# Novel method for in-situ drift velocity measurement in large volume TPCs:
the Geometry Reference Chamber of the NA61/SHINE experiment at CERN

András László  Ádám Gera  Gergő Hamar  Botond Pálfi  Piotr Podlaski  Brant
Rumberger  Dezső Varga

###### Abstract

This paper presents a novel method for low maintenance, low ambiguity in-situ
drift velocity monitoring in large volume Time Projection Chambers (TPCs). The
method was developed and deployed for the $40\,\mathrm{m}^{3}$ TPC tracker
system of the NA61/SHINE experiment at CERN, which has a one meter of drift
length. The method relies on a low-cost multi-wire proportional chamber (MWPC)
placed downstream of the TPCs to be monitored. The drift velocity is then
determined by matching the reconstructed tracks in the TPC to the hits of the
pertinent monitoring chamber, called Geometry Reference Chamber (GRC), which
is then used as a differential length scale. An important design requirement
on the GRC was minimal added complexity to the existing system, in particular,
compatibility with Front-End Electronics (FEE) cards already used to read out
the TPCs. Moreover, the GRC system was designed to operate both in large and
small particle flux. The system is capable of monitoring the evolution of the
in-situ drift velocity down to a one permil precision, with a few minutes of
time sampling.

##  1 Introduction

The NA61 experiment, also called the SPS Heavy Ion and Neutrino Experiment
(SHINE) [2], is a large-acceptance hadron spectrometer experiment at the Super
Proton Synchrotron (SPS) accelerator at CERN. Its physics program includes
measurements for heavy-ion physics, as well as measurements on particle
production in hadron-nucleus collisions, with an emphasis on the application
of those measurements in flux predictions for long-baseline neutrino
oscillation experiments (T2K, DUNE) and for cosmic ray observatories (Pierre
Auger Observatory). These physics programs demand the operability of
NA61/SHINE in both high and low particle multiplicity environments.

The main components of the NA61/SHINE experiment are two large superconducting
bending magnets and a system of large volume Time Projection Chambers (TPCs).
As the drift dimension of the TPCs is considerably large
(${\gtrsim}1\,$meter), in order to achieve necessary position resolution in
the drift direction, the in-situ drift velocity inside the TPCs needs to be
monitored needs to me monitored at the permil level. Since the drift velocity
slowly evolves with time, its evolution needs to be followed with a few
minutes of time sampling. In this paper, we describe a cost-efficient
supplementary detector system specifically developed for this purpose.

One of the default methods for monitoring drift velocity in a TPC is to
analyze its exhaust gas with a small probe chamber [1, 2, 3, 4] and then
predict the in-situ drift velocity of the TPC via the known pressure,
temperature and drift field in the TPC gas. This method dates back to the
large-volume TPCs at LEP. NA61/SHINE also uses this technique to monitor the
drift velocity in its TPCs, but due to the limited absolute systematic
accuracy of the method it is mainly used for monitoring the time stability of
the gas composition.111The so-called normalized drift velocity, estimated via
exhaust analyzer, is very sensitive to changes in the working gas mixture
composition. This quantity is the drift velocity extrapolated to normal
pressure ($101325\,k\mathrm{Pa}$) and normal temperature ($20\,{}^{\circ}C$),
and to a fixed drift field. The exhaust method is prone to uncertainties in
pressure and temperature measurements in the large-volume chambers, and is
also vulnerable to contaminations along the exhaust line and system
outgassing.

Due to the systematic uncertainties of the exhaust analyzer method, typically
it is complemented by absolute in-situ drift velocity measurements. For this,
typically a UV laser system is used: ionizing laser beam tracks are projected
into the sensitive volume at known locations, and the drift velocity can then
be measured from the apparent positions of these beams in terms of TPC drift
time. Direct measurement methods using laser systems are common [5, 3] and
very accurate. UV laser-based drift velocity and geometry monitoring systems,
however, are rather hard to implement or upgrade on existing chambers
retrospectively.222Due to historical reasons, NA61/SHINE is not equipped with
such a UV laser based system. Moreover, the cost and development time for such
systems is substantial.

Along with other CERN experiments, the NA61/SHINE facility was also
significantly upgraded during the Long Shutdown 2 (LS2) period (2019-2021) of
the accelerator complex. Since during the LS2 the subdetectors were moved,
precise alignment calibration of the chambers was required, as was a system
for accurately and reliably determining their drift velocities. As the TPCs
were already constructed, minimally-invasive measurement systems were
preferred. Our chose was a simple solution: a planar detector with fixed
segmentation along the TPC drift direction, called the Geometry Reference
Chamber (GRC), was to be placed downstream of the TPC system. TPC tracks
reconstructed with an approximately known drift velocity were extrapolated to
the GRC and paired with the GRC hits. If the drift velocity estimate differs
from the true drift velocity inside the TPC, the mismatch of the drift
coordinate of the hits becomes worse with increasing drift depth. The
correction between the drift velocity estimate and the true drift velocity
then can be calculated from the slope of this mismatch scatter plot. That is,
the GRC is used as a differential length scale in order to convert the drift
time, measured by the TPC readout, to drift depth. It is quite advantageous
that the GRC method is differential: an accurate alignment of the TPC against
the GRC is not needed. The concept is illustrated in Fig.1.333Once the drift
velocity has been calibrated with such method in one of the TPC chambers in
the detector complex, the other chambers can be calibrated against each-other,
using the already calibrated chambers as geometry reference. Moreover, the
remaining geometry calibration constants of the TPC system can also be
determined, such as the trigger latency ($t_{0}$), alignment shift and
alignment angles for each chamber, see Appendix A and B.

Figure 1: (Color online) Illustration of the Geometry Reference Chamber (GRC)
concept. The particle tracks in the TPC chamber are reconstructed with an
initial estimate for the drift velocity and are paired to the hits of a
segmented detector, the GRC. In case the drift velocity estimate differs from
the true in-situ drift velocity inside the TPC, the mismatch along the drift
coordinate gets gradually worse at increasing drift depths. The slope of the
drift coordinate mismatch against the drift coordinate as measured by the GRC
provides the correction between the true in-situ drift velocity versus the
initial estimate.

In order to implement the GRC concept, several design requirements were
considered. To reduce the development time and cost of the readout
electronics, a solution using the existing TPC readout Front-End Electronics
(FEE) was chosen. In order to reduce the number of readout channels, we
decided to implement a simple Multi-Wire Proportional Chamber (MWPC) with
cartesian readout. This required two issues to be addressed. Firstly, the
signal formation in the GRC had to be slowed down sufficiently such that the
signal persists until the TPC electronics begins to sample the charges. This
is not entirely trivial to achieve, since TPC readout electronics typically
have a microsecond-level trigger latency with respect to the particle passage
time, during which the signal may already disappear from the GRC. In order to
mitigate this issue, a drift volume with adjustable drift field was introduced
in the GRC design, transverse to the particle passage direction, and with that
the GRC signal duration could be controlled. Secondly, the cartesian readout
scheme tends to be problematic whenever the particle occupancy is large. In
order to overcome this limitation, the GRC acceptance was designed to be
variable: for low-multiplicity type of collisions, the entire GRC acceptance
can be used, whereas for high-multiplicity collisions, individual anode wires
(sense wires) of the GRCs can be disabled, down to a single amplifying wire.
This helps reduce the number of combinatorial ghost hits.

The structure of the paper is as follows. In Section 2 an overview of the
NA61/SHINE experimental facility is given. In Section 3 the GRC design
requirements, parameters, and implementation are discussed. In Section 4 the
operational performance is shown. In Section 5 a conclusion is provided. The
paper is closed by Appendix A and Appendix B discussing the details of
systematic errors and the calibration of further geometry-related TPC
calibration constants, where the GRC chamber plays a role.

##  2 The NA61/SHINE experimental facility

The NA61/SHINE [2] is a fixed-target hadron spectrometer experiment at the
CERN SPS accelerator. Large parts of its tracking devices were inherited from
a previous experiment called NA49 [3]. Its physics program covers the study of
strongly-interacting matter via heavy-ion collisions and measurements of
identified particle production spectra in hadron-nucleus collisions as
reference data for flux prediction in long baseline neutrino experiments (T2K,
DUNE) and large area cosmic ray observatories (Pierre Auger Observatory).

An overview of the NA61/SHINE experiment is shown in Fig.2. Two large
superconducting bending magnets (Vertex-I and II) are responsible for particle
deflection for charge and momentum determination. Their total maximum bending
power is ${\sim}9\,\mathrm{Tm}$ (up to $1.5\,\mathrm{Tesla}$ magnetic field in
Vertex-I and $1.1\,\mathrm{Tesla}$ in Vertex-II). A target holder with target-
in / target-out moving capability sits just upstream of the first Vertex TPC.
Thin targets can be placed inside a silicon Vertex Detector (VD) for precise
vertex determination. NA61/SHINE also has the ability to measure interactions
in extended replica targets for long baseline neutrino experiments. The
tracking devices for spectrometry are composed of eight large volume TPCs
(total ${\sim}40\,\mathrm{m}^{3}$ and ${\sim}1\,\mathrm{m}$ drift length),
capable of performing both tracking and $\mathrm{d}E/\mathrm{d}x$
measurements. An MRPC-based Time-of-Flight wall (L-ToF) provides further
particle identification (PID) capabilities around mid-rapidity, whereas a
scintillator based Time-of-Flight wall covers (F-ToF) covers the forward phase
space, enabling two-dimensional (ToF+$\mathrm{d}E/\mathrm{d}x$) separation of
particle species at large parts of the acceptance. A calorimeter is placed at
the end of the beamline, called the Projectile Spectator Detector (PSD), which
helps to characterize collision centrality in heavy-ion collisions, and
consists of two compartments (MPSD and FPSD) for optimal shower containment.
Upstream of the target position, a set of beam position detectors (BPDs),
scintillator and Cherenkov detectors serve as beam and beam PID trigger (not
shown in the figure). On the beamline between VTPC-1 and the GapTPC, a small
plastic scintillator with a $1\,\mathrm{cm}$ diameter serves as an interaction
trigger in most collision types (S4). In rare run settings, a very similar
scintillator (S5) just upstream of MPSD is used instead. For heavy-ion runs
the scintillator (S3) just downstream of VD is used for interaction definition
through detecting loss of charge (i.e., by sensing a decreased $Z^{2}$ with
respect to the beam nuclei). The VD, the new silicon BPDs, the MRPC-based
L-ToF, the MPSD and FPSD were introduced during the Long Shutdown 2 (LS2)
upgrade period between 2019-2021. Also during LS2, the data acquisition (DAQ)
was upgraded, allowing for an event recording rate up to $1.8\,k\mathrm{Hz}$
for the detector complex.

Figure 2: (Color online) The NA61/SHINE detector configuration, after the Long
Shutdown 2 (LS2) upgrade, i.e. since 2022. The drawing shows the top view of
the bending plane. That is, the approximately homogeneous magnetic field
inside the Vertex-I and Vertex-II magnets is perpendicular to the figure. The
electric drift field of the TPCs is also perpendicular to the figure. The GRC
system for in-situ drift velocity monitoring was also conceptualized and
introduced during the LS2. The GRC system, as of now, consists of two adjacent
identical chambers (GRC-1 and GRC-2), for redundancy reasons.

The concept of the Geometry Reference Chambers (GRCs), just downstream of
MTPC-L, was also developed during the LS2 upgrade period. Its lightweight
design was motivated by short development time, and relatively fast and low
risk implementability. In addition, the TPC chambers were moved during the LS2
upgrade, and therefore seven calibration constants apart from the drift
velocity must be re-determined. These are the trigger latency, the three-
dimensional alignment shifts, and the three alignment angles for each TPC.
These are too many unknowns to determine without an absolute reference. The
GRC enables unambiguous determination of all seven parameters and the drift
velocity.

##  3 The GRC system

The NA61/SHINE TPCs have a drift length of ${\sim}1\,\mathrm{m}$ and are
capable of resolving particle cluster positions at the ${\sim}1\,\mathrm{mm}$
at the extremum of the drift volume. Therefore, in order to reduce the
systematic errors of the position measurements below the position resolution,
the in-situ drift velocity has to be determined with permil accuracy. The
drift velocity in a typical TPC system is affected by the working gas
composition, the electric drift field applied to the TPC, the gas temperature,
and the gas pressure. Most large TPC systems, such as the TPCs in NA61/SHINE
experiment or the one in ALICE [5] at CERN’s LHC, or in STAR [6, 7] at BNL’s
RHIC, are equilibrated with the atmospheric pressure, with a very slight
constant overpressure applied in order to avoid air inflitration. Since the
gas composition, drift field and the temperature is typically stabilized, the
shortest time scale variations are caused by meteorological changes in the
ambient air pressure. This means that the time scale of anticipated
substantial drift velocity variations are not shorter than about $5$ minutes.
Therefore, the design goal of the GRC system was to have an in-situ drift
velocity measurement inside MTPC-L with a permil statistical error for about
every $5$ minute time window during data taking. Given the worst-case minimal
particle flux, this requirement gives the lower bound to the acceptance (area)
of the GRC in case of low-multiplicity collision types. In order to minimize
the necessary area of the GRC, a location was chosen with the maximum possible
particle flux passing through MTPC-L, but avoiding the vicinity of the beam
spot. This justifies the chosen placement of the GRC system, shown in Fig.2.
At the selected location, the worst-case minimum particle flux was estimated
using formerly recorded proton-proton collision data at $13\,\mathrm{GeV}/c$
beam momentum. With about a factor of two safety margin, this resulted in a
$40\,\mathrm{cm}$ $\times$ $120\,\mathrm{cm}$ sensitive area, with the longer
dimension fully covering the drift direction of MTPC-L. For redundancy
reasons, two identical GRC stations (GRC-1 and GRC-2) were placed in an
adjacent position (see Fig.2).

For cost efficiency reasons, the GRC implementation was chosen to be a robust
MWPC with cartesian readout loosely based on the design [8, 9], which was
originally optimized for cosmic muon imaging purposes. Gas amplification is
achieved on the anode wires (Sense Wires), and the transverse position
information is read from adjacent wires (Field Wires) parallel to these in the
same plane. For the vertical position information relevant for drift velocity
monitoring, the segmentation is achieved using wires with the same
functionality as cathode strips, perpendicular to the Sense Wire / Field Wire
direction (those are called Pickup Wires). The design choice to use such wires
instead of strips is due to easier construction of the whole chamber. For low-
multiplicity data taking campaigns, the entire GRC acceptance is read out as
is. For high-multiplicity runs, such as heavy-ion data taking periods, the
cartesian combinatorial background becomes an issue. In order to mitigate
this, the design allowed for high-voltage jumpers used to turn off individual
Sense Wires. This has the effect of narrowing the transverse acceptance down
to a single active sense wire. This feature completely eliminates the
cartesian combinatorics, preserving the drift velocity determination accuracy
in high multiplicity runs. The concept is sketched in the left panel of Fig.3.

Figure 3: (Color online) Left panel: sketch of a GRC chamber as seen from the
direction of the arriving particle flux (not to scale). The transverse
direction is covered by the Field Wire / Sense Wire plane, where the electron
multiplication takes place on the Sense Wires, and their mirror charges on the
Field Wires are read out. The TPC drift direction is covered by the Pickup
Wire plane, also seeing the mirror charges of the charge clouds on the Sense
Wires. This provides the other cartesian coordinate for readout. Individual
Sense Wires can be connected to high voltage or ground using configurable
high-voltage jumpers (not drawn on the figure). Right panel: sketch of a GRC
as seen from side view (not to scale). The thickness of the drift region is
larger in the direction of the particle flux, and a drift cathode electrode
with adjustable potential is used. This combination makes sure that the charge
collection can be sufficiently slowed down such that the signal in the chamber
persists until the TPC readout electronics, used for the GRC readout, begins
to measure.

In order to reduce cost and development time for the readount electronics, the
GRCs were designed to be read out using the existing TPC FEEs. Two TPC FEE
cards are enough to read out the induced charges on the $32$ Field Wire
channels $\times$ $192$ Pickup Wire channels in each GRC. A natural issue with
this solution is that the TPC, being a relatively slow device, allows for a
relatively large trigger latency. A typical TPC readout FEE is generally
optimized for making meaningful charge sampling measurements only
$1-2\,\mu\mathrm{s}$ after the beam passage. In order to make sure that the
signal in the GRC persists until the TPC FEE begins to take meaningful charge
samples, a $30\,\mathrm{mm}$ thick drift volume was added. This has the effect
of extending charge collection over time. In order to control the signal
collection timescale, an adjustable drift field electrode was added to the
design. The signal stretching concept is shown schematically in the right
panel of Fig.3. The GRC front-end readout system is fully integrated with the
NA61/SHINE Data Acquisition System (DAQ). GRC-specific event reconstruction
software is fully integrated with the NA61/SHINE offline software
(ShineOffilne) [10]. The fundamental parameters of the MWPC design of the GRCs
are listed in Table 1.

chassis | FR4 bars and typical PCB board material
---|---
typical working gas |  $\mathrm{Ar}(80):\mathrm{CO}_{2}(20)$ at $5\,\mathrm{l/h}$ flow
typical Drift Cathode potential | $0\,\mathrm{V}$
Field Wire and Pickup Wire potential | $0\,\mathrm{V}$
typical Sense Wire potential | $1580\,\mathrm{V}$
Pickup Wire | CuZn40 (EDM Tec) $100\,\mu{}\mathrm{m}$, $70\,\mathrm{g}$ tension
Field Wire | CuZn40 (EDM Tec) $100\,\mu{}\mathrm{m}$, $70\,\mathrm{g}$ tension
Sense Wire | Au coated W (Luma Metall) $20\,\mu{}\mathrm{m}$, $15\,\mathrm{g}$ tens.
Field Wire to Sense Wire distance | $6\,\mathrm{mm}$
distance between adjacent Pickup Wires |  $3\,\mathrm{mm}$ (read out pairwise)
FW/SW to Pickup Wire plane distance | $10\,\mathrm{mm}$
Pickup Wire plane to cathode backplane dist. | $2\,\mathrm{mm}$
Drift Cathode to FW/SW plane | $30\,\mathrm{mm}$
number of Sense Wires | $31$
number of Field Wires | $32$
number of Pickup Wire pairs | $192$
Table 1: Design parameters of the GRC MWPCs.

Since the GRC performs a differential measurement along the TPC drift
direction, precise alignment of the GRCs with respect to the TPC is not
critical. However, if the TPC-GRC alignment is known, the GRC system can be
used to determine a further calibration constant: the trigger latency
($t_{0}$). In order to make use of this possibility, care was taken in the
design such that the position of the optical survey target for geodesic
localization of the GRCs can be accurately related to the internal wire
positions. A CAD sketch of the sensor plane and a photograph of the installed
GRC system is seen in Fig.4.

Figure 4: (Color online) Left panel: the PCB CAD snapshot of the sensor layer
of a GRC chamber. The six holders for the optical geodesic survey targets are
well visible. By design, these are precisely relatable to the internal wire
geometry. Right panel: the photograph of the installed GRC system as seen
looking upstream. The black wall is the F-TOF, covering the downstream plane
of MTPC-L. The upstream chamber is GRC-1, whereas the downstream chamber is
GRC-2.

##  4 Drift velocity calibration using the GRC

Drift velocity calibration via the GRC is based on track matching between
MTPC-L and the GRC as sketched in Fig.1, and subsequently using GRC as a
differential length scale.444For combinatorial background suppressing reasons,
in practice global tracks are used, which are associated to a collision in the
target (global main-vertex tracks). These can be already reconstructed with
approximate estimates for the calibration parameters, such as the drift
velocity. Then, these global main-vertex tracks are dissected to local tracks,
and the parameter mismatch of the local track pieces in the adjacent TPC
chambers are evaluated for calibration. For MTPC-L, the mismatch against the
GRC hits are used for calibration. If the drift velocity estimate differs from
the true drift velocity, the TPC drift coordinate versus GRC coordinate
mismatch has a slope as a function of the GRC coordinate. This slope can be
used for calibration. Tracks are typically collected over the course of 5
minutes of data taking, providing a drift velocity estimate spanning those 5
minutes.

The calibration equations can be derived as follows. The NA61/SHINE coordinate
convention, as seen in Fig.2, is that the global cartesian $x$ coordinate is
from MTPC-R to MTPC-L, the global $y$ coordinate is from down to up, and the
global $z$ coordinate is along the nominal beamline from upstream toward
downstream. Similarly oriented $x$, $y$, $z$ cartesian local coordinates are
also used within each chamber. The TPC drift coordinate is the $y$ local
coordinate, by convention, and the drift is from negative to positive
direction, whereas the local $x$ coordinate is along the readout pads of a
padrow (along the Sense Wires of the TPC), and the local $z$ coordinate is
stepping in between pad rows. For a TPC chamber, one has that

$\displaystyle y_{\mathrm{nom}}$ $\displaystyle=$ $\displaystyle
y_{\mathrm{anode},\mathrm{nom}}-(t_{0,\mathrm{nom}}+t_{\mathrm{drift,raw}})\cdot
v_{\mathrm{drift,nom}},$ (4.1) $\displaystyle y_{\mathrm{true}}$
$\displaystyle=$ $\displaystyle
y_{\mathrm{anode},\mathrm{true}}-(t_{0,\mathrm{true}}+t_{\mathrm{drift,raw}})\cdot
v_{\mathrm{drift,true}}.$ (4.2)

Here $y_{\mathrm{nom}}$ is the reconstructed drift coordinate assuming a
nominal estimate during reconstruction for the position of the TPC
amplification plane at $y_{\mathrm{anode},\mathrm{nom}}$. The symbol
$t_{0,\mathrm{nom}}$ stands for a nominal estimate for the trigger latency
during reconstruction. $t_{\mathrm{drift,raw}}$ is the measured particle track
cluster position in terms of drift time as measured by the TPC readout.
$v_{\mathrm{drift,nom}}$ is the nominal estimate for the electron drift
velocity in the TPC during reconstruction. The quantities labeled by
$\,{()}_{\mathrm{true}}\,$ are the true but unknown values of these
calibration factors. The negative sign in Eq.(4.2) is due to the coordinate
convention in NA61/SHINE. The coordinate $y_{\mathrm{nom}}$ measured by the
TPC along the drift direction corresponds to a GRC measurement
$y_{\mathrm{true}}$, for each track matched to a GRC hit, as seen in Fig.1.
The matching is done via a typically ${\sim}3\,\mathrm{cm}$ wide tolerance
window. By eliminating the running variable $t_{\mathrm{drift,raw}}$ in
Eq.(4.2), one infers

$\displaystyle\Delta{y}$ $\displaystyle=$
$\displaystyle\underbrace{\left(v_{\mathrm{drift,nom}}/v_{\mathrm{drift,true}}-1\right)}_{\mathrm{slope}}\cdot
y_{\mathrm{true}}$ (4.3) .
$\displaystyle\qquad\;+\;\underbrace{\left(y_{\mathrm{anode,nom}}-v_{\mathrm{drift,nom}}/v_{\mathrm{drift,true}}\cdot
y_{\mathrm{anode,true}}+v_{\mathrm{drift,nom}}\cdot(t_{0,\mathrm{true}}-t_{0,\mathrm{nom}})\right)}_{\mathrm{offset}}$
(4.4)

for the ensemble of TPC-GRC tracks. Here
$\Delta{y}:=y_{\mathrm{nom}}-y_{\mathrm{true}}$ stands for the drift
coordinate TPC-GRC mismatch and $y_{\mathrm{true}}$ stands for the GRC
measurement. Occasionally, the label $()_{\mathrm{true}}$ will be suppressed
where not confusing. The slope of the $\Delta{y}$ versus $y$ scatter plot
determines the correction factor between the true drift velocity
$v_{\mathrm{drift,true}}$ and the estimate $v_{\mathrm{drift,nom}}$ assumed
during the reconstruction, thus $v_{\mathrm{drift,true}}$ can be determined
from the TPC-GRC mismatch data. This is shown in Fig.5 top panel for a typical
low-multiplicity data set (proton-carbon collisions at $120\,\mathrm{GeV}/c$
beam momentum, recorded in 2023). In Fig.5 bottom left panel the same is
shown, but for a typical high-multiplicity data set (minimum-bias Pb+Pb
collisions at $150\,A\mathrm{GeV}/c$ beam momentum, recorded in 2022). Fig.5
bottom right panel shows the same high-multiplicity data set with only a
single Sense Wire switched on, thus eliminating the cartesian combinatorial
ghost hits using the design feature explained in Section 3.

Figure 5: (Color online) Top panel: an example for the $\Delta{y}$ versus $y$
drift velocity calibration method on a real data sample in a $5$-minute time
window. The corresponding data set is a low multiplicity run (proton-carbon
collisions at $120\,\mathrm{GeV}/c$ beam momentum, recorded in 2023). Here,
the full GRC acceptance was used. Bottom left panel: the equivalent plot for a
high multiplicity run, with the full GRC acceptance (minimum-bias Pb+Pb
collisions at $150\,A\mathrm{GeV}/c$ beam momentum, recorded in 2022). Bottom
right panel: the same high multiplicity data set with only a single Sense Wire
used, thus eliminating the cartesian combinatorial ghost hits in GRC. Note the
substantially-reduced background. The color scale is merely to emphasize hit
density.

In order to further suppress the combinatorial background accompanying the
correlation signalsignal, the ROOT implementation of the Least Trimmed Squares
(LTS) fitting was used. After calibrating MTPC-L, the upstream chambers are
subsequently calibrated step-by-step using downstream chambers as geometric
references.

The statistical uncertainty of the method can be estimated from above by
treating the pickup wire electrodes as $6\,\mathrm{mm}$ wide strips. Thus, the
single-track position resolution of the GRC is not worse than
$6\,\mathrm{mm}/\sqrt{12}$, corresponding to the resolution of a
$6\,\mathrm{mm}$ wide uniform distribution. For a track sample of a
${\sim}5\,\mathrm{minute}$ time window, typically there are hundreds of MTPC-L
tracks hitting the GRC. It is thus possible to reach the desired statistical
uncertainty of $1\,\mathrm{mm}$, corresponding to $1$ per mil of the
${\sim}1\,\mathrm{m}$ TPC drift length. The systematic accuracy of the method
was quantified using a closure test. By re-reconstructing the data using the
calibrated drift velocities, the calculated correction on top of the
previously-corrected drift velocities was found to be better than 1-2 permil.
This can be attributed to the systematic uncertainty of the method. The
systematics were also double-checked by comparing the obtained drift
velocities to the exhaust analysis method, which agreed within 1-2 permil, up
to a normalization constant. Note that the exhaust analysis method is expected
to have worse systematics in comparison to the more-direct $\Delta{y}$ versus
$y$ method discussed in this paper. These validation plots are seen in Fig.6.

Figure 6: (Color online) Final validation of the GRC-based $\Delta{y}$ versus
$y$ drift velocity calibration method. It is seen that the closure test is
satisfied to the desired accuracy (1-2 permil). Moreover, good consistency is
seen with the exhaust analyzer method. (Note that the exhaust analyzer method
is expected to have some absolute systematic error, hence the necessity for
its normalization scale adjustment.)

##  5 Concluding remarks

In this paper a novel cost-efficient method was described for in-situ drift
velocity monitoring in large volume TPCs. The method was deployed and is
currently in use at the NA61/SHINE experiment at CERN. The key idea is to
place a low-cost segmented detector, called to be the Geometry Reference
Chamber (GRC), downstream of the TPC to be monitored. This monitoring solution
was added retrospectively to the existing TPC system without a surgical
operation to the chambers. A GRC was cost-effectively realized by using a MWPC
designed to be compatible with existing TPC readout electronics. In such a way
the development time and cost of the readout electronics could be spared. In
order to match such an MWPC to the TPC readout electronics, it was necessary
to slow down the signal formation in the GRC, since TPC readout electronics
typically posses a considerable trigger latency. This signal slowing was
achieved by using an adjustable drift field transverse to the particle passage
direction. The number of required readout channels was minimized using
cartesian readout. Since the NA61/SHINE experiment is used to study both low-
and high-multiplicity collisions, the acceptance of the GRC was designed to be
sufficiently large while addressing the issue of combinatorial ghost hits. The
latter was handled by making the acceptance adjustable: when studying high-
multiplicity collisions, individual Sense Wires can be switched off down to a
single wire, eliminating the cartesian ghosts, whereas for low-multiplicity
runs the full acceptance is used with all of the Sense Wires operational.

The GRC system has been demonstrated of being capable of monitoring the drift
velocity down to a one permil absolute precision in about $5$ minute time
windows. That is, the pertinent method is applicable to large drift length TPC
systems, with minimal design time, and relatively low manpower and cost.

Motivated by the GRC-based drift velocity monitoring method, a calibration
procedure for the other geometric calibration parameters of the NA61/SHINE TPC
chambers was also developed. Using that concept, all 8 parameters (time-
dependent drift velocity, trigger latency, alignment shifts, and alignment
angles) of the TPC system were successfully calibrated.

## Acknowledgements

We thank to the members of the REGARD gaseous R&D group at the Wigner RCP,
moreover for the support and cooperation from the members of the NA61/SHINE
Collaboration at CERN, in particular to Bartosz Maksiak, Wojciech Bryliński
and Kyle Allison.

This work was supported by the Hungarian Scientific Research Fund (NKFIH
OTKA-K138136-K138152, OTKA-FK135349 and TKP2021-NTKA-10). Detector
construction and testing was completed within the Vesztergombi Laboratory for
High Energy Physics (VLAB) at HUN-REN Wigner RCP.

## Appendix

##  Appendix A Trigger latency and drift-direction alignment of TPCs

After the drift velocity has been calibrated with the $\Delta{y}$ versus $y$
method, one has $v_{\mathrm{drift,nom}}\approx v_{\mathrm{drift,true}}$, and
therefore the TPC-GRC drift coordinate mismatch becomes $y$-independent,
expressable as

$\displaystyle\Delta{y}$ $\displaystyle=$
$\displaystyle\underbrace{(t_{0,\mathrm{true}}-t_{0,\mathrm{nom}})}_{\mathrm{slope}}\cdot
v_{\mathrm{drift}}\;+\;\underbrace{(y_{\mathrm{anode,nom}}-y_{\mathrm{anode,true}})}_{\mathrm{offset}}$
(A.1)

as seen from Eq.(4.4). Taking several data samples with in-situ drift velocity
differing by $1-2\%$, the slope of the $\Delta{y}$ versus $v_{\mathrm{drift}}$
scatter plot provides the additive correction between the true trigger latency
($t_{0,\mathrm{true}}$) against the trigger latency estimate applied during
the reconstruction ($t_{0,\mathrm{nom}}$). Thus, $t_{0,\mathrm{true}}$ can be
determined from the TPC-GRC mismatch data. After the $t_{0}$ is calibrated,
the TPC-GRC mismatch will be not only $y$-independent, but also
$v_{\mathrm{drift}}$-independent, and one is left with
$\Delta{y}=y_{\mathrm{anode,nom}}-y_{\mathrm{anode,true}}$. Thus, the drift
direction displacement ($y$-alignment correction)
$y_{0}:=y_{\mathrm{anode,nom}}-y_{\mathrm{anode,true}}$ of the TPC versus the
GRC can be determined from the TPC-GRC mismatch data. An example for the
$\Delta{y}$ versus $v_{\mathrm{drift}}$ calibration method is shown in Fig.7.
Having the $t_{0}$ and $y_{0}$ obtained, the upstream TPC chambers can also be
calibrated, using the already calibrated TPCs as geometry reference, in place
of the GRC.

Figure 7: (Color online) An example for the $\Delta{y}$ versus
$v_{\mathrm{drift}}$ analysis for trigger latency ($t_{0}$) and drift
direction shift ($y_{0}$) calibration, after the drift velocity
($v_{\mathrm{drift}}$) has been obtained. The slope of the pertinent scatter
data gives the $t_{0}$ correction, whereas the offset gives the $y_{0}$
correction. For the method to be workable, data samples with different drift
velocities (up to $1-2\%$) are used. The pertinent data set is a high
multiplicity collision type (minimum-bias Pb+Pb collisions at
$150\,A\mathrm{GeV}/c$ beam momentum, recorded in 2022).

In the calibration database, for each TPC, the $t_{0}$ information is stored
as follows. The $t_{0}$ for one of the TPCs (MTPC-L) is stored as a global
reference (called to be the global $t_{0}$). For the other TPCs, only their
relative $t_{0}$ difference with respect to this reference TPC is stored
(called to be the chamber $t_{0}$). The rationale in such separation of the
global and chamber $t_{0}$ contributions is the following. The chamber $t_{0}$
values are supposed to be detector constants, as they can only depend on cable
lengths and electronic delays with respect to the trigger system. The global
$t_{0}$, however, can also depend on the physics settings, such as beam time-
of-flight, custom settings of delays inside the trigger logic etc. Thus, it is
safer to recalibrate the global $t_{0}$ (MTPC-L versus GRC) for each data
taking campaign, whereas it is enough to determine the relative chamber
$t_{0}$ constants (other TPCs versus MTPC-L) only once, as they are detector
constants. The recalculated relative chamber $t_{0}$ values are used, though,
for consistency checks and quality monitoring.

##  Appendix B Systematic errors and alignment calibration of the TPCs

As summarized in Section 4, the final systematics of the in-situ drift
velocity calibration via the $\Delta{y}$ versus $y$ method was verified to be
not worse than 1-2 permil (or $1\,\mathrm{mm}$ overall, along the
${\sim}1\,\mathrm{m}$ drift length). This performance, however, was only
achievable after some detailed studies. The main systematics error comes from
a possible misalignment of the adjacent chambers, or misalignment of MTPC-L
versus the GRC. Assume that we have a pair of TPC chambers, to be calibrated
against each-other from the point of view of drift velocity. Assume moreover
that the downstream one is either the GRC, or is an already drift velocity
calibrated TPC. Using automatic formula manipulation programs (e.g. Maple)
evaluating affine transformations is not difficult, and it is rather
straightforward to show that if the upstream chamber has an imperfectly known
alignment, that contributes to the multiplicative drift velocity correction
$v_{\mathrm{drift,corr}}:=v_{\mathrm{drift,true}}/v_{\mathrm{drift,nom}}$
obtained by the $\Delta{y}$ versus $y$ in-situ drift velocity calibration
method, described in Section 4. The track-by-track systematics is seen to be

$\displaystyle\mathrm{syst}(v_{\mathrm{drift,corr}})=\Big{(}y\cdot\theta_{x}+x\cdot\theta_{y}+(y-y_{\mathrm{mainvertex}})\cdot\theta_{x}+z_{0}\Big{)}\cdot\frac{1}{z-z_{\mathrm{mainvertex}}}$
(B.1)

to the first order.555This can be derived analytically, for magnetic field-off
data, i.e. for straight tracks. Since the drift direction of the NA61/SHINE
TPCs approximately coincide with the direction of the magnetic bending field,
i.e. the drift and the bending does not interfere to the first order, the
formula holds for magnetic field-on data as well, to a good approximation.
Here, $x,y,z$ is the particle hit position at the $z=const$ reference plane
(e.g. the GRC plane) of the $\Delta{y}$ versus $y$ study, moreover
$x_{0},y_{0},z_{0}$ denote the (unkown) correction to the position alignment
of the upstream chamber, and $\theta_{x},\theta_{y},\theta_{z}$ denote the
(unknown) correction to the angular alignment of the upstream chamber, whereas
the $x,y,z$ coordinates indexed by $()_{\mathrm{mainvertex}}$ denote the
coordinates of the collision point.666The downstream chamber is assumed to be
either the alignment reference or to be already calibrated for alignment. In
our experiment, MTPC-L is taken as reference for alignment, and all the other
chambers are calibrated against this reference, pairwise. It is seen that the
drift velocity systematics decreases with distance from the main-vertex, and
therefore justifies our placement of the GRC downstream of all the TPCs, as
seen in Fig.2. Even if the angular or position alignment of the MTPC-L with
respect to the GRC were imperfectly known to $|y\cdot\theta_{x}|\lesssim
1\,\mathrm{cm}$ or $|x\cdot\theta_{y}|\lesssim 1\,\mathrm{cm}$ or
$|(y-y_{\mathrm{mainvertex}})\cdot\theta_{x}|\lesssim 1\,\mathrm{cm}$ or
$|z_{0}|\lesssim 1\,\mathrm{cm}$, since
$|z_{{}_{\mathrm{GRC}}}-z_{\mathrm{mainvertex}}|\approx 16\,\mathrm{m}$, this
would not disturb the MTPC-L versus GRC drift velocity calibration, beyond
half a permil. That is, the alignment imperfection does not give sizable
systematics to the MTPC-L versus GRC in-situ drift velocity calibration. When
propagating the drift velocity calibration toward the upstream chambers
(VTPC-1, VTPC-2), however, the systematics Eq.(B.1) by imperfectly known
alignment can give a sizable contribution. Therefore, in this appendix we
briefly describe, how the alignment is calibrated.

A TPC chamber has in total 8 unknown calibration coefficients related to its
geometry: the $x_{0},y_{0},z_{0}$ position misalignment correction
(transverse, drift direction, longitudinal), the
$\theta_{x},\theta_{y},\theta_{z}$ angular misalignment correction, the drift
velocity correction $(v_{\mathrm{drift,corr}}-1)$, and the correction to the
trigger latency $t_{0,\mathrm{corr}}:=t_{0,\mathrm{true}}-t_{0,\mathrm{nom}}$.
These unkown imperfection parameters, to be determined by the calibration
procedure, are small, and therefore, correction to the first order Taylor
expansion terms with respect to these is enough. We use special magnetic
field-off calibration runs in order to have only straight trajectories for
charged particle tracks throughout the chamber system. In other words, the
chamber system is tomographed by an ensemble of straight global tracks
associated to the main-vertex, and their mismatch pattern is studied when
dissected and refitted by straight lines in the local detectors, as
illustrated in Fig.8.

Figure 8: (Color online) Sketch of the alignment calibration strategy for
adjacent TPCs: special magnetic field-off alignment calibration runs are
recorded, i.e. the chamber system is tomographed by straight global main-
vertex tracks. If the relative alignment parameters for the chambers assumed
during the reconstruction are imperfect, the track parameters will have a
mismatch, when compared on a joint reference plane, for the adjacent chambers.

Quantitatively, the magnetic field-off data based alignment calibration goes
as follows. For MTPC-L, the $(v_{\mathrm{drift,corr}}-1)$ parameter is
determined as in Section 4, and the parameters $t_{0,\mathrm{corr}}$,$y_{0}$
are determined as in Appendix A, whereas $\theta_{x}=\theta_{y}=\theta_{z}=0$
and $x_{0}=z_{0}=0$ by convention, i.e. MTPC-L is taken as alignment
reference. The other chambers are calibrated against each-other pairwise, for
all the 8 parameters, assuming that e.g. the downstream one is already
calibrated. An imperfection in the assumption about the 8 parameters of the
upstream chamber causes a mismatch between locally refitted track pieces in
the adjacent chambers, when compared on a $z=const$ reference plane, see again
Fig.8. If the straight track segments are parameterized by the four parameters
$N_{x},N_{y},M_{x},M_{y}$ as $x=M_{x}+(z-z_{\mathrm{ref}})\cdot N_{x}$ and
$y=M_{y}+(z-z_{\mathrm{ref}})\cdot N_{y}$, then the locally refitted and
compared local tracks will give a track parameter mismatch vector field
$\Delta{N}_{x},\Delta{N}_{y},\Delta{M}_{x},\Delta{M}_{y}$ as a function of the
track parameters $N_{x},N_{y},M_{x},M_{y}$ in the downstream chamber (which is
already calibrated or reference). Evaluating affine transformations, one
arrives at the formula

$\displaystyle\Delta{N}_{x}$ $\displaystyle=$
$\displaystyle(1+N_{x}^{2})\cdot\theta_{y}+N_{y}\cdot(N_{x}\cdot\theta_{x}+\theta_{z}),$
(B.2) $\displaystyle\Delta{N}_{y}$ $\displaystyle=$ $\displaystyle
N_{y}^{2}\cdot(\theta_{x}+\theta_{y})+N_{y}\cdot(v_{\mathrm{drift,corr}}-1)-N_{x}\cdot\theta_{z}+\theta_{x},$
(B.3) $\displaystyle\Delta{M}_{x}$ $\displaystyle=$ $\displaystyle
N_{x}\cdot(M_{y}\cdot\theta_{x}+M_{x}\cdot\theta_{y}+z_{0})+z\cdot\theta_{y}+M_{y}\cdot\theta_{z}-x_{0},$
(B.4) $\displaystyle\Delta{M}_{y}$ $\displaystyle=$ $\displaystyle
N_{y}\cdot(M_{x}\cdot\theta_{y}+M_{y}\cdot\theta_{x}+z_{0})+z\cdot\theta_{x}-M_{x}\cdot\theta_{z}-y_{0}$
(B.6)
$\displaystyle\qquad+(M_{y}-y_{\mathrm{anode,nom}})\cdot(v_{\mathrm{drift,corr}}-1)-t_{0,\mathrm{corr}}\cdot
v_{\mathrm{drift,nom}}$

for the mismatch field, to the first order, given the 8 imperfection
parameters $\theta_{x},\theta_{y},\theta_{z}$, $x_{0},z_{0}$,
$(v_{\mathrm{drift,corr}}-1)$, $t_{0,\mathrm{corr}}$, $y_{0}$. For practical
reasons, the track sample satisfying $|N_{y}|\approx 0$, i.e. close to
horizontal tracks are used, since for these the first two lines of Eq.(B.6)
simplifies as

$\displaystyle\mathrm{for\;}|N_{y}|\approx 0,\mathrm{one\;has}:$ (B.7)
$\displaystyle\Delta{N}_{x}$ $\displaystyle=$
$\displaystyle(1+N_{x}^{2})\cdot\theta_{y},$ (B.8)
$\displaystyle\Delta{N}_{y}$ $\displaystyle=$ $\displaystyle-
N_{x}\cdot\theta_{z}+\theta_{x}.$ (B.9)

That is, for magnetic field-off data, for $|N_{y}|\approx 0$ main-vertex
tracks, from the $\Delta{N}_{x}$ versus $N_{x}$ plot the parameter
$\theta_{y}$ can be read off, whereas from the $\Delta{N}_{y}$ versus $N_{x}$
plot the parameters $\theta_{x}$ and $\theta_{z}$ can be read off, see Fig.9
top panels. After re-reconstructing with the obtained
$\theta_{x},\theta_{y},\theta_{z}$ angular alignment corrections, the third
line of Eq.(B.6) yields

$\displaystyle\Delta{M}_{x}$ $\displaystyle=$ $\displaystyle N_{x}\cdot
z_{0}-x_{0}.$ (B.10)

That is, for magnetic field-off data, from the $\Delta{M}_{x}$ versus $N_{x}$
plot the parameters $x_{0}$ and $z_{0}$ can be read off, see Fig.9 bottom
panel.

Figure 9: (Color online) Top panels: an example for the
$\theta_{x},\theta_{y},\theta_{z}$ angular alignment calibration using
$\Delta{N}_{x}$ versus $N_{x}$ and $\Delta{N}_{y}$ versus $N_{x}$ plots for
$|N_{y}|\approx 0$ tracks in magnetic field-off data (the pertinent sample is
minimum-bias Pb+Pb at $150\,A\mathrm{GeV}/c$ beam momentum, recorded in 2022).
Bottom panel: an example for the $x_{0},z_{0}$ position alignment calibration
using $\Delta{M}_{x}$ versus $N_{x}$ plot for tracks in magnetic field-off
data (using the same sample as for the top panels). The black vertical lines
indicate fiducial cuts, in order to avoid regions close to the TPC field
cages, where the drift electric field can be imperfect (inhomogeneus). The
color scale is merely for emphasizing hit density.

After re-reconstruction with the obtained $\theta_{x},\theta_{y},\theta_{z}$
angular alignment corrections and the $x_{0},z_{0}$ alignment shift
corrections, the fourth line of Eq.(B.6) yields

$\displaystyle\Delta{M}_{y}$ $\displaystyle=$ $\displaystyle
M_{y}\cdot(v_{\mathrm{drift,corr}}-1)-y_{\mathrm{anode,nom}}\cdot(v_{\mathrm{drift,corr}}-1)-t_{0,\mathrm{corr}}\cdot
v_{\mathrm{drift,nom}}-y_{0}$ (B.11)

which is nothing but the already known Eq.(4.4), to the first order in
$(v_{\mathrm{drift,corr}}-1)$, re-expressed in the notations of this appendix.
That is, the drift velocity correction $(v_{\mathrm{drift,corr}}-1)$ can be
determined via the usual $\Delta{y}$ versus $y$ method, as already described
in Section 4. Finally, after re-reconstruction, one arrives at

$\displaystyle\Delta{M}_{y}$ $\displaystyle=$ $\displaystyle-
t_{0,\mathrm{corr}}\cdot v_{\mathrm{drift,nom}}-y_{0},$ (B.12)

which is nothing but the already known Eq.(A.1), re-expressed in the notations
of this appendix. Therefore, the remaining corrections $t_{0,\mathrm{corr}}$
and $y_{0}$ can be determined via the usual $\Delta{y}$ versus
$v_{\mathrm{drift}}$ method, as already described in Appendix A. Thus, the
triangular system of calibration equations for the 8 TPC geometry unkowns is
saturated, and the entire TPC system can be calibrated.

After the full alignment correction, drift velocity corection, $t_{0}$
correction, $y_{0}$ correction, the overall systematics of the geometry
related calibration parameters of the large TPCs, as estimated via closure
tests, are listed in Table 2. The statistical errors are negligible.

|  $\theta_{x},\theta_{y},\theta_{z}$ [∘] |  $x_{0},z_{0}$ [$\mathrm{mm}$] |  $v_{\mathrm{drift}}$ [$\%$] |  $t_{0}$ [$\mathrm{ns}$] |  $y_{0}$ [$\mathrm{mm}$]
---|---|---|---|---|---
| syst. | syst. | syst. | syst. | syst.
MTPC-L |  $0$ (reference) |  $0$ (reference) | $0.2$ | $11$ | $0.15$
VTPC-2 | $0.038$ | $0.30$ | $0.2$ | $33$ | $0.25$
MTPC-R | $0.014$ | $0.20$ | $0.2$ | $33$ | $0.63$
VTPC-1 | $0.026$ | $0.25$ | $0.2$ | $32$ | $0.07$
Table 2: The table of overall systematics of the TPC geometry parameters, for
the large chambers, as estimated via closure tests. The calibration scheme was
GRC $\rightarrow$ MTPC-L, followed by MTPC-L $\rightarrow$ VTPC-2, then VTPC-2
$\rightarrow$ MTPC-R and VTPC-2 $\rightarrow$ VTPC-1.

## References

  * [1] M. Kuich, Kaon production in mid-rapidity in Be+Be collisions at the CERN SPS, PhD thesis, University of Warsaw (2019), sections D.4 and D.5 [EDMS:2150851].
  * [2] N. Abgrall et al. (the NA61 Collaboration), NA61/SHINE facility at the CERN SPS: beams and detector system, JINST 9 (2014) P06005 [arXiv:1401.4699].
  * [3] S. Afanasiev et al. (the NA49 Collaboration), The NA49 large acceptance hadron detector, Nucl. Instrum. Meth. A430 (1999) 210.
  * [4] A. Cattai, H. G. Fischer, A. Morelli, Drift velocity in Ar + CH4 + water vapours, DELPHI Internal Report (1989) 63.
  * [5] J. Alme et al., The ALICE TPC, a large 3-dimensional tracking device with fast readout for ultra-high multiplicity events, Nucl. Instrum. Meth. A622 (2010) 316 [arXiv:1001.1950].
  * [6] J. Abele et al., The Laser System for the STAR Time Projection Chamber, Nucl. Instrum. Meth. A499 (2003) 692.
  * [7] M. Anderson et al., The STAR Time Projection Chamber: a unique tool for studying high multiplicity events at RHIC, Nucl. Instrum. Meth. A499 (2003) 659.
  * [8] D. Varga, G. Nyitrai, G. Hamar, L. Oláh, High efficiency gaseous tracking detector for cosmic muon radiography, Advances in High Energy Physics 2016 (2016) 1962317 [arXiv:1607.08494].
  * [9] D. Varga, Sz. J. Balogh, Á. Gera, G. Hamar, G. Nyitrai, G. Surányi, Construction and readout systems for gaseous muography detectors, J. Adv. Instrum. in Science 2022 (2022) 307.
  * [10] R. Sipos, A. László, A. Marcinek, T. Paul, M. Szuba, M. Unger, D. Veberic, O. Wyszynski, The offline software framework of the NA61/SHINE experiment, J. Phys. Conf. Ser. 396 (2012) 022045.


[1]Swathi V V

[1]Department of Mathematics, National Institute of Technology Calicut,
Calicut, 673601, Kerala, India

#  $k$-Power Graphs of Finite Groups

<EMAIL_ADDRESS>M S Sunitha<EMAIL_ADDRESS>*

###### Abstract

For a finite group $G$ and for a fixed positive integer $k$, $k\geq 2$, the
$k$-power graph of $G$ is an undirected simple graph with vertex set $G$ in
which two distinct vertices $x$ and $y$ are adjacent if and only if $x^{k}=y$
or $y^{k}=x$. In this paper, we investigate some graph parameters such as
number of edges, clique number, connectedness, etc. of $k$-power graphs of
finite groups. Also find some properties of $k$-power graphs of finite cyclic
groups, and finally we present an application.

###### keywords:

$k$-power graphs, Power graphs, Graphs of groups.

##  1 Introduction

The directed power graph $\overrightarrow{P}(S)$ of a semigroup $S$ was
defined by Kelarev and Quinn kelarev2000combinatorial , as a digraph with
vertex set $S$ in which there is an arc from a vertex $x$ to another vertex
$y$ if and only if $y=x^{n}$ for some positive integer $n$. Following this,
the (undirected) power graph $P(S)$ of a semigroup $S$ was defined by
Chakrabarty et al.chakrabarty2009undirected  as an undirected simple graph
with vertex set $S$ in which two distinct vertices $x$ and $y$ are adjacent if
and only if $x^{n}=y$ or $y^{n}=x$ for some positive integer $n$. The authors
proved that power graph of a finite group $G$ is complete if and only if $G$
is cyclic of order $1$ or $p^{m}$ where $p$ is a prime and $m$ is a positive
integer.

We find a number of research publications where researchers describe various
graph parameters for power graphs completely or in part. abawajy2013power ;
cameron2011power ; cameron2019power ; cameron2020connectivity ;
cameron2022matching ; chattopadhyay2019vertex ; chelvam2018power ;
doostabadi2015connectivity ; ma2015chromatic  are some of the literatures
which discuss numerous properties of power graphs of finite groups.

For fixed positive integer $k$ with $k\geq 2$ and for a semigroup $S$,
Sriparna Chattopadhyay and Pratima Panigrahi chattopadhyay2017some  defined
the $k-$power graph $P(S,k)$ as a graph with vertex set $S$ in which two
distinct vertices $x$ and $y$ are adjacent if and only if $x^{k}=y$ or
$y^{k}=x$. They studied cycle structures, connectedness and symmetry of
$k-$power graphs of cyclic groups. Clearly, for every positive integer $k\geq
2$, $P(S,k)$ is a spanning subgraph of $P(S)$.

In swathi2021square  the authors defined square graph of a finite group which
is a particular $k$-power graph for $k=2$, and they investigated upper bounds
for clique number and chromatic number.

In this paper, notation and terminology of herstein2006topics  for graphs,
chartrand2006introduction  for groups and nathanson2008elementary  for number
theory are followed.

We denote the order of a group $G$ by $o(G)$ and order of an element $a$ in
$G$ by $o(a)$. The identity element in $G$ is denoted by $e$. Also, a cyclic
group of order $n$ is denoted by $\mathbb{Z}_{n}$.

The order of a graph $H$ is the number of vertices in that graph. We denote
$deg(x)$ as the degree of the vertex $x$, which is the number of edges
incident to $x$. The greatest distance between any two vertices of a connected
graph $H$ is called the diameter of $H$ and is denoted by $diam(H)$. The
complete graph of order $n$ is denoted by $K_{n}$ and a path of order $n$ by
$P_{n}$. We denote $x\sim y$ if the vertices $x$ and $y$ are adjacent in the
graph.

The greatest common divisor of two integers $m$ and $n$ are denoted as
$gcd(m,n)$. For a positive integer $n>1$ and for an integer $a$ such that
$gcd(a,n)=1$, the order of $a$ modulo $n$ is defined as the least positive
integer $k$ such that $a^{k}\equiv 1(mod\,n)$, which is denoted by
$ord_{n}(a)$. If $a$ has order $\phi(n)$, where $\phi(m)$ is the Euler totient
function of $m$, then $a$ is called the primitive root modulo $n$.

In this paper we find some structural properties of $k$-power graph of a
general finite group $G$ of order $n$ and $2\leq k\leq n$. Also we prove some
properties of $P(\mathbb{Z}_{n},k)$ and give an application also.

##  2 $k$-power graphs

Let $G$ be a finite group such that $o(G)=n$ and $P(G,k)$ be the $k$-power
graph of $G$. For any two integers $k_{1},k_{2}\geq 2,$ if $k_{1}\equiv
k_{2}(mod\,n)$ then clearly $P(G,k_{1})=P(G,k_{2})$. We begin this section by
noting that for a finite group $G$ of order $n$, the edges in $P(G,k)$ is at
most $n-1$, since only the $k-$th power of each element is considered. We
calculate the number of edges in $P(G,k)$ in the following theorem.

######  Theorem 1.

Let $G$ be a finite group group of order $n$. Then the number of edges in the
$k-$power graph of $G$ is given by $\mid E(P(G,k))\mid=n-\sum_{d\mid
k_{1}}t_{d}-\sum_{d\mid k_{2},d\nmid k_{1}}\frac{t_{d}}{2}$, where
$k_{1}=gcd(k-1,n),k_{2}=gcd(k^{2}-1,n)$ and $t_{d}=$number of elements of
order $d$ in $G$.

Proof: Corresponding to each element $a\in G$ there exists an edge in $P(G,k)$
between $a$ and $a^{k}$ if $a\neq a^{k}$. Now, if $x$ is an element $G$ which
has order $d$ and $d\mid k-1$, then $x^{k}=x$. Hence, if $d\mid gcd(k-1,n)$,
there are no edges corresponding to the elements of order $d$ in $G$. Now,
edges corresponding to two elements $x$ and $y$ coincide if $x^{k}=y$ and
$y^{k}=x$. In this case, $o(x)\mid{k^{2}-1}$ and $o(y)\mid{k^{2}-1}$. Also if
$x\in G$ has order $d$, $d\mid gcd(k^{2}-1,n)$ and $d\nmid gcd(k-1,n)$, then
$x\neq x^{k}$ and edges corresponding to $x$ and $x^{k}$ coincide. Note that
number of elements in $G$ of order $d$ is even if $d>2$, and if $d=2$ and
$d\mid k_{2}$, then $d\mid k_{1}$. Since there are $\phi(d)$ elements of order
$d$ in a cyclic group, the following corollary is obvious.

######  Corollary 2.

Let $\mathbb{Z}_{n}$ be the cyclic group of order $n$. Then $\mid
E(P(\mathbb{Z}_{n},k))\mid=n-\sum_{d\mid k_{1}}\phi(d)-\sum_{d\mid
k_{2},d\nmid k_{1}}\frac{\phi(d)}{2}$.

######  Example 1.

The $k$-power graphs of the symmetric group on 3 elements ($S_{3}$) for
$k=2,3,4$ and $5$ are given in figures 1,2,3 and 4 respectively, where
$\sigma_{0}=e,\sigma_{1}$=(1 2 3), $\sigma_{2}=$(1 3 2), $\tau_{1}=$(1 2),
$\tau_{2}$=(2 3) and $\tau_{3}$=(1 3) .

$\sigma_{0}$$\tau_{1}$$\tau_{2}$$\tau_{3}$$\sigma_{1}$$\sigma_{2}$Figure 1:
$P(S_{3},2)$$\sigma_{0}$$\tau_{1}$$\tau_{2}$$\tau_{3}$$\sigma_{1}$$\sigma_{2}$$\sigma_{0}$$\tau_{1}$$\tau_{2}$$\tau_{3}$$\sigma_{1}$$\sigma_{2}$Figure
2: $P(S_{3},3)$

$\sigma_{0}$$\tau_{1}$$\tau_{2}$$\tau_{3}$$\sigma_{1}$$\sigma_{2}$Figure 3:
$P(S_{3},4)$$\sigma_{0}$$\tau_{1}$$\tau_{2}$$\tau_{3}$$\sigma_{1}$$\sigma_{2}$Figure
4: $P(S_{3},5)$

######  Example 2.

The $k$-power graph of $\mathbb{Z}_{4}$ when $k=2$ is the following tree.

1203Figure 5: $P(\mathbb{Z}_{4},2$)

It is clear from the Figures 1-5 that $k$-power graphs of a finite group need
not be connected always. The following proposition gives a characterization to
the cyclic groups such that its $k-$power graph is connected.

######  Proposition 3.

chattopadhyay2017some  The graph $P(\mathbb{Z}_{n},k)$ is connected if and
only if $n\mid k^{m}$ for some $m\in\mathbb{N}.$ Also in this case
$P(\mathbb{Z}_{n},k)$ is a tree.

For any finite group $G$, the characterization for $P(G,k)$ to be connected is
given in the following theorem.

######  Theorem 4.

Let $G$ be a finite group. $P(G,k)$ is connected if and only if for every
$x\in G$, $o(x)\mid k^{n}$ for some $n\in\mathbb{N}$. For $x\in G$, let
$n_{x}=min\\{n\in\mathbb{N}:o(x)\mid k^{n}\\}$, then $diam(P(G,k))\leq
2max\\{n_{x}:x\in G\\}$.

Proof: Let $x\in G$ and $o(x)\mid k^{n}$ for some $n\in\mathbb{N}$. Suppose
$n_{x}=min\\{n\in\mathbb{N}:o(x)\mid k^{n}\\}$. Then $x$ is connected to $e$
through a path of length $n_{x}$ in $P(G,k)$. Hence $P(G,k)$ is connected if
for all $x\in G$, $o(x)\mid k^{n}$ for some $n\in\mathbb{N}$, and
$diam(P(G,k))\leq 2max\\{n_{x}:x\in G\\}$. The converse also follows. The
following theorem proves that the clique number is bounded above.

######  Theorem 5.

The clique number of $P(G,k)$, $\omega(P(G,k))\leq 3$ for any finite group $G$
and $\omega(P(G,k))=3$ if and only if there exist elements in $G$ of order $m$
such that $m>3,m\mid k^{3}-1$ and $m\nmid k-1.$

Proof: If possible, suppose there exist elements $x,y,z,w\in G$ which induces
$K_{4}$. Then the following six conditions must be satisfied.

  1. 1.

$x^{k}=y$ or $y^{k}=x$

  2. 2.

$y^{k}=z$ or $z^{k}=y$

  3. 3.

$z^{k}=w$ or $w^{k}=z$

  4. 4.

$w^{k}=x$ or $x^{k}=w$

  5. 5.

$x^{k}=z$ or $z^{k}=x$

  6. 6.

$y^{k}=w$ or $w^{k}=y$

Suppose $x^{k}=y$ in condition 1. Then conditions 4 and 5 implies that
$w^{k}=x$ and $z^{k}=x$, which is not possible by condition 3. Hence, $P(G,k)$
has no subgraph isomorphic to $K_{4}$. Hence, the clique number is atmost 3.

Now suppose $\omega(P(G,k))=3$. Then there exists an element $a\neq e$ such
that $a^{k}\neq a,e$, $a^{k^{2}}\neq a,e$ and $a^{k^{3}}=a$. Then $o(a)\mid
k^{3}-1$ and $o(a)\nmid k-1$. Note that, if $m\mid k^{3}-1$ and $m\mid
k^{2}-1$ then $m\mid k-1$. The converse also follows.

Next theorem states that the chromatic number also bounded above by 3.

######  Theorem 6.

Let $G$ be a finite group, then the chromatic number $\chi(P(G,k))\leq 3$.

Proof: Choose a vertex $x$ from a connected component of $P(G,k)$ and colour 1
is assigned. Assign colour 2 to the vertex $x^{k}$. Label $y$ to all vertices
adjacent to $x$ except $x^{k}$. Then clearly $y^{k}=x$. If $(x^{k})^{k}=y$,
colour 3 is assigned to $y$, otherwise assign colour 2. Now all the vertices
adjacent to $y$ are labelled as $z$. Then $z^{k}=y$ and colour $z$ using
colour 3. Now label $w$ to all the vertices adjacent to $z$, then $w^{k}=z$.
Again colour 3 is given to $w,$ if $(x^{k})^{k}=w$, otherwise give colour 2.
Proceed with the same process until every vertices in that component get a
colour. Apply the same procedure to each connected components of $P(G,k)$. So
we conclude that the minimum number of colours required to properly colour
$P(G,k)$ is 3.

The question of whether the $k$-power graph is perfect therefore arises
naturally. The following remark gives an answer for that.

######  Remark 1.

The $k$-power graph of a finite group need not be perfect. For example,
$P(\mathbb{Z}_{31},2)$ is a union of an isolated vertex and six $5-$cycles,
which has chromatic number $3$ and clique number equal to $2$.

The groups given by the presentation $Q_{4n}=\langle
a,b:a^{n}=b^{2},a^{2n}=1,b^{-1}ab=a^{-1}\rangle$ are the generalized
quaternion groups.

######  Theorem 7.

Let $G$ be a finite group, then $P(G,k)$ is a star graph if and only if one of
the following holds.

  * •

$o(x)\mid k\,\,\forall x\in G$

  * •

$G=\mathbb{Z}_{4}$ and $k=2$

  * •

$G=Q_{8}$ and $k=2\,\text{or}\,\,6$

Proof: Let $P(G,k)$ is a star graph and let there exist $x\in G$ such that
$o(x)\nmid k$, then $x^{k}\neq e$. Suppose $x^{k}=y$ for some $y\in G$. If
$y=x$, then $z^{k}=x$ for all $z\in G,z\neq x$ which is not possible since
$e^{k}\neq x$. If $y\neq x$, then $z^{k}=y$ for all $z\in G,z\neq y$ and
$y^{k}=e$.

If $x^{2}=y$, then $y^{2}=e$, which implies $G$ is a group with unique
involution and all other elements have order $4$. Hence $G=\mathbb{Z}_{4}$
with $k=2$ or $G=Q_{8}$ with $k=2\,\text{or}\,6$ [A $p$-group with unique
subgroup of order $p$ is either cyclic or generalized quaternion].
If $x^{2}\neq y$, then $x^{k}=y$ and $(x^{2})^{k}=y$ which implies $x^{k}=e$
and hence $o(x)\mid k$.

For the converse, if $o(x)\mid k,\forall x\in G$, then $x^{k}=e,\forall x\in
G$, and hence $P(G,k)$ is a star graph. The $k$-power graphs of
$\mathbb{Z}_{4}$ and $Q_{8}$ also star graphs when $k=2\,\text{and}\,6$.
Clearly, if $k-1\mid o(x)$ for all $x\in G$, then $x^{k}=x$ and every vertex
in $G$ is an isolated vertex in $P(G,k)$, and vice versa. Hence the following
theorem is immediate.

######  Theorem 8.

$P(G,k)$ is an empty graph if and only if $o(x)\mid k-1$ for every $x\in G$.

We have already seen that $P(G,k)$ need not be connected, and if $P(G,k)$ is
connected, then it is a tree. The following theorem characterizes groups whose
$k-$power graphs are forests.

######  Theorem 9.

Let $G$ be a finite group. $P(G,k)$ is a forest if and only if there exist no
element in $G$ of order $m>1$ such that $gcd(k,m)=1$ and $ord_{m}(k)>2$.

Proof: By Euler’s theorem in number theory, if $m$ and $k$ are co-prime
integers, then $a^{\phi(m)}\equiv 1(mod\,m)$. Hence, if there exists an
element $x$ in $G$ of order $m$ such that $gcd(m,k)=1$ and $ord_{m}(k)=l>2$,
then $x^{k^{l}}=x$, and $x,x^{k},x^{k^{2}},...x^{k^{l}}$ is a cycle in
$P(G,k)$ of length $l$ and hence $P(G,k)$ is not a forest.

Conversely, if $P(G,k)$ is not a forest, then there is a cycle
$x,x^{k},x^{k^{2}},..x^{k^{l}}=x$ in $P(G,k)$, which implies $o(x)\mid
k^{l}-1$. Hence $k^{l}\equiv 1(mod\,o(x))$ which holds only if
$gcd(k,o(x))=1$.

##  3 Cyclic groups

Let $gcd(n,k)=d$, then the congruent relation $kx\equiv a(mod\,n)$ has at most
$d$ solutions. Hence, if $G$ is a cyclic group of order $n$, then the maximum
degree of its $k-$power graph is at most $d+1$. Then following theorem
describes degree of each element in a cyclic group.

######  Theorem 10.

Consider the cyclic group $\mathbb{Z}_{n}=\\{0,1,2,...n-1\\}$. Let $d=(n,k)$
and $a\in\mathbb{Z}_{n}$. Then,

If $d\nmid a$, $deg(a)=\begin{cases}0&\text{if}\,\,o(a)\mid k-1\\\
\,1&\text{if}\,\,o(a)\nmid k-1\end{cases}$

If $d\mid a$, $deg(a)=\begin{cases}d-1&\text{if}\,\,ka\equiv
a(mod\,n)\,\,\text{and}\,\,\,o(a)\mid k-1\\\ d&\text{if}\,\,ka\not\equiv
a(mod\,n)\,\,and\,\,\,o(a)\mid k-1\\\ d&\text{if}\,\,ka\equiv
a(mod\,n)\,\,\text{and}\,\,\,o(a)\nmid k-1\\\ d+1&\text{if}\,\,ka\not\equiv
a(mod\,n)\,\,\text{and}\,\,\,o(a)\nmid k-1\end{cases}$

Proof: If $d\nmid a$, the congruence relation $kx\equiv a(mod\,n)$ has no
solution. Therefore, the vertex $a$ is adjacent to $a^{k}$ only. In this case,
if $o(a)\mid k-1$, then $a^{k}=a$ and $deg(a)=0$ and if $o(a)\nmid k-1$, then
$a^{k}\neq a$ and $deg(a)=1$.

Next suppose $d\mid a$. Then the congruence relation $kx\equiv a(mod\,n)$ has
exactly $d$ solutions. Now we have two cases,
case 1: $ka\equiv a(mod\,n)$
In this case, $a$ itself is a solution of the congruence relation. Hence
$deg(a)=d-1$ if $o(a)\mid k-1$ and $deg(a)=d$ if $o(a)\nmid k-1$.
case 2: $ka\not\equiv a(mod\,n)$
Here, all the $d$ solutions of the congruence relation are different from $a$,
Hence $deg(a)=d$ if $o(a)\mid k-1$ and $deg(a)=d+1$ if $o(a)\nmid k-1$.

For a positive integer $m$, let $\pi(m)=\\{p:p\mid m,p\,\text{is a prime}\\}$.
The following theorem gives another characterization for $P(\mathbb{Z}_{n},k)$
to be connected.

######  Theorem 11.

$P(\mathbb{Z}_{n},k)$ is connected if and only if
$\pi(n)\setminus\pi(k)=\phi$.

Proof: Suppose $\pi(n)\setminus\pi(k)\neq\phi$, and let $p\in\pi(n)$ such that
$p\notin\pi(k)$. Then $gcd(n,k)=1$ and $k^{p-1}\equiv 1(mod\,p).$ Let
$m=exp_{p}(k)$ and let $x$ be an element in $G$ such that $o(x)=p$. Then
$x^{k^{m}}=x$ and $x$ is not connected to $e$ through any path in
$P(\mathbb{Z}_{n},k)$. Hence $P(\mathbb{Z}_{n},k)$ is not connected.

Conversely suppose $\pi(n)\setminus\pi(k)=\phi$. Let
$n=p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}...p_{s}^{\alpha_{s}}$ and
$k=p_{1}^{\beta_{1}}p_{2}^{\beta_{2}}...p_{r}^{\beta_{r}}$ be the prime
factorization of $n$ and $k$ respectively, where $r\leq s$. Let
$m=lcm(\beta_{1},\beta_{2},...\beta_{r})$. Then $n\mid k^{m}$, and hence by
Proposition 3, $P(\mathbb{Z}_{n},k)$ is connected.

The identity element $e$ is an isolated vertex in $P(\mathbb{Z}_{n},k)$ if
$gcd(n,k)=1$ since, if $x^{k}=e$ for some $x\in\mathbb{Z}_{n}$ then $o(x)\mid
k$ and hence $o(x)\mid gcd(n,k)$.

######  Proposition 12.

Let $gcd(n,k)=1$. If $x\sim y$ in $P(\mathbb{Z}_{n},k)$, then $o(x)=o(y)$.

Proof: Let $x\sim y$, then $kx\equiv y(mod\,n)\implies o(x)kx\equiv
o(x)y(mod\,n)\implies 0\equiv o(x)y(mod\,n)\implies o(y)\mid o(x)$. Also
$o(y)kx\equiv o(y)y(mod\,n)\implies o(y)kx\equiv 0(mod\,n)\implies o(x)\mid
o(y)k\implies o(x)\mid o(y)$, since $gcd(o(x),k)=1$.

The following theorem states that the number of connected components is
bounded above.

######  Theorem 13.

If $gcd(n,k)=1$, then the number of connected components of
$P(\mathbb{Z}_{n},k)$, $c(P(\mathbb{Z}_{n},k))\geq\tau(n)$, where $\tau(n)$ is
the number of divisors of $n$, and $c(P(\mathbb{Z}_{n},k))=\tau(n)$ if and
only if $k$ is a primitive root modulo $d$, for every divisor $d$ of $n$.

Proof: By Proposition $3.4$, all the vertices in a component have same order.
Hence, the inequality follows since an element of order $d$ exists in
$\mathbb{Z}_{n}$ if $d\mid n$.

Suppose $k$ is a primitive root modulo $d$, for every $d\mid n$. Then
$ord_{d}(k)=\phi(d)$. Hence, if $x$ is an element in $G$ of order $d$, then
$\phi(d)$ is the smallest positive integer such that $x^{k^{\phi(d)}}\equiv
x(mod\,d)$. Hence $x,x^{k},x^{k^{2}},...,x^{k^{\phi(d)-1}}$ are the distinct
$\phi(d)$ elements of order $d$ in $G$ and are in the same component of
$P(\mathbb{Z}_{n},k)$.

Conversely, suppose $k$ is not a primitive root modulo $d$ for some $d\mid n$.
Then, $ord_{d}(k)=m<\phi(d)$ and hence $k^{m}\equiv 1(mod\,d)$ and
$x^{k^{m}}=x$ for every element $x$ of order $d$. Hence, the component
containing $x$ has exactly $m$ vertices. Therefore, the elements in
$\mathbb{Z}_{n}$ of order $d$ contribute at least two components.

######  Proposition 14.

chattopadhyay2017some  If $gcd(n,k)=1$ then any component of
$P(\mathbb{Z}_{n},k)$ is an isolated vertex, the complete graph $K_{2}$ or a
cycle of length at least 3.

The following theorem shows that the converse of the Proposition 3 is also
satisfies.

######  Theorem 15.

Any component of $P(\mathbb{Z}_{n},k)$ is an isolated vertex, the complete
graph $K_{2}$ or a cycle of length at least $3$ if and only if $gcd(n,k)=1$.

Proof: Suppose $gcd(n,k)=d>1$. Then, the congruence $kx\equiv 1(mod\,n)$ has
no solutions. Considering $1$ and $k$ as elements of $\mathbb{Z}_{n}$, $1\sim
k$ in $P(\mathbb{Z}_{n},k)$. Hence, $deg(1)=1$. Also, $kx\equiv k(mod\,n)$ has
$d$ solutions and so $deg(k)\geq 2$. Therefore, the edge between $1$ and $k$
in $P(\mathbb{Z}_{n},k)$ is not a part of a cycle nor $K_{2}$ as a component.
Hence, the necessary part follows. The sufficient part follows from
Proposition $14$.

######  Theorem 16.

Let $n$ be even and $\frac{n}{2}$ odd, then
$P(\mathbb{Z}_{n},\frac{n}{2})=2K_{1,\frac{n}{2}-1}$ and
$P(\mathbb{Z}_{n},\frac{n}{2}+1)=\frac{n}{2}P_{2}$.

Proof: Let $n$ be even and $\frac{n}{2}$ odd. If $k=\frac{n}{2}$, then using
Theorem $10$, $deg(a)=\frac{n}{2}-1$ if $a=0,\frac{n}{2}$ and $deg(a)=1$ if
$a\neq 0,\frac{n}{2}.$ Also, $\frac{n}{2}a\equiv 0(mod\,n)$ if $a$ is even and
hence all the even elements are adjacent to the vertex 0, and
$\frac{n}{2}a\equiv\frac{n}{2}(mod\,n)$ if $a$ is odd and hence all the odd
elements are adjacent to the vertex $\frac{n}{2}$.

If $k=\frac{n}{2}+1$, for $a\in\mathbb{Z}_{n}$, $(\frac{n}{2}+1)a\equiv
a(mod\,n)$ if $a$ is even, and $(\frac{n}{2}+1)a\equiv\frac{n}{2}+a(mod\,n)$
if $a$ is odd. Hence, the edges in $P(\mathbb{Z}_{n},k)$ are between the
vertices $a$ and $\frac{n}{2}+a$ only.

##  4 Application

Consider the directed $k$-power graph of a group $G$,
$\overrightarrow{P}(G,k)$, defined as a digraph with vertex set $G$ in which
there is an arc from a vertex $a$ to another vertex $b$ if and only if
$b=a^{k}$. In this section, we present a riddle and solve that riddle using
directed $k$-power graphs of cyclic groups.

Shifting Chair Problem: In a game, $n$ chairs are evenly spaced in a circle
and $n$ people are assigned unique numbers from $1$ to $n$. People should sit
in ascending order in the clockwise direction when the first whistle blows.
For each of the whistles thereafter, the individual who is assigned with
number $i$ should acquire his next chair by skipping exactly $i-1$ number of
chairs in the clockwise direction. The problem is to determine the smallest
number of whistles required to ensure that each chair is occupied by exactly
one person.

Consider the above problem, representing the movement of the individuals from
their initial seats to the position after $k^{\text{th}}$ whistle by a
directed edge, we get the directed $k$-power graph of $\mathbb{Z}_{n}$. Then
the problem is to find the smallest $k>1$ such that, $od(a)=id(a)=1$ in
$\overrightarrow{P}(\mathbb{Z}_{n},k)$ for all $a\in\mathbb{Z}_{n}$, where
$od(a)$ is the outdegree of $a$ and $id(a)$ is the indegree of $a$, and the
following theorem present the solution.

######  Theorem 17.

In $\overrightarrow{P}(\mathbb{Z}_{n},k)$, $id(a)=od(a)=1$ for all
$a\in\mathbb{Z}_{n}$ if and only if $gcd(n,k)=1$.

Proof: If $gcd(n,k)=1$, then for each $a\in\mathbb{Z}_{n}$, the congruence
$kx\equiv a(mod\,n)$ has exactly one solution, so $id(a)=1$. Also $od(a)=1$
for all $a\in\mathbb{Z}_{n}$ since $a$ is adjacent to $a^{k}$.

Conversely suppose $gcd(n,k)=d>1$. Then $kx\equiv 1(mod\,n)$ has no solutions,
so $id(1)=0$.

##  5 Declarations

Conflict of interest On behalf of all authors, the corresponding author states
that there is no conflict of interest.

## Acknowledgements

The first author gratefully acknowledges the financial support of Council of
Scientific and Industrial Research, India (CSIR) (Grant
No-09/874(0029)/2018-EMR-I). The authors would like to thank the DST,
Government of India, for providing support to carry out the work under the
scheme ‘FIST’ (No.SR/FST /MS-I/2019/40).

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