# Individual and Averaged Power Density Spectra of X-ray bursts from SGR
J1935+2154: Quasiperiodic Oscillation Search and Slopes

Shuo Xiao  Guizhou Provincial Key Laboratory of Radio Astronomy and Data
Processing, Guizhou Normal University, Guiyang 550001, People’s Republic of
China  School of Physics and Electronic Science, Guizhou Normal University,
Guiyang 550001, People’s Republic of China;
<EMAIL_ADDRESS>Xiao-Bo Li  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China;
<EMAIL_ADDRESS>Wang-Chen Xue  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China;
<EMAIL_ADDRESS>University of Chinese Academy of Sciences, Chinese Academy of
Sciences, Beijing 100049, China  Shao-Lin Xiong  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China;
<EMAIL_ADDRESS>Shuang-Nan Zhang  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China;
<EMAIL_ADDRESS>University of Chinese Academy of Sciences, Chinese Academy of
Sciences, Beijing 100049, China  Wen-Xi Peng  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China;
<EMAIL_ADDRESS>Ai-Jun Dong  Guizhou Provincial Key Laboratory of Radio
Astronomy and Data Processing, Guizhou Normal University, Guiyang 550001,
People’s Republic of China  School of Physics and Electronic Science, Guizhou
Normal University, Guiyang 550001, People’s Republic of China;
<EMAIL_ADDRESS>You-Li Tuo  Institut für Astronomie und Astrophysik,
University of Tübingen, Sand 1, 72076 Tübingen, Germany  Ce Cai  College of
Physics, Hebei Normal University, 20 South Erhuan Road, Shijiazhuang, 050024,
China  Xi-Hong Luo  Department of Physics, Chongqing University, Chongqing
400044, Peopleʼs Republic of China  Jiao-Jiao Yang  Guizhou Provincial Key
Laboratory of Radio Astronomy and Data Processing, Guizhou Normal University,
Guiyang 550001, People’s Republic of China  School of Physics and Electronic
Science, Guizhou Normal University, Guiyang 550001, People’s Republic of
China;
<EMAIL_ADDRESS>Yue Wang  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China;
<EMAIL_ADDRESS>University of Chinese Academy of Sciences, Chinese Academy of
Sciences, Beijing 100049, China  Chao Zheng  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China;
<EMAIL_ADDRESS>University of Chinese Academy of Sciences, Chinese Academy of
Sciences, Beijing 100049, China  Yan-Qiu Zhang  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China;
<EMAIL_ADDRESS>University of Chinese Academy of Sciences, Chinese Academy of
Sciences, Beijing 100049, China  Jia-Cong Liu  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China;
<EMAIL_ADDRESS>University of Chinese Academy of Sciences, Chinese Academy of
Sciences, Beijing 100049, China  Wen-Jun Tan  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China;
<EMAIL_ADDRESS>University of Chinese Academy of Sciences, Chinese Academy of
Sciences, Beijing 100049, China  Chen-Wei Wang  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China;
<EMAIL_ADDRESS>University of Chinese Academy of Sciences, Chinese Academy of
Sciences, Beijing 100049, China  Ping Wang  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China;
<EMAIL_ADDRESS>Cheng-Kui Li  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China;
<EMAIL_ADDRESS>Shu-Xu Yi  Key Laboratory of Particle Astrophysics, Institute
of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China;
<EMAIL_ADDRESS>Shi-Jun Dang  Guizhou Provincial Key Laboratory of Radio
Astronomy and Data Processing, Guizhou Normal University, Guiyang 550001,
People’s Republic of China  School of Physics and Electronic Science, Guizhou
Normal University, Guiyang 550001, People’s Republic of China;
<EMAIL_ADDRESS>Lun-Hua Shang  Guizhou Provincial Key Laboratory of
Radio Astronomy and Data Processing, Guizhou Normal University, Guiyang
550001, People’s Republic of China  School of Physics and Electronic Science,
Guizhou Normal University, Guiyang 550001, People’s Republic of China;
<EMAIL_ADDRESS>Ru-shuang Zhao  Guizhou Provincial Key Laboratory of
Radio Astronomy and Data Processing, Guizhou Normal University, Guiyang
550001, People’s Republic of China  School of Physics and Electronic Science,
Guizhou Normal University, Guiyang 550001, People’s Republic of China;
<EMAIL_ADDRESS>Qing-Bo Ma  Guizhou Provincial Key Laboratory of Radio
Astronomy and Data Processing, Guizhou Normal University, Guiyang 550001,
People’s Republic of China  School of Physics and Electronic Science, Guizhou
Normal University, Guiyang 550001, People’s Republic of China;
<EMAIL_ADDRESS>Wei Xie  Guizhou Provincial Key Laboratory of Radio
Astronomy and Data Processing, Guizhou Normal University, Guiyang 550001,
People’s Republic of China  School of Physics and Electronic Science, Guizhou
Normal University, Guiyang 550001, People’s Republic of China;
<EMAIL_ADDRESS>Jian-Chao Feng  Guizhou Provincial Key Laboratory of
Radio Astronomy and Data Processing, Guizhou Normal University, Guiyang
550001, People’s Republic of China  School of Physics and Electronic Science,
Guizhou Normal University, Guiyang 550001, People’s Republic of China;
<EMAIL_ADDRESS>Bin Zhang  Guizhou Provincial Key Laboratory of Radio
Astronomy and Data Processing, Guizhou Normal University, Guiyang 550001,
People’s Republic of China  School of Physics and Electronic Science, Guizhou
Normal University, Guiyang 550001, People’s Republic of China;
<EMAIL_ADDRESS>Zhen Zhang  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China;
<EMAIL_ADDRESS>Ming-Yu Ge  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China;
<EMAIL_ADDRESS>Shi-Jie Zheng  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China;
<EMAIL_ADDRESS>Li-Ming Song  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China;
<EMAIL_ADDRESS>Qi-Jun Zhi  Guizhou Provincial Key Laboratory of Radio
Astronomy and Data Processing, Guizhou Normal University, Guiyang 550001,
People’s Republic of China  School of Physics and Electronic Science, Guizhou
Normal University, Guiyang 550001, People’s Republic of China;
<EMAIL_ADDRESS>
###### Abstract

The study of quasi-periodic oscillations (QPOs) and power density spectra
(PDS) continuum properties can help shed light on the still illusive emission
physics of magnetars and as a window into the interiors of neutron stars using
asteroseismology. In this work, we employ a Bayesian method to search for the
QPOs in the hundreds of X-ray bursts from SGR J1935+2154 observed by Insight-
HXMT, GECAM and Fermi/GBM from July 2014 to January 2022. Although no
definitive QPO signal (significance $>3\sigma$) is detected in individual
bursts or the averaged periodogram of the bursts grouped by duration, we
identify several bursts exhibiting possible QPO at $\sim$ 40 Hz, which is
consistent with that reported in the X-ray burst associated with FRB 200428.
We investigate the PDS continuum properties and find that the distribution of
the PDS slope in the simple power-law model peaks $\sim$ 2.5, which is
consistent with other magnetars but higher than 5/3 commonly seen in gamma-ray
bursts. Besides, the distribution of the break frequency in the broken power-
law model peaks at $\sim$ 60 Hz. Finally, we report that the power-law index
of PDS has an anti-correlation and power-law dependence on the burst duration
as well as the minimum variation timescale.

magnetars – methods: data analysis

1.5in

##  1 Introduction

Quasi-periodic oscillations (QPOs) have been reported in some magnetars,
including SGR 1806-20 (Israel et al., 2005; Watts & Strohmayer, 2006;
Strohmayer & Watts, 2006; Huppenkothen et al., 2014), SGR 1900+14 (Strohmayer
& Watts, 2005), SGR J1550-5418 (Huppenkothen et al., 2014a), SGR J1935+2154
(Li et al., 2022), an extragalactic magnetar (GRB 200415A) (Castro-Tirado et
al., 2021) and even in a magnetar in a compact binary merger (GRB 211211A)
(Xiao et al., 2022a). Roberts et al. (2023) recently reported 42 Hz QPO
candidates in the $vF_{v}$ spectrum peak energy ($E_{\rm p}$) of a burst from
SGR J1935+2154, though statistically may not be very significant (e.g. Pumpe
et al., 2018). The QPOs can be explained as originating from oscillations in
the crustal movement of magnetars (Huppenkothen et al., 2014b; Miller et al.,
2019), therefore, in principle, QPO can be used to constrain both the equation
of state and the interior magnetic field of the neutron star through the
identification of QPO frequency via particular global seismic modes
(Huppenkothen et al., 2013). These QPO studies have opened up the possibility
of studying neutron star interiors via asteroseismology.

In our recent work, we performed a systematic analysis of the spectral lags
(Xiao et al., 2023a) and minimum variation timescales (MVTs) (Xiao et al.,
2023b) of the X-ray burst from SGR J1935+2154, and reported that the
characteristics are significantly different from gamma-ray bursts (GRBs).
Besides, the MVT of the burst associated with FRB 200428 is larger compared to
most other bursts, which implies its special radiation mechanism (Xiao et al.,
2023b). Therefore, regarding the recently reported interesting QPO in the
burst associated with FRB 200428 from SGR J1935+2154 (Li et al., 2022), we
wonder if this magnetar is special compared to others. For example, are there
QPOs in other X-ray bursts from this magnetar? Do their power density spectra
(PDSs) have similar continuum properties (Huppenkothen et al., 2013) or QPO
candidates like other magnetars in their average PDSs (e.g. Huppenkothen et
al., 2014a)? However, to date these analyses have not been systematically
studied, which is the focus of this work.

PDS is not only the most common method for QPOs search in frequency domain
(Vaughan, 2010; Huppenkothen et al., 2013; Guidorzi et al., 2016; Xiao et al.,
2022b), the PDS continuum properties also contain valuable information,
Huppenkothen et al. (2013) and Guidorzi et al. (2016) indicated the
differences between the distributions of the PDS slopes of gamma-ray bursts
(GRBs), active galactic nuclei (AGNs) and 21 magnetar bursts, Guidorzi et al.
(2016) then suggest the PDS slope depends on the relative strength of the fast
component with timescale $<\sim 1$ s (Shen & Song, 2003; Vetere et al., 2006;
Gao et al., 2012) in the light curve. For the very active SGR J1935+2154,
there are about 700 bursts from it observed by Insight-HXMT, GECAM and
Fermi/GBM from July 2014 to Jan 2022, which are beneficial for PDS analyses.
Besides, these three satellites have dead times and time resolutions in the
order of microseconds or sub-microsecond (Liu et al., 2020; Xiao et al., 2020,
2022c; Meegan et al., 2009; Liu et al., 2021), which allows high-frequency QPO
searches.

In this work, we present our sample selection and analysis methods in Section
2 and give the results of the PDS slopes and QPO searches in Section 3.
Finally, a discussion and summary are given in Section 4.

##  2 SAMPLE SELECTION and Methodology

###  2.1 Data selection

The bursts from SGR J1935+2154 are collected from our previous work (Xiao et
al., 2023b). After removing the bursts with saturation or “timing glitches”, a
total of 40, 82 and 503 bursts from SGR J1935+2154 are collected for HXMT,
GECAM, and GBM, respectively.

The energy range used in our analysis is 8-100 keV. For GECAM and GBM, those
detectors with incident angles of SGR J1935+2154 less than 60 degrees are
selected to improve the statistics. Note that the observed energy range of
GECAM may in fact be greater than 10 keV due to instrumental effects.

The time range of the light curve in the QPO search for an individual burst is
selected using the Bayesian block method, which is a nonparametric modeling
technique to detect and characterize local variability in time series of
sequential data and can find the optimal segmentation of the burst (Scargle et
al., 2013). The time bin size of the light curves is selected as 0.1 ms, which
can achieve a Nyquist frequency of 5000 Hz.

Figure 1: Distribution of the PDS slopes (i.e. power-law index $\alpha$) in
PLC model for bursts from SGR J1935+2154 observed by GBM, GECAM and HXMT,
respectively, the red dotted line represents the peak 2.5 of the distribution.

Figure 2: Left panel: distributions of $p$-values corresponding to the
likelihood ratios of the PLC and BPLC models, the black dotted line represents
the 0.05 threshold we set, i.e., BPLC is considered the optimal model when
$p$-value $<$0.05. Right panel: distributions of the break frequency (i.e. the
break frequency between the low-frequency and high-frequency power law) for
the optimal model is BPLC, the black dotted line represents the peak 60 Hz of
the distribution.  Figure 3: Consistency check of the power-law index measured
by GBM and GECAM. The red dashed lines represent the equality line.

Figure 4: Left panel: power-law index of PDS vs. burst duration ($T_{\rm
BB}$), the fitted yellow line $\alpha=2.27_{-0.02}^{+0.02}\times T_{\rm
BB}^{-0.06_{-0.01}^{+0.01}}$ is obtained by MCMC. Right panel: power-law index
of PDS vs. minimum variation timescale, the fitted yellow line is
$\alpha=1.28_{-0.06}^{+0.06}\times\Delta t_{\rm MVT}^{-0.11_{-0.01}^{+0.01}}$.

Figure 5: Left panel: the 2 ms binned light curve of a burst (UTC
2021-09-11T20:13:40.550) from SGR J1935+2154 observed by GECAM. Right panel:
unbinned PDS of the burst, the blue line is fitted by the PLC model, and the
red line represents the 3 $\sigma$ level. The red dotted line represents the
possible signal around 40 Hz.

Figure 6: Left panel: the 2 ms binned light curve of a burst (UTC
2020-04-27T21:15:36.400) from SGR J1935+2154 observed by the GBM for NaIs 3,
4, 6, 7 and 8. Right panel: unbinned PDS of the burst, the blue line is fitted
by the PLC model, and the red line represents the 3 $\sigma$ level. The 40 Hz
and 80 Hz are QPO candidates.

Figure 7: Left panel: the 2 ms binned light curve of a burst (UTC
2019-11-04T09:17:53.480) from SGR J1935+2154 observed by GBM for NaIs 3, 4, 6,
7 and 8. Right panel: the blue line is fitted by the PLC model, and the red
line represents the 3 $\sigma$ level. The 40 Hz is a QPO candidate.

Figure 8: Left panel: the 1 ms binned light curve of a burst (UTC
2020-05-16T18:12:52.080) from SGR J1935+2154 observed by HXMT. Right panel:
the blue line is fitted by the PLC model, and the red line represents the 3
$\sigma$ level. Note that there are also QPO candidates around 40 Hz and 80
Hz.

Figure 9: Averaged periodograms of the bursts with different durations
observed by GBM, the red lines are obtained by fitting with BPLC model. The
bottom right panel shows the average power spectrum obtained by choosing
1-second duration for all bursts.

Figure 10: Averaged periodograms of the bursts with different durations
observed by GECAM, the red lines are obtained by fitting with BPLC model. The
bottom right panel shows the average power spectrum obtained by choosing
1-second duration for all bursts.

Figure 11: Averaged periodogram of the bursts with different duration observed
by HXMT, the red lines are obtained by fitting with BPLC model. The bottom
right panel shows the average power spectrum obtained by choosing 1-second
duration for all bursts.

Figure 12: Comparison of PDS slopes of SGRs and GRBs. Distributions of the PDS
slopes ($\alpha$) for bursts from SGR J1935+2154, short and long GRBs. The
results of SGRBs and LGRBs are from Dichiara et al., 2013 and Guidorzi et al.,
2016, respectively.

###  2.2 Methodology

####  2.2.1 PDS Slopes

As the power fluctuations of an unbinned PDS follow a $\chi^{2}$ distribution,
we first fit the periodogram using a combination of red noise at low
frequencies and white noise at high frequencies with a power law plus a
constant (PLC) (Vaughan, 2010; Huppenkothen et al., 2013),

$S_{\rm plc}(f)=\beta f^{-\alpha}+\gamma,$ (1)

where $\alpha$ is the power-law index (i.e. slope), or fitted with a broken
power law plus a constant (BPLC) (Huppenkothen et al., 2013),

$S_{\rm bplc}(f)=\beta
v^{-\alpha_{1}}(1+(v/\delta)^{(\alpha_{2}-\alpha_{1})})^{-1}+\gamma,$ (2)

where $\alpha_{1}$ and $\alpha_{2}$ are the power-law indices in low and high
frequencies (the frequency $\delta$ is the change point), respectively, and
$\gamma$ is a constant that represents the white (Poisson) noise in the
periodogram.

To compare the goodness of fit for the PLC and BPLC models, we use the
likelihood ratio test (LRT) in the Bayesian implementation (Vaughan, 2010) and
accept the BPLC model when the probability of chance improvement $p$-value
$<$0.05. We perform the above calculations using $Stingray$ package
111https://docs.stingray.science/index.html (Huppenkothen et al., 2019).

####  2.2.2 Search for QPO signals

For the periodic signal search, we adopt the steps described in Huppenkothen
et al., 2014, that is, comparing the maximum residuals $R_{j}^{\rm
obj}=2I_{j}^{\rm obs}/S_{j}(\theta)$ of a large number of simulations to the
maximum residual for the best-fit broadband model (i.e. PLC or BPLC) in the
observed periodogram. To achieve a reasonable coverage of the frequency range,
we use 1, 3, 5, 7 and 10 times the original frequency resolution, which can
ensure that the searches are sensitive to periodic signals with smaller
widths, approximately equal to the unbinned frequency resolution and broader
signals for the potential periodic component.

When the probability of no QPO signal $p<1\times 10^{-5}$ is in at least one
frequency bin for observing the recorded maximum power residual under the
assumption of pure noise, we consider it has a potential QPO signal. It is
worth noting that the threshold $p<1\times 10^{-5}$ is conservative because
there are 628 bursts across 5 different frequency resolutions, that is, the
modified threshold is $p<0.031$, corresponding to a Gaussian significance of
the threshold of only 1.9$\sigma$.

We note that there is another approach to searching for QPOs based on a model
selection problem, that is, the broadband noise model is compared to a more
complex model combining both the broadband noise model and a Lorentzian
(account for the QPO) (Huppenkothen et al., 2013). Although this method may
better represent the shape of the PDS, but it requires more calculations on
intricate model selection criteria; we thus do not utilize the method in this
work.

##  3 Result

###  3.1 PDS slopes

As shown in Fig. 1, the PDS slopes (i.e. power-law index) in the PLC model of
the bursts from SGR J1935+2154 are around 2.5 and the distribution ranges
observed by the three satellites are consistent (Note only samples with PDS
slope errors less than 0.5 are selected). Furthermore, A Kolmogorov–Smirnov
(K-S) test comparing the populations observed by GBM and GECAM gives a p-value
of 0.04, supporting the null hypothesis that the two populations are drawn
from the same underlying distribution. However, the p-value of comparing the
populations observed by GBM and HXMT is 0.003, which indicates that they may
come from different distributions; however, since the sample of HXMT is small,
no significant conclusion can be given.

The left panel in Fig. 2 shows the p-value distribution of the likelihood
ratios for the PLC and BPLC models, where the proportions of BPLC being the
optimal model (i.e. the $p$-value¡0.05) observed by HXMT, GECAM and GBM are
0.31, 0.28, and 0.25, respectively, which are consistent. The right panel in
Fig. 2 shows that the distribution of the break frequency for the optimal
model is BPLC and peaks at $\sim$ 60 Hz. As discussed in Guidorzi et al.,
2016, the PDS reflects the effect of pulses superposition of different time
scales in the light curve; if the total variance is mostly dominated by some
specific timescale, the PDS will be best fit with BPLC with the timescale
corresponding to the break frequency $f_{\rm b}$. Therefore, the distribution
of the break frequency peaks at $\sim$ 60 Hz implies that several pulses in
some bursts have a duration of $\sim$ 17 ms.

On the other hand, we investigate whether the PDS slopes for the same bursts
observed by GECAM and GBM are consistent. As shown in Fig. 3, the results are
consistent within the error range. It is worth noting here that the slopes of
several bursts are smaller inferred with GECAM data, due to their lower
signal-to-noise ratios observed by GECAM.

Huppenkothen et al. (2014) reported a weak anti-correlation (the power-law
index is -0.1) between the power-law index of the PDS and duration in bursts
from SGR J1550-5418, that is, shorter bursts usually have steeper power-law
indices than longer bursts. Therefore, we also investigate the power-law index
and duration relationships of bursts from SGR J1935+1935; as shown in the left
panel of Fig. 4, they can also be fitted well with a power law
$\alpha=2.27_{-0.02}^{+0.02}\times T_{\rm BB}^{-0.06_{-0.01}^{+0.01}}$.
Besides, we find a similar anti-correlation power-law relationship between the
power-law index and the minimum variation timescale; as shown in the right
panel of Fig. 4, they can be fitted well with a power law
$\alpha=1.28_{-0.06}^{+0.06}\times\Delta t_{\rm MVT}^{-0.11_{-0.01}^{+0.01}}$,
which suggests that the more drastically variable bursts usually have steeper
PDS slopes.

###  3.2 QPO Searches in Individual Burst

We find no QPO candidate with significance above 3 $\sigma$; however, this
does not exclude that QPO actually exists especially in low frequencies due to
the limitations of this PDS-based method used in this work. Since the burst
envelope and red noise dominate in the low frequencies, Huppenkothen et al.
(2013) indicated through simulations that even strong signals below 70 Hz will
be very difficult to detect. Since Li et al. (2022) recently reported a quasi-
periodic oscillation (QPO) with 3.4$\sigma$ significance at $\sim$ 40 Hz in
the X-ray burst associated with FRB 200428 from SGR J1935+2154, we carefully
examine the bursts with maximum residuals corresponding to frequencies in the
range of 30-50 Hz and find that in some bursts (e.g. UTC
2021-09-11T20:13:40.550, UTC 2020-04-27T21:15:36.400, UTC
2019-11-04T09:17:53.480 and UTC 2020-05-16T18:12:52.080) possible QPO at
$\sim$ 40 Hz may exist in the data of HXMT, GECAM and GBM. As shown in Figs.
5, 6, 7 and 8, their light curves are all multi-pulses and have a period of
about $\sim$ 25 ms, resulting in possible signals at $\sim$ 40 Hz in each PDS.
Interestingly, we find features at $\sim$ 80 Hz in bursts UTC
2020-04-27T21:15:36.400 and UTC 2020-05-16T18:12:52.080, which may be
harmonics of the 40 Hz QPO. However, none of them are significant enough to be
claimed as a QPO; thus we only consider them as candidates.

###  3.3 QPO Searches in Average PDS

Due to the low efficiency of QPO detection at lower frequencies (Huppenkothen
et al., 2013), we need to hypothesize the possibility that the majority of
bursts have QPOs at some frequency (e.g. excite the same star quakes), but
cannot be detected in individual bursts due to sensitivity. Therefore, we
divide the bursts into five groups based on duration, i.e. $<50$ ms, 50-100
ms, 100-250 ms,250-500 ms and 500-1000 ms. We select the longest duration
(i.e. 50 ms, 100 ms, 250 ms, 500 ms and 1000 ms) in each group as the duration
of light curves for each burst in the same group; thus each periodogram would
have the same number of frequencies to average periodograms.

The average PDSs for the five groups observed by GECAM, GECAM and HXMT are
shown in Fig. 9,10 and 11, which are all fitted with BPL. No definitive QPO is
detected in the groups; however, some features are detected at about 40 Hz in
the longer duration groups 250-500 ms and 500-1000 ms observed by GBM.

##  4 Discussion and conclusion

In this work, we investigate the PDS slopes (including the distribution, and
the dependence on the duration and the minimum variation timescale) and QPO
searches (including individual bursts and average PDS) for hundreds of bursts
from SGR J1935+2154 observed by HXMT, GECAM and GBM.

If a p-value of 0.05 is taken as the threshold, about 30% of the bursts can be
better fitted with the BPLC model and the others should be preferred with the
PLC model. The PDS slope distribution for PLC model peaks at $\sim$ 2.5, which
are consistent with other magnetars, such as SGR J0501+4516 (Huppenkothen et
al., 2013) and SGR J1550-5418 (Huppenkothen et al., 2014). The distributions
obtained from the three satellites are not significantly different. Besides,
we also report a power law relationship $\alpha=2.27_{-0.02}^{+0.02}\times
T_{\rm BB}^{-0.06_{-0.01}^{+0.01}}$ between the power-law index of PDS and the
burst duration, and the power-law index is roughly consistent with the result
in J1550-5418 (Huppenkothen et al., 2014). Interestingly, we discover a power-
law relationship $\alpha=1.28_{-0.06}^{+0.06}\times\Delta t_{\rm
MVT}^{-0.11_{-0.01}^{+0.01}}$ between the power-law index of PDS and the
minimum variation timescale, which suggests that the more drastically variable
bursts usually have a steeper PDS slope. On the other hand, the distribution
of the break frequency for the optimal model is BPLC and peaks at $\sim$ 60
Hz, which implies that the pulses in some bursts have a duration of $\sim$ 17
ms.

We also compare the PDS slope distributions of SGR J1935+2154 and LGRBs,
SGRBs, where the results of SGRBs and LGRBs are from Dichiara et al., 2013 and
Guidorzi et al., 2016. As shown in Fig. 12, their distributions are different,
and the peaks are around 2.5, 5/3, and 1.9, respectively. Based on the anti-
correlation between the PDS slope and the minimum variation timescale, we
speculate that the reason for the steeper slope of SGR J1935+2154 is due to
its smaller minimum variation timescale (the median value of the distribution
is $\sim$ 2 ms) (Xiao et al., 2023b), but the median values for the SGRB and
LGRB are 40 ms and 480 ms (Golkhou et al., 2015), respectively.

Although QPOs with sufficient significance ($>3\sigma$) are not found in the
bursts from SGR J1935+2154, we can not exclude that QPOs actually exist
especially in low frequencies, due to the detection difficulties of the PDS-
based method (Huppenkothen et al., 2014) used in this work. In fact, The most
straightforward way is to model the shape of the light curve directly, and
then compare the observed periodogram to the periodograms of the simulated
light curves (e.g. Fox et al., 2001). However, due to the absence of knowledge
about the emission processes in magnetar bursts (i.e. the light curves of
magnetar bursts cannot be well modeled), we adopt the conservative method that
models the burst light curve in PDS as a pure red noise process.

We detect some features around 40 Hz in the PDS of multiple bursts, the
frequency of which is consistent with that in the burst in association with
FRB 200428 (Li et al., 2022). Furthermore, the features also appear in the
average power density spectra of bursts with durations from 250 to 1000 ms.
These imply that the 40 Hz QPO in the burst associated with FRB 200428 may be
not peculiar. In addition, the frequency of 40 Hz also lies in the range of
18-155 Hz of the QPO of the giant flares from other magnetars such as SGR
1900+14 (Strohmayer & Watts, 2005) and SGR 1806-20 (Watts & Strohmayer, 2006).
According to previous studies, the 40 Hz QPO can be explained as torsional
Alfven modes of the highly magnetized fluid core (Thompson & Duncan, 2001) or
torsional shear modes of the neutron star crust (Piro, 2005). Therefore, the
detected QPOs are potentially powerful to constrain the interior structures of
magnetars (Lattimer & Prakash, 2007).

## Acknowledgments

We acknowledge the public data from Fermi/GBM. This work is supported by the
National Key R&D Program of China (2022YFF0711404, 2021YFA0718500), the
Scientific Research Project of the Guizhou Provincial Education (Nos.
KY[2022]123, KY[2022]132, KY[2022]137, KY[2021]303, KY[2020]003 and
KY[2023]059), and partially supported by International Partnership Program of
Chinese Academy of Sciences (Grant No. 113111KYSB20190020). The authors also
thank supports from the Strategic Priority Research Program on Space Science,
the Chinese Academy of Sciences (Grant No. XDA15010100, XDA15360100,
XDA15360102, XDA15360300, XDA15052700), the National Natural Science
Foundation of China (Projects: 12273042, 12061131007, Grant No. 12173038, Nos.
12273008 and 12103013), the Foundation of Education Bureau of Guizhou
Province, China (Grant No. KY (2020) 003), Guizhou Provincial Science and
Technology Foundation (Nos. ZK[2022]304, ZK[2022]322, [2021]023, [2023]024),
the National SKA Program of China (Nos. 2022SKA0130100, 2022SKA0130104), the
Major Science and Technology Program of Xinjiang Uygur Autonomous Region
(No.2022A03013-4), the Joint Research Fund in Astronomy (Grant Nos. U1931101)
under cooperative agreement between the National Natural Science Foundation of
China (NSFC) and Chinese Academy of Sciences (CAS). S. Xiao is grateful to W.
Xiao, G. Q. Wang, J. H. Li and Q. Q. Xiao for their useful comments.

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2022

S. Ganesamurthy

These authors contributed equally to this work.

These authors contributed equally to this work.

[1]Department of Mathematics, Periyar University, Salem, 636011, Tamil Nadu,
India

2]Department of Computer science and Engineering, Kalasalingam Academy of
Research and Education, Krishnankoil, Srivilliputhur, 626128, Tamil Nadu,
India

3]Department of Mathematics, Idhaya College of Arts and Science for Women,
Lawspet, 605008, Puducherry, India

# Connected power domination number of product graphs

<EMAIL_ADDRESS>J. Jeyaranjani<EMAIL_ADDRESS>R. Srimathi
<EMAIL_ADDRESS>*  [  [

###### Abstract

In this paper, we consider the connected power domination number
($\gamma_{P,c}$) of three standard graph products. The exact value for
$\gamma_{P,c}(G\circ H)$ is obtained for any two non-trivial graphs $G$ and
$H.$ Further, tight upper bounds are proved for the connected power domination
number of the Cartesian product of two graphs $G$ and $H.$ Consequently, the
exact value of the connected power domination number of the Cartesian product
of some standard graphs is determined. Finally, the connected power domination
number of tensor product of graphs is discussed.

###### keywords:

Connected Power domination number, Power domination number, Product graphs.

###### pacs:

[

MSC Classification]05C38, 05C76, 05C90.

##  1 Introduction

We only consider non-trivial simple connected graphs of finite order, unless
otherwise stated. For a vertex $v\in V(G),$ the open neighborhood of $v$ is
$N(v)=\\{u\,:\,uv\in E(G)\\}$ and the closed neighborhood of $v$ is
$N[v]=\\{v\\}\cup N(v).$ For a set $A\subset V(G),$ the open neighborhood of
$A$ is $N(A)=\cup_{v\in A}N(v)$ and the closed neighborhood of $A$ is
$N[A]=\cup_{v\in A}N[v].$ The subgraph of the graph $G$ induced by the subset
$A$ of the vertices of $G$ is denoted by $\langle A\rangle.$ A vertex $v\in
V(G)$ is called universal vertex of $G$ if $v$ is adjacent to each vertex of
the graph $G.$

Let $K_{n},\,P_{n},\,C_{n},\,W_{n},\,F_{n},$ and $K_{m,\,n},$ respectively,
denote complete graph, path, cycle, wheel, fan, and complete bipartite graph.
For $k\geq 3$ and $1\leq m_{1}\leq m_{2}\leq\dots\leq m_{k},$ the complete
multipartite graph with each partite set of size $m_{i}$ is denoted by
$K_{m_{1},\,m_{2},\,\dots,\,m_{k}}.$

Let $S\subset V(G).$ If $N[S]=V(G),$ then $S$ is called a domination set. If
the subgraph induced by the dominating set is connected, then we say $S$ is a
connected dominating set. For each vertex $v\in V(G),$ if a dominating set $S$
satisfies the property $N(v)\cap S\neq\emptyset,$ then we call the set $S$ is
a total dominating set. The minimum cardinality of dominating set (connected
dominating set) of $G$ is called domination number (connected domination
number) and it is denoted by $\gamma(G)$ ($\gamma_{c}(G)$).

_Algorithm:_dmks22  For the graph $G$ and a set $S\subset V(G),$ let $M(S)$ be
the collection of vertices of $G$ monitored by $S.$ The set $M(S)$ is built by
the following rules:

  1. 1.

(Domination)

  2. Set $M(S)\leftarrow S\cup N(S).$

  3. 2.

(Propagation)

  4. As long as there exists $v\in M(S)$ such that $N(v)\cap(V(G)-M(S))=\\{w\\},$ set $M(S)\leftarrow M(S)\cup\\{w\\}.$

In other words, initially the set $M(S)=N[S],$ and then repeatedly add to
$M(S)$ vertices $w$ that has a neighbor $v$ in $M(S)$ such that all the other
neighbors of $v$ are already in $M(S).$ After no such vertex $w$ exists, the
set monitored by $S$ is constructed.

For a subset $S$ of $V(G),$ if $M(S)=V(G),$ then the set $S$ is called a power
dominating set (PDS). The minimum cardinality of power dominating set of $G$
denoted by $\gamma_{p}(G).$ If the subgraph of $G$ induced by the vertices of
a PDS $S$ is connected, then the set $S$ is connected power domination set
(CPDS), and its minimum cardinality is denoted by $\gamma_{P,\,c}(G).$

laa428  Color-change rule: If $G$ is a graph with each vertex colored either
white or black, $u$ is a black vertex of $G,$ and exactly one neighbor $v$ of
$u$ is white, then change the color of $v$ to black. Given a coloring of $G,$
the derived coloring is the result of applying the color-change rule until no
more changes are possible. A zero forcing set for a graph G is a set $Z\subset
V(G)$ such that if initially the vertices in $Z$ are colored black and the
remaining vertices are colored white, the entire graph G may be colored black
by repeatedly applying the color-change rule. The zero forcing number of
$G,Z(G),$ is the minimum cardinality of a zero forcing set.

If a zero forcing set $Z$ satisfies the connected condition, then we call such
set as connected zero forcing set (CZFC) and it is denoted by $Z_{c}.$ The
connected zero forcing number of $G,Z_{c}(G),$ is the minimum cardinality of a
connected zero forcing set.

For a graph $G$ and a set $X\subseteq V(G),$ the set $X_{i},\,i>0,$ denotes
the collection of all vertices of the graph $G$ monitored by the propagation
up to step $i,$ that is, $X_{1}=N[X]$ (dominating step) and
$X_{i+1}=\cup\\{N[v]\,:\,v\in X_{i}$ such that $|N[v]\setminus X_{i}|\leq
1\\}$ (propagation steps). Similarly, for a connected zero forcing set
$Z_{c}\subseteq V(G)$ and $i\geq 1,$ let $Z_{c}^{i}$ denote the collection of
all vertices of the graph $G$ whose color changed from white to black at step
$i$ (propagation steps).

For two graphs $G$ and $H,$ the vertex set of the Cartesian product ($G\square
H$), tensor product $(G\times H)$ and lexicographic product ($G\circ H$) is
$V(G)\times V(H).$ The adjacency relationship between the vertices $u=(a,\,b)$
and $v=(x,\,y)$ of these products are as follows:

  * •

Cartesian product: $uv\in E(G\square H)$ if either $a=x$ and $by\in E(H),$ or
$b=y$ and $ax\in E(G).$

  * •

Tensor product: $uv\in E(G\times H)$ if $ax\in E(G)$ and $by\in E(H).$

  * •

Lexicographic product: $uv\in E(G\circ H)$ if $ax\in E(G),$ or $a=x$ and
$by\in E(H).$

Let $G\ast H$ be any of the three graph products defined above. Then the
subgraph of $G\ast H$ induced by $\\{g\\}\times V(H)$ ($V(G)\times\\{h\\})$ is
called an $H$-fiber ($G$-fiber) and it is denoted by ${}^{g}H$ ($G^{h}$).
Notation and definitions which are not presented here can be found in rbbook ;
hikbook .

The problem of computing the power domination number of $G$ is NP-hard in
general. The complexity results for power domination in graphs are studied in
ajco19 ; gnr52 ; hhhh15 ; lllncs . Further, some upper bound for the power
domination number of graphs is obtained in zkc306 . Furthermore, the power
domination number of some standard families of graphs and product graphs are
studied in bf58 ; bgpv38 ; dmks22 ; dh154 ; ks13 ; ks16 ; skp18 ; sk11 ; sk48
; vthesis ; vvlncs ; vvh38 . Recently, Brimkvo et al. bms38  introduced the
concept of connected power domination number of graph and obtained the exact
value for trees, block graph, and cactus graph. Further, in gplncs , the
complexity results for split graph, chain graph, and chordal graph are
considered.

In this paper, we extend the study of connected power domination number for
three standard products.

##  2 The Lexicographic Product

The exact value of the power domination number of the lexicographic product of
graphs obtained in dmks22 . In this section, we have obtained the exact value
of the connected power domination number of $G\circ H.$ The assumption of the
connected condition for graph $H$ is relaxed in this section.

######  Theorem 1.

For any two graphs $G$ and $H,$

$\gamma_{P,c}(G\circ
H)=\left\\{\begin{array}[]{rl}\mbox{$\gamma_{c}(G)\mathchar
24635\relax\;$}&\mbox{ if $\gamma_{c}(G)\geq 2,$}\\\ \mbox{$1\mathchar
24635\relax\;$}&\mbox{either $\gamma(G)=\gamma(H)=1$ or $\gamma(G)=1$ and
$H\cong\overline{K_{2}},$}\\\ \mbox{$2\mathchar 24635\relax\;$}&\mbox{if
$\gamma(G)=1$ and $\gamma(H)>1$ with $|V(H)|\geq 3.$}\end{array}\right.$

Proof: First we complete the proof for the case $\gamma_{c}(G)\geq 2.$ Let $X$
be a minimum connected dominating set of $G$ and let $u\in V(H).$ Set
$S=X\times\\{u\\}.$ As $X$ is a connected dominating set of $G,$ it is a total
dominating set of $G\mathchar 24635\relax\;$ consequently, each vertex of $G$
is a neighbor of some vertex in $X.$ Thus each vertex $(g,\,h)\in V(G\circ H)$
is a neighbour of some vertex in $S.$ Since $\langle S\rangle$ is connected
and which monitors each vertex of $G\circ H,$ $\gamma_{P,c}(G\circ
H)\leq\gamma_{c}(G).$

Assume that $S$ is a connected power dominating set of $G\circ H$ whose
cardinality is strictly less than $\gamma_{c}(G).$ Then there exists a vertex
$u\in V(G)$ such that $\\{u\\}\times V(H)\cap N[S]=\emptyset.$ Hence the
vertices in $\\{u\\}\times V(H)$ are monitored by the propagation. Let
$A=\\{u\\}\times V(H).$ Clearly, each vertex in $V(G\circ H)\setminus A$ has
either zero or $|A|$ neighbours in $\langle A\rangle\cong\,^{u}H$-fiber.
Therefore propagation on ${}^{u}H$-fiber is not possible as $|V(H)|\geq 2.$
Therefore $\gamma_{P,c}(G\circ H)\geq\gamma_{c}(G).$

Let $\gamma(G)=\gamma(H)=1.$ Then the graphs $G$ and $H$ have universal
vertices, namely, $u$ and $v,$ respectively. Consequently, the vertex
$(u,\,v)\in V(G\circ H)$ is a universal vertex of the graph $G\circ H.$ Thus
$\gamma_{P,c}(G\circ H)=1.$

Consider $\gamma(G)=1$ and $H\cong\overline{K_{2}}.$ Let $u$ be a universal
vertex of $G$ and let $V(H)=\\{x,\,y\\}.$ Then the vertex $(u,\,x)\in V(G\circ
H)$ dominates all the vertices of the graph $G\circ H$ except $(u,\,y).$
Clearly, the vertex $(u,\,y)$ is monitored by the propagation as $(u,\,y)$ is
the only unmonitored vertex of $G\circ H.$ Therefore, $\gamma_{P,c}(G\circ
H)=1.$

Assume that $\gamma(G)=1$ and $\gamma(H)>1.$ It is easy to observe that a
$\gamma_{P,c}(G\circ H)\geq 2$ as $|V(H)|\geq 3$ and $\gamma(H)>1.$ Let $u$ be
a universal vertex of the graph $G.$ Then the set $\\{(u,\,a),\,(v,\,a)\\}$
dominates all the vertices of the graph $G\circ H.$ Since $u$ is a universal
vertex, $\langle\\{(u,\,a),\,(v,\,a)\\}\rangle\cong K_{2}.$ Hence,
$\gamma_{P,c}(G\circ H)\leq 2.$

##  3 The Cartesian Product

We begin this section by proving a general upper bound for the connected power
domination number of $G\square H.$

######  Theorem 2.

For any two graphs $G$ and $H,$

$\gamma_{P,c}(G\,\square\,H)\leq$
min$\\{\gamma_{P,c}(G)|V(H)|,\gamma_{P,c}(H)|V(G)|\\}.$

Proof: Let $X$ be a CPDS of $G.$ Consider $X^{\prime}=X\times V(H).$ Clearly,
for each vertex $u\in X,\,^{u}H$-fiber is observed as $\\{u\\}\times V(H)\in
X^{\prime}.$ Also, by our choice of $X^{\prime},$ for each vertex $v\in
N(X),\,^{v}H$-fiber is observed (dominating step). To complete the proof, it
is enough to show that if $w\in X_{i},$ then $V(^{w}H)\in X_{i}^{\prime}.$ We
proceed with the proof by induction. The result is true for $i=1.$ Assume that
the result holds for some $i>0.$ Let $w\in X_{i+1}.$ If $w\in X_{i},$ then
$V(^{w}H)\in X_{i}^{\prime}$ by induction hypothesis. If $w\notin X_{i},$ then
there exists a vertex $y\in X_{i}$ which is the neighbour of $w$ such that
$|N[y]\setminus X_{i}|\leq 1.$ This gives $V(^{y}H)\in X_{i}^{\prime},$ by
induction hypothesis. Hence, for fixed $h\in V(H),\,|N[(y,\,h)]\setminus
X_{i}^{\prime}|=|N[y]\setminus X_{i}|\leq 1.$ Thus, $N[(y,\,h)]\in
X_{i+1}^{\prime}$ which implies that $(w,\,h)\in X_{i+1}^{\prime}.$ As it is
true for each $h\in V(H),\,V(^{w}H)\in X_{i+1}^{\prime}.$ Therefore,
$\gamma_{P,c}(G\,\square\,H)\leq\gamma_{P,c}(G)|V(H)|.$ It is easy to prove
that $\gamma_{P,c}(G\,\square\,H)\leq\gamma_{P,c}(H)|V(G)|$ as $G\square H$ is
commutative.

From the definitions of CPDS and CZFS, it is clear that if $X\subseteq V(G)$
is a CPDS, then $N[X]$ is a CZFS. From this observation, we prove the
following upper bound for $\gamma_{P,c}(G\square H)$ in terms of the product
of Connected zero forcing number and connected domination number.

######  Theorem 3.

For any two graphs $G$ and $H,$

$\gamma_{P,c}(G\,\square\,H)\leq$
min$\\{Z_{c}(G)\gamma_{c}(H),Z_{c}(H)\gamma_{c}(G)\\}.$

Proof: Let $Z_{c}$ be a CPDS of $G$ and let $S$ be a connected dominating set
of $H.$ Consider $X=Z_{c}\times S.$ Clearly, for each vertex $u\in
Z_{c},\,^{u}H$-fiber is observed as $\\{u\\}\times S\in X.$ We proceed with
the proof by induction. The result is true for $i=0.$ Assume that the result
holds for some $i\geq 0.$ Let $w\in Z_{c}^{i+1}.$ If $w\in Z_{c}^{i},$ then
$V(^{w}H)\in X_{i}$ by induction hypothesis. If $w\notin Z_{c}^{i},$ then
there exists a vertex $y\in Z_{c}^{i}$ which is the neighbour of $w$ such that
$|N[y]\setminus Z_{c}^{i}|\leq 1.$ This gives $V(^{y}H)\in X_{i},$ by
induction hypothesis. Hence, for fixed $h\in V(H),\,|N[(y,\,h)]\setminus
X_{i}|=|N[y]\setminus Z_{c}^{i}|\leq 1.$ Thus, $N[(y,\,h)]\in X_{i+1}$ which
implies that $(w,\,h)\in X_{i+1}.$ As it is true for each $h\in
V(H),\,V(^{w}H)\in X_{i+1}.$ Therefore, $\gamma_{P,c}(G\,\square\,H)\leq
Z_{c}(G)\gamma_{c}(H).$ In a similar way, it is easy to prove that
$\gamma_{P,c}(G\,\square\,H)\leq Z_{c}(H)\gamma_{c}(G).$

The upper bound in the above theorem is tight if $G$ has a universal vertex
and $H\in\\{P_{n},\,C_{n},\,W_{n},\,F_{n}\\}.$ Also, if we replace $Z_{c}=Z$
and $\gamma_{c}=\gamma$ in the above theorem, then we have the upper bound for
$\gamma_{P}(G\square H)$ in terms of zero forcing number and domination
number.

######  Corollary 4.

For any two graphs $G$ and $H,$

$\gamma_{P}(G\,\square\,H)\leq$ min$\\{Z(G)\gamma(H),Z(H)\gamma(G)\\}.$

The following corollaries are immediate from Theorem 3 as $Z_{c}(P_{n})=1,$
$Z_{c}(C_{n})=2,$ $Z_{c}(W_{n})=3$ and $Z_{c}(F_{n})=2.$

######  Corollary 5.

For a graph $G,$ $\gamma_{P,c}(G\,\square\,P_{n})\leq\gamma_{c}(G).$

######  Corollary 6.

For a graph $G,$ $\gamma_{P,c}(G\,\square\,C_{n})\leq 2\gamma_{c}(G),$ where
$|V(G)|\geq 3.$

######  Corollary 7.

For $n\geq 4$ and a graph $G,\,\gamma_{P,c}(G\,\square\,W_{n})\leq
3\gamma_{c}(G),$ where $|V(G)|\geq 3.$

######  Corollary 8.

For a graph $G,$ $\gamma_{P,c}(G\,\square\,F_{n})\leq 2\gamma_{c}(G),$ where
$|V(G)|\geq 3$ and $n\geq 3.$

As mentioned earlier, the upper bounds in the above four corollaries are tight
if $G$ has a universal vertex. Some of their consequences are listed in the
following table.

Result | $G$ | $\gamma_{P,c}$
---|---|---
Corollary 6 | $C_{m}\square K_{n},\,m,\,n\geq 3$ | 2
Corollary 6 |  $C_{m}\square W_{n},\,m\geq 3$ and $m\geq 4$ | 2
Corollary 6 | $C_{m}\square K_{1,\,m},\,m,\,n\geq 3$ | 2
Corollary 6 | $C_{m}\square F_{n},\,m,\,n\geq 3$ | 2
Corollary 7 | $W_{m}\square W_{n},\,m,\,n\geq 4$ | 3
Corollary 7 | $W_{m}\square K_{1,\,m},\,m,\,n\geq 4$ | 3
Corollary 7 | $W_{m}\square K_{n},\,m,\,n\geq 4$ | 3
Corollary 8 | $F_{m}\square F_{n},\,m,\,n\geq 3$ | 2
Corollary 8 | $F_{m}\square K_{n},\,m,\,n\geq 3$ | 2
Corollary 8 | $F_{m}\square K_{1,\,n},\,m,\,n\geq 3$ | 2
Corollary 8 |  $F_{m}\square W_{n},\,m\geq 3$ and $n\geq 4$ | 2

######  Observation 1.

For any graph $G,$ $\gamma_{p}(G)\leq\gamma_{P,c}(G).$

######  Theorem 9.

sk11  For $2\leq m\leq n,$ $\gamma_{p}(K_{m}\square K_{n})=m-1.$

######  Theorem 10.

For $2\leq m\leq n,$ $\gamma_{P,c}(K_{m}\square K_{n})=m-1.$

Proof: By Theorem 9 and Observation 1, we have
$m-1\leq\gamma_{P,c}(K_{m}\square K_{n}).$ Let
$V(K_{m})=\\{v_{1},\,v_{2},\,\dots,\,v_{m}\\}$ and
$V(K_{n})=\\{u_{1},\,u_{2},\,\dots,\,u_{n}\\}.$ It is easy to observe that the
set $S=\\{(v_{1},\,u_{1}),\,(v_{2},\,u_{1}),\,\dots,\,(v_{m-1},\,u_{1})\\}$ is
a CPDS of $K_{m}\square K_{n}.$ Thus, $\gamma_{P,c}(K_{m}\square K_{n})=m-1$
as $|S|=m-1.$

######  Theorem 11.

ks16  For $m,\,n\geq 3,$ $\gamma_{P}(K_{m}\square
K_{1,\,n})=min\\{m-1,\,n-1\\}.$

######  Theorem 12.

For $m,\,n\geq 3,$ $\gamma_{P,c}(K_{m}\square K_{1,\,n})=min\\{m-1,\,n\\}.$

Proof: Let $V(K_{m})=Z_{m}$ and $V(K_{1,n})=Z_{n+1},$ where the vertex $0$ is
the universal vertex of $K_{1,\,n}.$ Then $V(K_{m}\square
K_{1,\,n})=Z_{m}\times Z_{n+1}.$

Case 1: $m\leq n+1$

By Theorem 3, we have $\gamma_{P,c}(K_{m}\square K_{1,\,n})\leq m-1$ as
$Z_{c}(K_{m})=m-1$ and $\gamma_{c}(K_{1,\,n})=1.$ By Theorem 11 and
Observation 1, $m-1\leq\gamma_{P,c}(K_{m}\square K_{1,\,n}).$ Hence,
$\gamma_{P,c}(K_{m}\square K_{1,\,n})=m-1.$

Case 2: $m>n+1$

Since $\gamma(K_{m})=1$ and $Z_{c}(K_{1,n})=n,\,\gamma_{P,c}(K_{m}\square
K_{1,\,n})\leq n$ (By Theorem 3). To prove the lower bound, first we need to
observe that any minimum CPDS $X$ of $K_{m}\square K_{1,\,n}$ must contains at
least one of the vertices of the form $(i,\,0)$ for some $i\in Z_{m}\mathchar
24635\relax\;$ otherwise, all the vertices in any CPDS $X\subset
V(K_{m}^{j}),$ for some fixed $j,$ where $j\in(Z_{m}\setminus\\{0\\}),$ and
hence $|X|>n$ as $m>n+1.$ Suppose there exists a minimum CPDS $X$ of
$K_{m}\square K_{1,\,n}$ with $|X|\leq n-1.$ Then the vertices in at least
three ${}^{i}K_{1,\,n}$-fiber and two $K_{m}^{j}$-fiber do not belong to $X.$
WLOG let $i\in\\{m-1,\,m,\,m+1\\}$ and $j\in\\{n-1,\,n\\}.$ Let
$A=\\{(i,\,j)\,|\,i\in\\{m-1,\,m,\,m+1\\}\,\,\mbox{and}\,\,j\in\\{n-1,\,n\\}\\}.$
Since $|N(x)\cap A|>1$ for any vertex $x\notin X$ and $x\in N(A)\setminus A,$
propagation is not possible to observe any vertices in the set $A.$ This leads
to the contradiction for the cardinality of the minimum CPDS is $n-1.$ Thus,
$\gamma_{P,c}(K_{m}\square K_{1,\,n})\geq n.$ This completes the proof.

From Case $1$ and $2,$ we have $\gamma_{P,c}(K_{m}\square
K_{1,\,n})=min\\{m-1,\,n\\}.$

######  Theorem 13.

For $3\leq x\leq y,\,\gamma_{P,\,c}(K_{1,\,x}\square K_{1,\,y})=x.$

Proof: Let $V(K_{1,\,x})=Z_{x}$ and $V(K_{1,\,y})=Z_{y}.$ Consider the vertex
with label $0$ is the universal vertex of the graph $K_{1,\,x}$ (respectively,
$K_{1,\,y}$). By Theorem 3, we have $\gamma_{P,c}(K_{1,\,x}\square
K_{1,\,y})\leq x$ as $Z_{c}(K_{1,\,x})=x$ and $\gamma_{c}(K_{1,\,y})=1.$

To attain the lower bound, we claim that any set $X\subset V(K_{1,\,x}\square
K_{1,\,y})$ with cardinality $x-1$ does not satisfy the CPDS condition. Note
that any minimum CPDS contains at least one of the vertex of the form
$(0,\,i)$ or $(j,\,0)\mathchar 24635\relax\;$ otherwise, the connected
condition fails. Suppose $X$ is a minimum CPDS of $K_{1,\,x}\square K_{1,\,y}$
with size $x-1.$ Since $|X|=x-1,$ the vertices in at least two
${}^{i}K_{1,\,y}$-fiber and two $K_{1,\,x}^{j}$-fiber do not belong to $X.$
WLOG let $i\in\\{x-1,\,x\\}$ and $j\in\\{y-1,\,y\\}.$ Let
$Y=\\{(a,\,b):a\in\\{x-1,\,x\\}\,\,\mbox{and}\,\,b\in\\{y-1,\,y\\}\\}.$ It is
clear that the vertices in $Y$ are monitored only by propagation set. But it
is not possible as $|N((0,\,b))\cap Y|>1$ and $|N((a,\,0))\cap Y|>1.$ Which is
a contradiction for $|X|=x-1.$ Hence, $\gamma_{P,\,c}(K_{1,\,x}\square
K_{1,\,y})=x.$

######  Theorem 14.

Let the order of two graphs $G$ and $H$ be at least four and let
$\gamma(G)=1.$ $Z_{c}(H)=2$ if and only if $\gamma_{P,c}(G\square H)=2.$

Proof: By hypothesis and Theorem 3, $\gamma_{P,c}(G\square H)\leq 2.$ Also,
$\gamma_{P,c}(G\square H)>1$ as $Z_{c}(H)=2.$ Hence $\gamma_{P,c}(G\square
H)=2.$

Conversely, assume that $\gamma(G)=1$ and $\gamma_{P,c}(G\square H)=2.$ By our
assumption, it is clear that $H\not\cong P_{m}.$ Let $v$ be a universal vertex
of $G$ and let $X$ be a CPDS for $G\square H.$ If $(a,\,b)$ and $(c,\,d)$ are
the vertices in $X,$ then $a=c=v$ and $b\neq d$ as $\langle X\rangle\cong
K_{2}\mathchar 24635\relax\;$ otherwise $a\neq b$ and $b=d,$ then the vertices
in $G\square H$ cannot be observed by propagation as $H\not\cong P_{m}.$
Consequently, propagation occurs from one $G$-fiber to another $G$-fiber only
if $Z_{c}(H)\leq 2.$ Since $H\not\cong P_{m},$ $Z_{c}(H)>1.$ Thus,
$Z_{c}(H)=2.$

######  Theorem 15.

Let $\gamma(G)=1$ and let $H=G\circ\overline{K_{n}}.$ For $n,\,m\geq
2,\,\gamma_{P,\,c}(H\square P_{m})=2.$

Proof: It is easy to observe that if $\gamma(G)=1,$ then
$\gamma(G\circ\overline{K_{n}})=2$ for all integer $n\geq 2.$ That is,
$\gamma_{c}(H)=2.$ By Theorem 3, we have $\gamma_{P,\,c}(H\square P_{m})\leq
2$ as $Z_{c}(P_{m})=1.$ On the other hand, $\gamma_{P,\,c}(H\square P_{m})>1$
as $\gamma(H)\neq 1.$ Thus, $\gamma_{P,\,c}(H\square P_{m})=2.$

##  4 The Tensor Product

Throughout this section, for a graph $G$ and $H,$ let
$V(G)=\\{u_{1},\,u_{2},\,\dots,\,u_{a}\\}$ and
$V(H)=\\{v_{1},\,v_{2},\,\dots,\,v_{b}\\}.$ Let $U_{i}=u_{i}\times V(H)$ and
$V_{j}=V(G)\times v_{j}.$ Then $V(G\times
H)=\\{\bigcup_{i=1}^{a}U_{i}\\}=\\{\bigcup_{j=1}^{b}V_{j}\\}.$ The sets
$U_{i}$ and $V_{j}$ are called the $i^{th}$-row and $j^{th}$-column of the
graph $G\times H,$ respectively.

The following theorem is proved for power domination number $G\times H$ but it
is true for connected power domination number of $G\times H$ also.

######  Theorem 16.

skp18  If $\gamma_{P}(G\times H)=\gamma_{P,\,c}(G\times H)=1,$ then $G$ or $H$
is isomorphic to $K_{2}.$

######  Theorem 17.

Let $G$ and $H$ be two non-bipartite graphs with at least two universal
vertices. Then $\gamma_{P,\,c}(G\times H)=2.$

Proof: Let $\\{u_{1},\,u_{2}\\}$ and $\\{v_{1},\,v_{2}\\}$ be universal
vertices of the graphs $G$ and $H,$ respectively. Consider the set
$X=\\{(u_{1},\,v_{1}),\,(u_{2},\,v_{2})\\}\subset V(G\times H).$ Clearly,
$\langle X\rangle\cong K_{2}.$ Since $u_{1}$ and $v_{1}$ are the universal
vertices of the graphs $G$ and $H,$ respectively, the vertex $(u_{1},\,v_{1})$
dominates the vertices in the set
$\\{\bigcup_{i=2}^{a}(U_{i}\setminus(u_{i},\,v_{1}))\\}.$ The vertex
$(u_{2},\,v_{2})$ dominates the vertices in the set
$(V_{1}\setminus(u_{1},\,v_{2}))\cup\\{\bigcup_{j=3}^{b}(V_{j}\setminus(u_{2},\,v_{j}))\\}$
as $u_{2}$ and $v_{2}$ are the universal vertices of the graphs $G$ and $H,$
respectively. Hence, the only unmonitored vertices of the graph $G\times H$
are $(u_{1},\,v_{2})$ and $(u_{2},\,v_{1}).$ These vertices are monitored by
the propagation step as $|N(u_{1},\,v_{2})\setminus
X_{1}|=|N(u_{2},\,v_{1})\setminus X_{1}|=1.$ Thus, $\gamma_{P,\,c}(G\times
H)\leq 2.$ By Theorem 16, we have $\gamma_{P,\,c}(G\times H)\neq 1.$
Therefore, $\gamma_{P,\,c}(G\times H)=2.$

######  Corollary 18.

  1.   2. 1.

For $m,\,n\geq 3,\,\gamma_{P,\,c}(K_{m}\times K_{n})=\gamma_{P}(K_{m}\times
K_{n})=2.$

  3. 2.

For $a\geq 1$ and $b\geq
1,\,\gamma_{P,\,c}(K_{1,\,1,\,m_{1},\,m_{2},\dots,\,m_{a}}\times
K_{1,\,1,\,n_{1},\,n_{2},\dots,\,n_{b}})=$

  4. $\gamma_{P}(K_{1,\,1,\,m_{1},\,m_{2},\dots,\,m_{a}}\times K_{1,\,1,\,n_{1},\,n_{2},\dots,\,n_{b}})=2.$

######  Theorem 19.

Let $G$ be a non-bipartite graph. For $2\leq x\leq y,\,\gamma_{P,c}(G\times
K_{x,\,y})=\gamma_{c}(G\times K_{2}).$

Proof: Let the bipartition of $K_{x,\,y}$ be
$A=\\{a_{1},\,a_{2},\,\dots,\,a_{x}\\}$ and
$B=\\{b_{1},\,b_{2},\,\dots,\,b_{y}\\}$ and let
$V(G)=\\{u_{1},\,u_{2},\,\dots,\,u_{t}\\}.$ Clearly, $G\times K_{x,\,y}$ is a
bipartite graph with bipartition $V_{A}$ and $V_{B},$ where $V_{A}=V(G)\times
A$ and $V_{B}=V(G)\times B.$ Let $U_{i}^{A}=u_{i}\times A$ and
$U_{i}^{B}=u_{i}\times B.$ Then $V(G\times K_{x,\,y})=V_{A}\cup
V_{B}=\\{\bigcup_{i=1}^{t}U_{i}^{A}\\}\cup\\{\bigcup_{i=1}^{t}U_{i}^{B}\\}.$
Observe that, if $u_{i}u_{j}\in E(G),$ then $\langle U_{i}^{A}\cup
U_{j}^{B}\rangle\cong\langle U_{j}^{A}\cup U_{i}^{B}\rangle\cong K_{x,\,y}.$

Let $X$ be a minimum connected dominating set of $G\times K_{2}.$ Now we claim
that $X$ is CPDS of $G\times K_{x,\,y}.$ If $(u_{i},\,a_{i})$ dominates
$(u_{j},\,b_{1}),$ then $(u_{i},\,a_{i})$ dominates all the vertices in
$U_{j}^{B}$ as $\langle U_{i}^{A}\cup U_{j}^{B}\rangle\cong K_{x,\,y}.$
Further, each vertex in $G\times K_{2}$ is adjacent to at least one of the
vertices in $X.$ Consequently, $X$ is connected dominating set of $G\times
K_{x,\,y}$ and hence $X$ is a CPDS of $G\times K_{x,\,y}.$ From this we have
$\gamma_{P,c}(G\times K_{x,\,y})\leq\gamma_{c}(G\times K_{2}).$

Assume that $X$ is a minimum CPDS of $G\times K_{x,\,y}$ with
$|X|<\gamma_{c}(G\times K_{2}).$ Then we can find $i$ or $j$ such that the
vertex $(u_{i},\,a_{1})$ or $(u_{j},\,b_{1})$ is not dominated by the vertices
in $X.$ This implies that all the vertices in $U_{i}^{A}$ or $U_{j}^{B}$ are
monitored only by propagation step (not dominating step). But it is not
possible as $U_{i}^{A}=x\geq 2$ or $U_{j}^{B}=y\geq 2.$ Hence,
$\gamma_{P,c}(G\times K_{x,\,y})=\gamma_{c}(G\times K_{2}).$

In fact, from the proof of the above theorem, it is easy to observe that
$\gamma_{P,c}(G\times K_{x,\,y})=\gamma_{c}(G\times K_{x,\,y})$ for $2\leq
x\leq y.$ This observation is used in the proof of the following theorem.

######  Theorem 20.

Let $G$ be a non-bipartite graph with at least two universal vertices. Then
$\gamma_{P,c}(G\times K_{x,\,y})=\left\\{\begin{array}[]{rl}1\mathchar
24635\relax&\mbox{if $G\cong C_{3}$ and $x=y=1,$}\\\ 2\mathchar
24635\relax&\mbox{if $G\not\cong C_{3}$ and $x=y=1,$}\\\ 3\mathchar
24635\relax&\mbox{if $x=1$ and $y\geq 2,$}\\\ 4\mathchar 24635\relax&\mbox{if
$x,\,y\geq 2.$}\end{array}\right.$

Proof: Consider the vertex set of $G\times K_{x,\,y}$ is as in Theorem 19. Let
$u_{1}$ and $u_{2}$ be two universal vertices of $G.$

First we complete the proof for $x=y=1.$ If $G\cong C_{3},$ then $G\times
K_{2}\cong C_{6}$ and hence $G\times K_{2}=1.$ Now we assume that $G\not\cong
C_{3}.$ Let $X=\\{(u_{1},\,a_{1}),\,(u_{2},\,b_{1})\\}.$ The vertices
$(u_{1},\,a_{1})$ and $(u_{2},\,b_{1})$ dominates the vertices in
$V_{B}\setminus(u_{1},\,b_{1})$ and $V_{A}\setminus(u_{2},\,a_{1}),$
respectively. The vertices $(u_{1},\,b_{1})$ and $(u_{2},\,a_{1})$ are
monitored by the propagation step as $|N((u_{1},\,b_{1}))\setminus
X_{1}|=|N((u_{2},\,b_{1}))\setminus X_{1}|=1.$ Hence, $\gamma_{P,\,c}(G\times
K_{2})\leq 2.$ Since $G$ has two universal vertices, minimum degree of $G$ is
at least two and two vertices have degree $t-1.$ As a consequence
$\gamma_{P,\,c}(G\times K_{2})\neq 1.$ Thus, $\gamma_{P,\,c}(G\times
K_{2})=2.$

Now we consider $x=1$ and $y\geq 2.$ For this, let
$X=\\{(u_{1},\,a_{1}),\,(u_{2},\,b_{1}),\,(u_{3},\,a_{1})\\}.$ The set $X$
dominates all the vertices of $G\times K_{1,\,y}$ except $(u_{2},\,a_{1}).$
This vertex is observed by the propagation step and hence
$\gamma_{P,\,c}(G\times K_{1,\,y})\leq 3.$ To prove the equality, assume that
$\gamma_{P,\,c}(G\times K_{1,\,y})=2.$ Then the CPDS contains two vertices,
namely, $X=\\{(u_{i},\,a_{1}),\,(u_{j},\,b_{m})\\},$ where $i\neq j.$ WLOG we
assume that $i=1$ and $j=2$ as this choice of $i$ and $j$ dominates maximum
number of vertices of $G\times K_{1,\,y}.$ The vertices which are dominated by
the vertices in $X$ are the vertices in $U_{1}^{B}$ and the vertex
$(u_{2},\,a_{2}.)$ Since $|U_{1}^{B}|=y\geq 2,$ propagation step from
$(u_{i},\,a_{1})\in V^{A}$ to the vertices in $U_{1}^{B}$ is not possible.
This implies that $\gamma_{P,\,c}(G\times K_{1,\,y})\neq 2.$ Thus,
$\gamma_{P,\,c}(G\times K_{1,\,y})=3.$

Let $2\leq x\leq y.$ Recall that $\gamma_{P,c}(G\times
K_{x,\,y})=\gamma_{c}(G\times K_{x,\,y})$ for $2\leq x\leq y.$ Form this, it
is enough to find $\gamma_{c}(G\times K_{x,\,y}).$ Let
$X=\\{(u_{1},\,a_{1}),\,(u_{2},\,b_{1}),\,(u_{3},\,a_{1}),\,(u_{1},\,b_{1})\\}.$
Clearly, the vertices in the set $X$ dominate all the vertices $G\times
K_{x,\,y}$ and $\langle X\rangle\cong P_{4}$ and hence $\gamma_{c}(G\times
K_{x,\,y})\leq 4.$ Since $G\times K_{x,\,y}$ is bipartite, connected subgraph
induced by any three vertices of $G\times K_{x,\,y}$ is isomorphic to $P_{3}.$
Clearly, the end vertices of $P_{3}$ belong to either $V^{A}$ or $V^{B}.$ We
assume that the end vertices of $P_{3}$ belong to $V^{A}.$ Then the two degree
vertex belongs to $V^{B}.$ Let the two degree vertex be $(u_{i},\,b_{j}).$
Clearly, this vertex does not dominates the vertices in the set $U_{i}^{A}.$
Consequently, three vertices do not form the connected dominating set.
Therefore, $\gamma_{c}(G\times K_{x,\,y})\geq 4.$

######  Theorem 21.

Let $G$ be a graph with at least two universal vertices. For $k\geq 3$ and
$1\leq m_{1}\leq m_{2}\leq\dots\leq m_{k},$

$\gamma_{P,\,c}(G\times
K_{m_{1},\,m_{2},\,\dots,\,m_{k}})=\left\\{\begin{array}[]{rl}2\mathchar
24635\relax&\mbox{if $m_{1}=m_{2}=1,$}\\\ 3\mathchar
24635\relax&\mbox{otherwise}\end{array}\right.$

Proof: Let $V(G)=\\{u_{1},\,u_{2},\,\dots,\,u_{t}\\}.$ For $1\leq i\leq k,$
let $V_{i}=\\{a_{1}^{i},\,a_{2}^{i},\,\dots,\,a_{m_{i}}^{i}\\}$ denote the
$i^{th}$ partite set of the graph $K_{m_{1},\,m_{2},\,\dots,\,m_{k}}$ with
size $m_{i}.$ Let $U_{i}=\\{\bigcup_{j=1}^{k}U_{i}^{V_{j}}\\},$ where
$U_{i}^{V_{j}}=u_{i}\times V_{j}.$ Then $V(G\times
K_{m_{1},\,m_{2},\,\dots,\,m_{k}})=\bigcup_{i=1}^{i}U_{i}=\bigcup_{i=1}^{i}\\{\bigcup_{j=1}^{k}U_{i}^{V_{j}}\\}.$
Let the universal vertices of $G$ be $u_{1}$ and $u_{2}.$ Let $H=G\times
K_{m_{1},\,m_{2},\,\dots,\,m_{k}}.$

If $m_{1}=m_{2}=1,$ then the result follows by Corollary 18. Now we assume
that $m_{2}\geq 2.$ Consider the set
$X=\\{(u_{1},\,a_{1}^{1}),\,(u_{2},\,a_{1}^{2}),\,(u_{3},\,a_{1}^{3})\\}.$ The
vertices in $V(H)\setminus(U_{2}^{V_{1}}\cup U_{1}^{V_{2}})$ are dominated by
the vertices in $\\{(u_{1},\,a_{1}^{1}),\,(u_{2},\,a_{1}^{2})\\}$ and the
vertices in $U_{2}^{V_{1}}\cup U_{1}^{V_{2}}$ are dominated by the vertex
$(u_{3},\,a_{1}^{3}).$ Hence $X$ is CPDS of $H.$ This gives
$\gamma_{P,\,c}(H)\leq 3.$ To obtain the reverse inequality, we claim that any
set $X$ contains two vertices of $H$ is not a CPDS of $H.$ Let
$X=\\{(u_{i},\,a_{1}^{x}),\,(u_{j},\,a_{1}^{y})\\}.$ Then $X_{1}=N[X].$
Clearly, the set $X_{1}$ does not contain the vertices in the set
$U_{i}^{V_{x}}\cup U_{j}^{V_{y}}.$ The propagation step from any vertex in
$X_{1}$ to any vertex in $U_{i}^{V_{x}}\cup U_{j}^{V_{y}}$ is not possible as
$|U_{i}^{V_{x}}|$ and $|U_{j}^{V_{y}}|$ are at least two. Consequently,
$\gamma_{P,\,c}(H)>2.$ Hence, $\gamma_{P,\,c}(G\times
K_{m_{1},\,m_{2},\,\dots,\,m_{k}})=3.$

######  Theorem 22.

For $t\geq 3,\,k\geq 3,\,1\leq n_{1}\leq n_{2}\leq\dots\leq n_{t}$ and $1\leq
m_{1}\leq m_{2}\leq\dots\leq m_{k},$ we have

$\gamma_{P,\,c}(K_{n_{1},\,n_{2},\,\dots,\,n_{t}}\times
K_{m_{1},\,m_{2},\,\dots,\,m_{k}})=\left\\{\begin{array}[]{rl}2\mathchar
24635\relax&\mbox{if $n_{1}=n_{2}=1$ and $m_{1}=m_{2}=1,$}\\\ 3\mathchar
24635\relax&\mbox{otherwise}\end{array}\right.$

Proof: Let $U_{i}=\\{u_{1}^{i},\,u_{2}^{i},\,\dots,\,u_{n_{i}}^{i}\\}$ and
$V_{j}=\\{v_{1}^{j},\,v_{2}^{j},\,\dots,\,v_{m_{j}}^{j}\\},$ respectively,
denote the partite sets of the graphs $K_{n_{1},\,n_{2},\,\dots,\,n_{t}}$ and
$K_{m_{1},\,m_{2},\,\dots,\,m_{k}}$ of sizes $n_{i}$ and $m_{j}.$ If
$n_{1}=n_{2}=m_{1}=m_{2}=1,$ then the result is immediate by Corollary 18. Now
we assume that $n_{2}\geq 2.$ Consider the set
$X=\\{(u_{1}^{1},\,v_{1}^{1}),\,(u_{1}^{2},\,v_{1}^{2}),\,(u_{1}^{3},\,v_{1}^{3})\\}.$
The vertices in $X$ dominates all the vertices of
$K_{n_{1},\,n_{2},\,\dots,\,n_{t}}\times K_{m_{1},\,m_{2},\,\dots,\,m_{k}}$
and hence $\gamma_{P,\,c}(K_{n_{1},\,n_{2},\,\dots,\,n_{t}}\times
K_{m_{1},\,m_{2},\,\dots,\,m_{k}})\leq 3.$ By employing similar argument as in
Theorem 21, we can conclude that
$\gamma_{P,\,c}(K_{n_{1},\,n_{2},\,\dots,\,n_{t}}\times
K_{m_{1},\,m_{2},\,\dots,\,m_{k}})\geq 3.$

Acknowledgment The first author is supported by Dr. D. S. Kothari Postdoctoral
Fellowship, University Grand Commission, Government of India, New Delhi
through the Grand No.F.4-2/2006(BSR)/MA/20-21/0067.

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