{3}(4,2)$ ($\cong P_{4}$). If $n\geq 5$, the inequality
in (3.15) is strict.

For $\Delta(T)\geq 3$, we take an arbitrary diameter in $T$, denoted by
$v_{1}v_{2}\dots v_{d}$ ($d\geq 4$). From the definition of domination number,
it holds $\Delta(T)\leq n-\gamma$. Then, we prove this theorem by induction on
$n$. Suppose the inequality in (3.15) holds for $|V_{T}|=n-1$. If $|V_{T}|=n$,
we let $T_{-1}=T-\left\\{v_{1}\right\\}$ and consider the following two cases.

Case 1. $\gamma(T_{-1})=\gamma(T)$. By calculation, we get

${}^{0}R_{\alpha}(T)$
$\displaystyle={{}^{0}}R_{\alpha}(T_{-1})-\left(d_{v_{2}}-1\right)^{\alpha}+\left(d_{v_{2}}\right)^{\alpha}+1$
$\displaystyle\geq(n-\gamma-1)+(n-\gamma-1)^{\alpha}+(\gamma-1)\cdot
2^{\alpha}-\left(d_{v_{2}}-1\right)^{\alpha}+\left(d_{v_{2}}\right)^{\alpha}+1$
(3.16) $\displaystyle=(n-\gamma)+(n-\gamma)^{\gamma}+(\gamma-1)\cdot
2^{\alpha}+\left[(n-\gamma-1)^{\alpha}-(n-\gamma)^{\alpha}\right]-\left[(d_{v_{2}}-1)^{\alpha}-(d_{v_{2}})^{\alpha}\right]$
$\displaystyle\geq(n-\gamma)^{\alpha}+(n-\gamma)+(\gamma-1)\cdot
2^{\alpha},~{}~{}\alpha\in(0,1).$ (3.17)

Equalities in both (3.16) and (3.17) hold if and only if
$d_{v_{2}}=\Delta(T)=n-\gamma$, implying $T\in\mathcal{F}_{3}(n,\gamma)$.

Case 2. $\gamma(T_{-1})=\gamma(T)-1$. Combining the definition of domination
set, yields $d_{v_{2}}=2$. Hence,

${}^{0}R_{\alpha}(T)$
$\displaystyle={{}^{0}}R_{\alpha}(T_{-1})-\left(d_{v_{2}}-1\right)^{\alpha}+\left(d_{v_{2}}\right)^{\alpha}+1$
$\displaystyle\geq(n-\gamma)^{\alpha}+(n-\gamma)+(\gamma-1)\cdot
2^{\alpha}+\left[(d_{v_{2}})^{\alpha}-(d_{v_{2}}-1)^{\alpha}+1-2^{\alpha}\right]$
(3.18) $\displaystyle=(n-\gamma)^{\alpha}+(n-\gamma)+(\gamma-1)\cdot
2^{\alpha},~{}~{}\alpha\in(0,1).$

Equality in (3.18) holds if and only if
$T_{-1}\in\mathcal{F}_{3}(n-1,\gamma-1)$, i.e.,
$T\in\mathcal{F}_{3}(n,\gamma)$.

This completes the proof. ∎

##  4 Bounds for the ${}^{0}R_{\alpha\in(-\infty,0)\cup(1,\infty)}$ of trees
in terms of domination number

Next, we will show two theorems to clarify the sharp upper and lower bounds on
${}^{0}R_{\alpha\in(-\infty,0)\cup(1,\infty)}$. Based on the Corollary (2.2)
and proofs in Section 3, we can derive the following two theorems by a
straightforward procedure.

######  Theorem 4.1.

Let $T$ be an $n$-vertex tree with domination number $\gamma$,
$\alpha\in(-\infty,0)\cup(1,\infty)$, then

  * (i)

${}^{0}R_{\alpha}\geq\left[\left(\lfloor\frac{n-1}{\gamma}\rfloor\right)^{\alpha}-\left(\lfloor\frac{n-1}{\gamma}\rfloor-1\right)^{\alpha}\right]\left(n-\gamma\lfloor\frac{n-1}{\gamma}\rfloor\right)+\gamma\left(\lfloor\frac{n-1}{\gamma}\rfloor-1\right)^{\alpha}+2(2^{\alpha}-1)(\gamma-1)+(n-\gamma),$

for $1\leq\gamma\leq\frac{n}{3}$, with equality holding if and only if
$T\in\mathcal{F}_{1}(n,\gamma)$.

  * (ii)

${}^{0}R_{\alpha}\geq\begin{cases}(n-2)\cdot 2^{\alpha}+2,&\mbox{for
}\gamma=\lceil\frac{n}{3}\rceil,\\\ (-3^{\alpha}+3\cdot
2^{\alpha}-1)n+3(3^{\alpha}-2\cdot
2^{\alpha}+1)\gamma+2(2^{\alpha}-3^{\alpha}),&\mbox{for
}\frac{n+3}{3}\leq\gamma\leq\frac{n}{2},\end{cases}$

with equality holding if and only if $T\in\mathcal{F}_{2}(n,\gamma)$.

######  Theorem 4.2.

Let $T$ be an $n$-vertex tree with domination number $\gamma$,
$\alpha\in(-\infty,0)\cup(1,\infty)$, then

${}^{0}R_{\alpha}\leq(n-\gamma)^{\alpha}+(n-\gamma)+(\gamma-1)\cdot
2^{\alpha},$

with equality holding if and only if $T\in\mathcal{F}_{3}(n,\gamma)$.

Combining the results in [2] and [10], Theorem 4.1 and 4.2 can be directly
verified.

## Acknowledgements

The authors would like to express their sincere gratitude to all the referees
for their careful reading and insightful suggestions.

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