# XUV ionization of the H2 molecule studied with attosecond angular streaking

Vladislav V. Serov1 Anatoli S. Kheifets2 1General, Theoretical and Computer
Physics, Saratov State University, Saratov 410012, Russia  2Research School of
Physics, The Australian National University, Canberra ACT 2601, Australia
<EMAIL_ADDRESS>
###### Abstract

We study orientation and two-center interference effects in attosecond time-
resolved photoionization of the H2 molecule. Time resolution of XUV ionization
of H2 is gained through the phase retrieval capability of attosecond angular
streaking demonstrated earlier by Kheifets  et al [arXiv:2202.06147 (2022)].
Once applied to H2 , this technique delivers an anisotropic phase and time
delay which both depend sensitively on the molecular axis orientation. In
addition, the photoelectron momentum distribution displays a very clear two-
center interference pattern. When the interference formula due to Walter and
Briggs [J. Phys. B 32 2487 (1999)] is applied, an effective photoelectron
momentum appears to be greater than the asymptotic momentum at the detector.
This effect is explained by a molecular potential well surrounding the
photoemission center.

###### pacs:

32.80.Rm 32.80.Fb 42.50.Hz

Attosecond time resolved studies of molecular photoionization have become a
rapidly growing field. Starting from the pioneering experiment of Huppert et
al. (2016) on H2O and N2O, the method of attosecond interferometry has been
progressively used combining an extreme-ultraviolet (XUV) attosecond pulse
train (APT) and a synchronized infrared (IR) pulse. This technique has also
been known as reconstruction of attosecond beating by interference of two-
photon transitions (RABBITT) Muller (2002); Toma and Muller (2002). Recent
applications of RABBITT to molecular photoionization include attosecond
resolution of coupled electron and nuclear dynamics in dissociative ionization
of H2 Cattaneo et al. (2018) and orientation-dependent time delay and electron
localization studies in CO Vos et al. (2018). Nandi et al. (2020) resolved
attosecond timing of electron emission from a shape resonance in N2. Kamalov
et al. (2020) recorded electron correlation effects in attosecond
photoionization of CO2. Wang et al. (2021) explored the role of nuclear-
electronic coupling in attosecond photoionization of H2 .

The roadmap of atomic and molecular physics Young et al. (2018) has identified
X-ray free-electron lasers (XFELs) as a promising tool for resolving ultrafast
molecular dynamics. Attosecond time-energy structure of XFEL pulses has been
recently demonstrated Hartmann et al. (2018); Duris et al. (2020). This
demonstration makes XFEL sources potentially suitable for attosecond time
resolution of atomic and molecular photoionization. The only stumbling block
preventing such an application is a stochastic nature and an inherent time
jitter of XFEL radaition.

The method of attosecond angular streaking of XUV ionization was developed to
overcome this obstacle. Prompted by theoretical works Zhao et al. (2005);
Kazansky et al. (2016); Li et al. (2018); Kazansky et al. (2019), this method
was eventually implemented in practice for a shot-to-shot characterization of
isolated attosecond pulses (IAP) at XFEL Hartmann et al. (2018); Duris et al.
(2020). Angular streaking of XUV ionization (ASXUVI or ASX for brevity) has
common elements with the two previously developed techniques: attosecond
angular streaking known as the attoclock Eckle et al. (2008a, b); Pfeiffer et
al. (2012) and the attosecond streak camera (ASC) Constant et al. (1997);
Itatani et al. (2002); Goulielmakis et al. (2004); Kienberger et al. (2004);
Yakovlev et al. (2005); Frühling et al. (2009); Zhang and Thumm (2011); Ivanov
and Smirnova (2011). As in ASC, ASX uses XUV pulses to ionize the target.
Then, similarly to the attoclock, the photoelectrons are steered by a
circularly polarized laser field which makes its imprint on the photoelectron
momentum distribution (PMD). This imprint is most visible in the plane
perpendicular to the laser propagation direction. In its original form
Kazansky et al. (2016); Li et al. (2018); Kazansky et al. (2019); Hartmann et
al. (2018); Duris et al. (2020), ASX employed an intense IR laser field and
was interpreted within the strong field approximation (SFA) Zhao et al.
(2022). In these strong field settings, the phase of the XUV ionization is
usually neglected and the timing information associated with this phase is
lost. An alternative view within the lowest order perturbation theory (LOPT)
Dahlström et al. (2012); Dahlström et al (2013); Maquet et al. (2014)
considers IR streaking as an interference phenomenon which opens a natural
access to the streaking phase $\Phi_{S}$. The latter is typically decomposed
into the XUV ionization phase (or Wigner phase) and the continuum-continuum
(CC) phase from the IR interaction. These two phases can be converted to the
corresponding time delay components, which add up to the atomic time delay
$\tau_{a}$.

Phase retrieval capability of ASX based on this analysis was demonstrated
recently by Kheifets et al. (2022). In their numerical simulations on the
hydrogen atom, they recovered accurately the streaking phase and the atomic
time delay across a wide range of photon energies starting from the threshold
and exceeding it many times. Most importantly, this phase retrieval could be
conducted from a single XUV shot. This is a significant advantage over the
existing interferometric techniques which require a systematic and
controllable variation of the XUV/IR pulse delay in one set of measurements in
order to record a streaking spectrogram or a RABBITT trace. This recording
require a precise and stable temporal synchronization of the XUV/IR pulses
which is not feasible at XFEL at present.

In this paper, we extend ASX to molecular photoionization. We solve
numerically the time-dependent Schrödinger equation (TDSE) describing the
hydrogen molecule driven by a combination of the linearly polarized XUV and
circularly polarized IR pulses. In our simulations, the XUV/IR pulse delay is
incremented in several steps. By augmenting the isochrone analysis proposed by
Kazansky et al. (2016) with the energy dependent XUV ionization phase, we are
able to interpret the molecular TDSE results in terms of the atomic time
delay. While the phase and time delay determination is most accurate combining
several increments of the XUV/IR delay, the accuracy is not significantly
compromised with just a single XUV/IR pulse delay. We make a comparison with
the previous RABBITT simulations on H2 Serov and Kheifets (2017) and confirm
validity of our interpreatation and accuracy of our numerical results. We also
demonstrate a strong dependence of the time delay on the molecular axis
orientation discovered earlier in H${}_{2}^{+}$ ion Ning et al. (2014); Serov
and Kheifets (2016).

The paper is organized into the following sections. In Sec. I we outline
basics of the ASX method. In Sec. II we describe our computational procedure.
In Sec. III we analyze and interpret our numerical results. In Sec. IV we give
our concluding remarks.

##  I Basic considerations

The proposed phase retrieval by ASX is outlined in our preceding work Kheifets
et al. (2022). The basics of the molecular ASX are essentially the same as for
atoms. We proceed as follows. We apply the SFA and write the photoionization
amplitude as Kitzler et al. (2002)

$a({\bm{k}},\tau)=i\int_{t_{0}}^{\infty}\\!\\!dt\
E_{x}(t-\tau)D_{x}\left[{\bm{k}}-{\bm{A}}(t)\right]e^{-i\Phi(t)}\ .$ (1)

Here the electric field of the XUV pulse $E_{x}$ is advancing the streaking
pulse by the time $\tau$. The streaking field is described by its vector
potential

${\bm{A}}(t)=A_{0}\cos(\omega t)\hat{{\bm{x}}}+A_{0}\sin(\omega
t)\hat{{\bm{y}}}\ .$

The photoelectron momentum is confined to the polarization plane
${\bm{k}}=k\cos\phi~{}\hat{{\bm{x}}}+k\sin\phi~{}\hat{{\bm{y}}}$, where $\phi$
is the emission angle.

The exponential term contains the phase factor

$\Phi(t)=\frac{1}{2}\int_{t}^{\infty}dt^{\prime}\left[{\bm{k}}-{\bm{A}}(t^{\prime})\right]^{2}-E_{0}\,t\
,$ (2)

which contains the photoelectron energy in the absence of streaking
$E_{0}=\Omega-I_{p}$. The most probable photoelectron trajectory, starting at
the time $t_{\rm st}$, keeps the phase stationary:

$\Phi^{\prime}(t_{\rm st})=\frac{1}{2}|{\bm{k}}-{\bm{A}}(t_{\rm
st})|^{2}-E_{0}=0$ (3)

We assume that the XUV pulse is short relative to the IR pulse and shifted
relative to its peak position by the time $\tau$. Under these conditions, Eq.
(3) is transformed to the following  isochrone equation Kazansky et al.
(2016):

$k^{2}/2-E_{0}=kA_{0}\cos(\phi-\omega\tau)$ (4)

Here we neglect the ponderomotive energy $U_{p}=A_{0}^{2}/2$ in a weak
streaking field.

The above stationary phase analysis should be modified to account for the
photoelectron energy dependence of the dipole matrix element Schultze et al.
(2010)

$\arg\left\\{D\left[{\bm{k}}-{\bm{A}}(t)\right]\right\\}\propto\alpha|{\bm{k}}-{\bm{A}}(t)|^{2}/2\
,$ (5)

where

$\alpha=\partial\arg D(\sqrt{2E})/\partial E$ (6)

The modified stationary phase equation reads

$\frac{1}{2}\left|{\bm{k}}-{\bm{A}}(t_{st})\right|^{2}-E_{0}+\frac{\alpha}{2}\frac{d}{dt}\left[\left({\bm{k}}-{\bm{A}}(t_{\rm
st})\right)^{2}\right]=0$ (7)

This leads to a generalized isochrone equation

$\displaystyle k^{2}/2-E_{0}$ $\displaystyle=$ $\displaystyle
kA_{0}\left[\cos(\phi-\omega\tau)-\alpha\omega\sin(\phi-\omega\tau)\right]$
(8) $\displaystyle\approx$ $\displaystyle
kA_{0}\cos[\phi-\omega\tau+\omega\alpha]$

Here $\alpha=\Phi_{S}/\omega=\tau_{a}$ under certain XUV and IR pulse
parameters as demonstrated in Kheifets et al. (2022).

##  II Computational details

We solve numerically the molecular TDSE equation using the computer code Serov
(2011) to obtain the ionization amplitude $f({\bm{k}})$. We use an angular
basis that included spherical harmonics up to $l_{max}=7$ and $|m_{max}|=7$.
Unlike the dipole selection rules in atomic XUV photoionization, the quantum
numbers $l,m$ adhere to the parity conservation.

The photoelectron momentum spectrum $P({\bm{k}})$ is obtained as the modulus
squared of the ionization amplitude

$P({\bm{k}})\propto|f({\bm{k}})|^{2}\ .$ (9)

The PMD is restricted to the polarization plane $P(k_{x},k_{y},k_{z}=0)$ and
converted to the polar coordinates $P(k,\phi)$ where

$k=(k_{x}^{2}+k_{y}^{2})^{1/2}\ ,\ \phi=\tan^{-1}(k_{y}/k_{x})\ .$ (10)

In these coordinates, we define the directional probability of the
photoelectron emission

$P(\phi)=\int dk\ P(k,\phi)$ (11)

and the mean (central) radial momentum in the given direction

$\bar{k}(\phi)=\int kP(k,\phi)dk/P(\phi)\ .$ (12)

The TDSE is driven by the XUV and IR pulses with the following parameters. The
XUV pulse with a Gaussian envelope has a FWHM of 2 fs and the intensity of
$\rm{6}\times 10^{{13}}~{}W/cm^{2}$. The XUV photon energy $\Omega$ ranges
from 0.7 au to 3 au. A relatively low XUV field intensity is required to
remain within the LOPT framework. A fairly large pulse duration is employed to
ensure a moderately narrow spectral width to probe XUV ionization sufficiently
close to the threshold at 15.6 eV (0.57 au). At the same time, the spectral
width $\Gamma$ should be kept sufficiently large to make sure the IR assisted
XUV absorption process overlaps spectrally with unassisted XUV ionization
Kheifets et al. (2022). This requires $\Gamma>2\omega$, where $\omega$ is the
laser photon energy. To satisfy this requirement, we chose a mid-IR laser
pulse with $\omega=0.038$ au corresponding to $\lambda=1200$ nm. The pulse has
a cosine squred envelope with FWHM of 25 fs and the intensity of
$\rm{1.5}\times 10^{{11}}~{}W/cm^{2}$. The XUV pulse is linearly polarized
along the $\hat{\bm{x}}$ axis whereas the IR pulse is circularly polarized in
the $(xy)$ plane. At each XUV photon energy, we scan the delay between the XUV
pulse and the IR laser field ($\tau$) in the range of 0 to 60 au in 7
increments.

##  III Numerical results

We identify three regions in the photoelectron energies which display
distinctively different PMD in the polarization plane. These regions can be
characterized by the strength of the molecular two-center interference. The
theory of this interference was proposed by Cohen and Fano (1966) and Kaplan
and Markin (1969) and further developed for diatomic molecules fixed in space
by Walter and Briggs (1999). In the latter formulation, the ionization
amplitude is approximated by the expression

$f_{\rm
WB}({\bm{k}})\propto({\bm{e}}\cdot{\bm{k}})\cos({\bm{k}}\cdot{\bm{R}}/2)\ ,$
(13)

where $\bm{e}$ is the polarization vector of light and ${\bm{R}}$ is the
vector connecting the nuclei. The first term in the RHS of Eq. (13) is the
atomic hydrogen dipole factor whereas the second term represents the molecular
two-center interference. In the following, we will use a scalar coefficient
$c=kR/2$ to identify the strength of this interference.

Figure 1: Top: PMD of H2 at $\Omega=0.7$ au in the parallel field orientation
with the XUV only pulse (top) and the XUV+IR pulses (middle). The horizontal
dashed line visualize the photoelectron momentum $k_{0}=\sqrt{2(\Omega-
I_{p})}$ from the energy conservation. The vertical dashed line mark the half
of the angular width.

###  III.1 Weak interference

At low photoelectron energy when $c\ll 1$, the PMD of H2 looks essentially
atomic like with very little anisotropy seen between the parallel and
perpendicular orientation of the molecular axis ${\bm{R}}$ relative to the
linear polarization axis $\bm{e}$ of the XUV pulse. This behavior is featured
in Fig. 1 which displays the PMD at $\Omega=0.7$ au. The top and middle panels
both illustrate the case of the parallel orientation with the XUV only pulse
(top) and XUV+IR pulses (middle). The bottom panel displays the radially
integrated PMD of the middle panel in the form of the angular distribution
$P(\phi)$ which is overlapped with the analogous distribution for the
perpendicular orientation. Except for an overall magnitude factor $\times
1.8$, the $\perp$ angular distributions look essentially the same as the
$\parallel$ one.

Meanwhile, the PMD of the top panel (XUV only) and the middle panel (XUV+IR)
differ by a noticeable displacement of the radial momentum by the vector-
potential $A_{\small\tt IR}$ of the streaking field. To quantify this
displacement, we use the central photoelectron momenta (12) in the downwards
(-) and upwards (+) shifted lobes of the PMD

$k_{-}\equiv\bar{k}(\phi=0)\ \ ,\ \ k_{+}\equiv\bar{k}(\phi=\pi)\ ,$

These momenta $k_{\pm}(\tau)$, which depend sensitively on the XUV/IR time
delay $\tau$, are then used to obtain the isochrone phase offset:

$k_{\pm}^{2}(\tau)/2-E_{0}=\pm A_{0}\,k_{\pm}(\tau)\cos(\omega\tau+\Phi_{S})\
.$ (14)

This determination is illustrated in the top panel of Fig. 2. Here we
determine $\Phi_{S}=-0.216\pm 0.003$ rad by fitting either of the
$k_{\pm}(\tau)$ branches with a common streaking phase value over the whole
set of the time delays $\tau$. Alternatively, we can apply Eq. (14) to
individual $\tau$ values and to determine the instantaneous $\Phi_{S}(\tau)$.
These values are displayed along with the average streaking phase on the
bottom panel of Fig. 2. Even though the variation of $\Phi_{S}(\tau)$ exceeds
the error bars of the average value, the accuracy of the instantaneous
streaking phase determination is not significantly compromised.

Figure 2: Top: Radial momentum displacements $k_{\pm}^{2}/2-k_{0}^{2}/2$ are
shown at various XUV/IR delays $\tau$. The dashed line represents the fit with
Eq. (14). The arrow indicates the streaking phase $\Phi_{S}$. Bottom: the fit
with Eq. (14) is applied to individual $\tau$ values to determine the
instantaneous $\Phi_{S}(\tau)$. The average $\Phi_{s}$ is shown as a solid
line with error bars visualized by dotted lines.  Figure 3: Top: PMD of H2 at
$\Omega=1.5$ au for the parallel (top) and perpendicular (middle) field
orientation with the XUV only pulse The horizontal dashed line visualize the
photoelectron momentum $k_{0}$ while the vertical line marks half of the
angular width.

###  III.2 Moderate interference

This region is characterized by a moderate factor $c\lesssim 1$. A typical PMD
in this region is presented in the top and middle panels of Fig. 3. Here the
XUV photon energy $\Omega=1.5$ au and the molecule is oriented parallel (top)
and perpendicular (middle) to the polarization axis. Both panels visualize
single-photon XUV ionization. Adding the IR streaking field does not change
the PMD structure except for a vertical up and down displacement by the amount
of $A_{\small\tt IR}$ as in the middle panel of Fig. 1.

The case of $c\lesssim 1$ differs from $c\ll 1$ by a significant deviation of
the PMD shapes corresponding to the parallel and perpendicular orientations.
The PMD lobes are noticeably elongated for the parallel orientation and
acquire a greater angular width. The photoelectron angular distribution shown
in the bottom panel is markedly different for the $\parallel$ and $\perp$
orientations. While the latter retains the atomic like structure, the former
widens significantly and becomes drastically, by a factor $\times 10$,
suppressed. This parallel emission suppression is documented in the literature
and termed the ”confinement effect” Fernández et al. (2007, 2009). This
corresponds to the dominant photoelectron $p$-wave trapped inside a one-
dimensional box of length $R$ when the momentum quantization condition
$kR=\pi$ satisfied at $c=\pi/2$.

###  III.3 Strong interference

This region is characterized by a large interference factor
$c\gtrapprox\pi/2$. In this region, the shape distortion of PMD is most
graphical as shown in Fig. 4 for $\Omega=2.5$ au. While the perpendicular
orientation (middle panel) retains an atomic like shape, the parallel
orientation (top panel) displays very clear interference fringes. These
fringes are also seen in the angular resolved cross-section exhibited in the
bottom panel of Fig. 4.

Figure 4: Same as Fig. 3 for $\Omega=2.5$ au  Figure 5: Top: expansion
coefficients of the ionization amplitude over the spherical harmonics (16)
plotted as functions of the interference factor $c=kR/2$. Bottom: angular half
width of the PMD lobes as a function of the photoelectron energy $E=\Omega-
I_{p}$. The upper horizontal scale marks the corresponding interference
factors.

To quantify the two-center interference effects across a wide range of the
photon energies, we plot in the bottom panel of Fig. 5 the half width of the
PMD lobes. The atomic like half width of $45^{\circ}$ corresponds to the
dipole $\cos^{2}\phi$ angular shape. It is retained consistently over the
whole photon energy range in the perpendicular molecular orientation for XUV
only photoionization. Adding a streaking IR field reduces this width
insignificantly for the $\perp$ orientation. Meanwhile, the $\parallel$
orientation, both in XUV and XUV+IR fields, displays a wide oscillation of the
width in the range of moderate to strong two-center interference.

To understand the nature of this oscillation, we note that the amplitude (13)
for the parallel orientation is reduced to

$f^{\parallel}_{\rm WB}(\phi)\propto\cos\phi\cdot\cos(0.5kR\cos\phi)\ .$ (15)

This amplitude can be expanded over the spherical harmonics with the expansion
coefficients given by the following expression Serov et al. (2012)

$A_{\ell}(c)=\left\langle Y_{\ell
0}|f^{\parallel}_{\text{WB}}\right\rangle=\sqrt{2\pi}\int_{-1}^{1}\bar{P}_{\ell}(\eta)\eta\cos(c\eta)d\eta.$
(16)

Here $\bar{P}_{\ell}(\eta)$ are the normalized Legendre polynomials which
depend on $\eta=\cos\phi$. The expansion coefficients (16) for various $\ell$
are plotted in the top panel of Fig. 5. From this graph we see clearly that
$c\simeq 1$ corresponds to a noticeable contribution of the $f$-wave whereas
at $c\simeq\pi/2$ the $p$\- and $f$-wave contributions become of the same
magnitude. These two boundaries correspond to the region of moderate and
strong two-center interference according to our classification in Sec. III.2
and Sec. III.3. In the meantime, the weak interference $c\ll 1$ considered in
Sec. III.1 corresponds to a nearly sole contribution of the $p$-wave.

Fitting the numerical TDSE results for the photoelectron angular distributions
with the squared amplitude (15) gives systematically higher effective momenta
$k_{\rm eff}$ in comparison with the nominal momenta $k$ determined by the
energy conservation. We find $k_{\rm eff}$ from the moduli ratio of the $f$\-
and $p$-waves

$\left|\frac{A_{3}(c_{a})}{A_{1}(c_{a})}\right|=\left|\frac{\left\langle
Y_{30}|f({\bm{k}})\right\rangle}{\left\langle
Y_{10}|f({\bm{k}})\right\rangle}\right|.$ (17)

This ratio equates the expansion coefficients $A_{\ell}$ from Eq. (16)
evaluated at $c_{a}=k_{\rm eff}R/2$ with the corresponding expansion
coefficients of the exact numerical amplitude $f({\bm{k}})$ found by the TDSE
solution.

The deviation $k_{\rm eff}$ from $k$ displayed in the top panel of Fig. 6 can
be explained by the effective potential of the ion remainder. Due to this
potential, the momentum of the electron near the nucleus is greater, and,
accordingly, a larger phase difference between the emitting centers is
accumulated. We can introduce an average effective potential related to the
effective momentum through the following expression

$k_{\rm eff}/k=\sqrt{1+2|\bar{U}_{\rm eff}|/k^{2}}\ .$ (18)

The values of $\bar{U}_{\rm eff}$ are presented in the bottom panel of Fig. 6.
A gradual reduction of $\bar{U}_{\rm eff}$ with a decreasing XUV photon energy
can be understood as follows. By the uncertainty principle, a slower
photoelectron has a larger birthplace area across which the ionic potential is
sampled. Therefore, its effective depth becomes smaller.

###  III.4 Streaking phase and time delay

The streaking phase results for the H2 molecule in the $\parallel$ and $\perp$
orientations are summarized in the top panel of Fig. 7 where they are compared
with the corresponding values of the H atom. While the molecular $\Phi_{S}$ in
the $\perp$ orientation is very similar to the atomic one, the $\parallel$
orientation displays a systematically higher values, especially at the onset
of the strong interference when the $c$ factor approaching $\pi/2$. The atomic
time delay derived from the streaking phase $\tau_{a}=\Phi_{S}/\omega$ is
shown in the bottom panel of Fig. 7 where it is compared with the
corresponding values returned by the RABBITT simulations Serov and Kheifets
(2017). Numerical $\tau_{a}$ values from the ASX and RABBITT simulations are
slightly different because of a difference in the wavelength $\lambda=1200$ nm
in the former and 800 nm in the latter. The IR photon wavelength and energy
affect the CC component of the atomic time delay Dahlström et al (2013); Serov
et al. (2015) which becomes particularly noticeable close to the threshold.
Nevertheless, the qualitative behavior of $\tau_{a}$ is very similar in both
sets of simulations. The atomic time delay in the H atom and the H2 molecule
in the $\perp$ orientation remain negative in the studied XUV photon energy
range. At the same time, the $\parallel$ orientation displays a sharp rise of
the time delay to positive values. This effect is also recorded in the
H${}_{2}^{+}$ ion Ning et al. (2014); Serov and Kheifets (2016). It was
attributed in Ning et al. (2014) to the destructive two-center interference.
We offer a more physically appealing interpretation of the positive time delay
due to the trapping the photoelectron in the molecular potential well. From
the condition of this trapping $k_{\rm eff}R=\pi$ occurring at $kR\simeq 2.4$
we can estimate $|U_{\rm eff}|\simeq 1$ au. This determination is consistent
with the values of $U_{\rm eff}$ presented in the bottom panel of Fig. 6.

Figure 6: Top: Effective momentum $k_{\rm eff}/k$. Bottom: effective potential
$U_{\rm eff}$.  Figure 7: Top: Streaking phase $\Phi_{S}$ as a function of the
photoelectron energy for the hydrogen atom and the H2 molecule in the
$\parallel$ and $\perp$ orientations. Bottom: the atomic time delay derived
from the streaking phase $\tau_{a}=\Phi_{S}/\omega$ is compared with the
corresponding values returned from the RABBITT simulations Serov and Kheifets
(2017).

##  IV Conclusions

In the present work, we employed the angular streaking of XUV ionization of
the H2 molecule to determine the streaking phase and time delay corresponding
to various orientations of the inter-nuclear axis relative to the polarization
axis of ionizing radiation. The ASX technique was originally developed to
characterize isolated attosecond pulses from XFEL source on the shot-to-shot
basis. This technique was adapted to determine the streaking phase and applied
in our previous work Kheifets et al. (2022) to the atomic hydrogen. In the
present work we expand this technique to diatomic homonuclear molecules. We
converted the streaking phase to the atomic time delay and found it in good
agreement with our earlier RABBITT simulations Serov and Kheifets (2017).
Unlike RABBITT, which requires an accurate and stable synchronization of the
ionizing XUV and probing IR pulses, ASX can determine the streaking phase and
time delay from a single XUV shot. This is essential in XFEL sources with
their inherent time jitter.

As in earlier works Ning et al. (2014); Serov and Kheifets (2016, 2017) we
observe a strong orientation dependence of the molecular time delay. In most
cases, $\tau_{a}$ remains negative in H, H2 and H${}_{2}^{+}$ due to a large
negative CC component. However, $\tau_{a}$ becomes positive in H2 and
H${}_{2}^{+}$ in the parallel orientation ${\bm{R}}\parallel\bm{e}$. This
happens when the photoelectron in the dominant $p$-wave becomes trapped in the
molecular potential well. From the condition of this trapping we can estimate
the depth of this well $U_{\rm eff}$.

While the streaking phase retrieval by ASX was demonstrated for a diatomic
homo-nuclear molecule H2 , the proposed method should work for arbitrary
molecular targets. Its application in XFEL will be particularly beneficial for
studying inner shell ionization in atomic and molecular targets which cannot
be ionized at present with conventional laser HHG sources.

#### Acknowledgment:

We thank Rickson Wielian for reviewing the literature and useful discussions.
This work is supported by the Discovery grant DP190101145 of the Australian
Research Council. Resources of National Computational Infrastructure facility
(NCI Australia) have been employed.

## References

  * Huppert et al. (2016) M. Huppert, I. Jordan, D. Baykusheva, A. von Conta, and H. J. Wörner, _Attosecond delays in molecular photoionization_ , Phys. Rev. Lett. 117, 093001 (2016).
  * Muller (2002) H. Muller, _Reconstruction of attosecond harmonic beating by interference of two-photon transitions_ , Appl. Phys. B 74, s17 (2002).
  * Toma and Muller (2002) E. S. Toma and H. G. Muller, _Calculation of matrix elements for mixed extreme-ultraviolet-infrared two-photon above-threshold ionization of argon_ , J. Phys. B 35(16), 3435 (2002).
  * Cattaneo et al. (2018) L. Cattaneo, J. Vos, R. Y. Bello, A. Palacios, S. Heuser, L. Pedrelli, M. Lucchini, C. Cirelli, F. Martín, and U. Keller, _Attosecond coupled electron and nuclear dynamics in dissociative ionization of H 2_, Nature Physics 14, 733 (2018).
  * Vos et al. (2018) J. Vos, L. Cattaneo, S. Patchkovskii, T. Zimmermann, C. Cirelli, M. Lucchini, A. Kheifets, A. S. Landsman, and U. Keller, _Orientation-dependent stereo Wigner time delay in a small molecule_ , Science 360(6395), 1326 (2018).
  * Nandi et al. (2020) S. Nandi, E. Pl’esiat, S. Zhong, A. Palacios, D. Busto, M. Isinger, L. Neoricic, C. L. Arnold, R. J. Squibb, R. Feifel, et al., _Attosecond timing of electron emission from a molecular shape resonance_ , Science Advances 6(31), eaba7762 (2020).
  * Kamalov et al. (2020) A. Kamalov, A. L. Wang, P. H. Bucksbaum, D. J. Haxton, and J. P. Cryan, _Electron correlation effects in attosecond photoionization of CO 2_, Phys. Rev. A 102, 023118 (2020).
  * Wang et al. (2021) A. L. Wang, V. V. Serov, A. Kamalov, P. H. Bucksbaum, A. Kheifets, and J. P. Cryan, _Role of nuclear-electronic coupling in attosecond photoionization of ${\mathrm{h}}_{2}$_, Phys. Rev. A 104, 063119 (2021).
  * Young et al. (2018) L. Young, K. Ueda, M. Gühr, P. H. Bucksbaum, M. Simon, S. Mukamel, N. Rohringer, K. C. Prince, C. Masciovecchio, M. Meyer, et al., _Roadmap of ultrafast x-ray atomic and molecular physics_ , J. Phys. B 51(3), 032003 (2018).
  * Hartmann et al. (2018) N. Hartmann, G. Hartmann, R. Heider, M. S. Wagner, M. Ilchen, J. Buck, A. O. Lindahl, C. Benko, J. Grünert, J. Krzywinski, et al., _Attosecond time-energy structure of x-ray free-electron laser pulses_ , Nature Photonics 12, 215 (2018).
  * Duris et al. (2020) J. Duris, S. Li, T. Driver, E. G. Champenois, J. P. MacArthur, A. A. Lutman, Z. Zhang, P. Rosenberger, J. W. Aldrich, R. Coffee, et al., _Tunable isolated attosecond x-ray pulses with gigawatt peak power from a free-electron laser_ , Nature Photonics 14, 30 (2020).
  * Zhao et al. (2005) Z. X. Zhao, Z. Chang, X. M. Tong, and C. D. Lin, _Circularly-polarized laser-assisted photoionization spectra of argon for attosecond pulse measurements_ , Opt. Express 13(6), 1966 (2005).
  * Kazansky et al. (2016) A. K. Kazansky, A. V. Bozhevolnov, I. P. Sazhina, and N. M. Kabachnik, _Interference effects in angular streaking with a rotating terahertz field_ , Phys. Rev. A 93, 013407 (2016).
  * Li et al. (2018) S. Li, Z. Guo, R. N. Coffee, K. Hegazy, Z. Huang, A. Natan, T. Osipov, D. Ray, A. Marinelli, and J. P. Cryan, _Characterizing isolated attosecond pulses with angular streaking_ , Opt. Express 26(4), 4531 (2018).
  * Kazansky et al. (2019) A. K. Kazansky, I. P. Sazhina, and N. M. Kabachnik, _Fast retrieval of temporal characteristics of fel pulses using streaking by thz field_ , Opt. Express 27(9), 12939 (2019).
  * Eckle et al. (2008a) P. Eckle, M. Smolarski, P. Schlup, J. Biegert, A. Staudte, M. Schoffler, H. G. Muller, R. Dorner, and U. Keller, _Attosecond angular streaking_ , Nat. Phys. 4, 565 (2008a).
  * Eckle et al. (2008b) P. Eckle, M. Smolarski, P. Schlup, J. Biegert, A. Staudte, M. Schoffler, H. G. Muller, R. Dorner, and U. Keller, _Attosecond angular streaking_ , Nat Phys 4, 565 (2008b).
  * Pfeiffer et al. (2012) A. N. Pfeiffer, C. Cirelli, M. Smolarski, D. Dimitrovski, M. Abu-samha, L. B. Madsen, and U. Keller, _Attoclock reveals natural coordinates of the laser-induced tunnelling current flow in atoms_ , Nat Phys 8, 76 (2012).
  * Constant et al. (1997) E. Constant, V. D. Taranukhin, A. Stolow, and P. B. Corkum, _Methods for the measurement of the duration of high-harmonic pulses_ , Phys. Rev. A 56, 3870 (1997).
  * Itatani et al. (2002) J. Itatani, F. Quéré, G. L. Yudin, M. Y. Ivanov, F. Krausz, and P. B. Corkum, _Attosecond streak camera_ , Phys. Rev. Lett. 88, 173903 (2002).
  * Goulielmakis et al. (2004) E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, et al., _Direct measurement of light waves_ , Science 305(5688), 1267 (2004).
  * Kienberger et al. (2004) R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, et al., _Atomic transient recorder_ , Nature 427, 817 (2004).
  * Yakovlev et al. (2005) V. S. Yakovlev, F. Bammer, and A. Scrinzi, _Attosecond streaking measurements_ , J. Mod. Opt. 52(2-3), 395 (2005).
  * Frühling et al. (2009) U. Frühling, M. Wieland, M. Gensch, T. Gebert, B. Schütte, M. Krikunova, R. Kalms, F. Budzyn, O. Grimm, J. Rossbach, et al., _Single-shot terahertz-field-driven x-ray streak camera_ , Nature Photonics 3, 523 (2009).
  * Zhang and Thumm (2011) C.-H. Zhang and U. Thumm, _Streaking and Wigner time delays in photoemission from atoms and surfaces_ , Phys. Rev. A 84, 033401 (2011).
  * Ivanov and Smirnova (2011) M. Ivanov and O. Smirnova, _How accurate is the attosecond streak camera?_ , Phys. Rev. Lett. 107, 213605 (2011).
  * Zhao et al. (2022) X. Zhao, S. Li, T. Driver, V.-H. Hoang, A.-T. Le, J. P. Cryan, A. Marinelli, and C. D. Lin, _Characterization of single-shot attosecond pulses with angular streaking photoelectron spectra_ , Phys. Rev. A 105, 013111 (2022).
  * Dahlström et al. (2012) J. M. Dahlström, A. L. Huillier, and A. Maquet, _Introduction to attosecond delays in photoionization_ , J. Phys. B 45(18), 183001 (2012).
  * Dahlström et al (2013) J. M. Dahlström et al, _Theory of attosecond delays in laser-assisted photoionization_ , Chem. Phys. 414, 53 (2013).
  * Maquet et al. (2014) A. Maquet, J. Caillat, and R. Taïeb, _Attosecond delays in photoionization: time and quantum mechanics_ , J. Phys. B 47(20), 204004 (2014).
  * Kheifets et al. (2022) A. S. Kheifets, R. Wielian, I. A. Ivanov, A. L. Wang, A. Marinelli, and J. P. Cryan, _Phase retrieval from angular streaking of XUV atomic ionization_ (2022), URL https://arxiv.org/abs/2202.06147.
  * Serov and Kheifets (2017) V. V. Serov and A. S. Kheifets, _Time delay in XUV/IR photoionization of H 2O_, J. Chem. Phys. 147(20), 204303 (2017).
  * Ning et al. (2014) Q.-C. Ning, L.-Y. Peng, S.-N. Song, W.-C. Jiang, S. Nagele, R. Pazourek, J. Burgdörfer, and Q. Gong, _Attosecond streaking of Cohen-Fano interferences in the photoionization of H ${}_{2}^{+}$_, Phys. Rev. A 90, 013423 (2014).
  * Serov and Kheifets (2016) V. V. Serov and A. S. Kheifets, _Angular anisotropy of time delay in XUV+IR photoionization of H ${}_{2}^{+}$_, Phys. Rev. A 93, 063417 (2016).
  * Kitzler et al. (2002) M. Kitzler, N. Milosevic, A. Scrinzi, F. Krausz, and T. Brabec, _Quantum theory of attosecond XUV pulse measurement by laser dressed photoionization_ , Phys. Rev. Lett. 88, 173904 (2002).
  * Schultze et al. (2010) M. Schultze, M. Fiess, N. Karpowicz, J. Gagnon, M. Korbman, M. Hofstetter, S. Neppl, A. L. Cavalieri, Y. Komninos, T. Mercouris, et al., _Delay in Photoemission_ , Science 328(5986), 1658 (2010).
  * Serov (2011) V. V. Serov, _Calculation of intermediate-energy electron-impact ionization of molecular hydrogen and nitrogen using the paraxial approximation_ , Phys. Rev. A 84, 062701 (2011).
  * Cohen and Fano (1966) H. D. Cohen and U. Fano, _Interference in the photo-ionization of molecules_ , Phys. Rev. 150, 30 (1966).
  * Kaplan and Markin (1969) I. G. Kaplan and A. P. Markin, _Interference phenomena in photoionization of molecules_ , Sov. Phys. Dokl. 14, 36 (1969).
  * Walter and Briggs (1999) M. Walter and J. Briggs, _Photo-double ionization of molecular hydrogen_ , J. Phys. B 32(11), 2487 (1999).
  * Fernández et al. (2007) J. Fernández, O. Fojón, A. Palacios, and F. Martín, _Interferences from fast electron emission in molecular photoionization_ , Phys. Rev. Lett. 98, 043005 (2007).
  * Fernández et al. (2009) J. Fernández, F. L. Yip, T. N. Rescigno, C. W. McCurdy, and F. Martín, _Two-center effects in one-photon single ionization of H ${}_{2}^{+}$, H2, and Li${}_{2}^{+}$ with circularly polarized light_, Phys. Rev. A 79, 043409 (2009).
  * Serov et al. (2012) V. V. Serov, I. A. Ivanov, and A. S. Kheifets, _Single-photon double ionization of H 2 away from equilibrium: A showcase of two-center electron interference_, Phys. Rev. A 86, 025401 (2012).
  * Serov et al. (2015) V. V. Serov, V. L. Derbov, and T. A. Sergeeva, _Interpretation of the Time Delay in the Ionization of Coulomb Systems by Attosecond Laser Pulses_ , _Advanced Lasers: Laser Physics and Technology for Applied and Fundamental Science_ (Springer Netherlands, Dordrecht, 2015), pp. 213–230, ISBN 978-94-017-9481-7.


# On Intersection Graph of Dihedral Group

Sanhan M.S. Khasraw
Department of Mathematics, College of Basic Education,
Salahaddin University-Erbil, Erbil, Kurdistan Region, Iraq
<EMAIL_ADDRESS>
Abstract

Let $G$ be a finite group. The intersection graph of $G$ is a graph whose
vertex set is the set of all proper non-trivial subgroups of $G$ and two
distinct vertices $H$ and $K$ are adjacent if and only if $H\cap
K\neq\\{e\\}$, where $e$ is the identity of the group $G$. In this paper, we
investigate some properties and exploring some topological indices such as
Wiener, Hyper-Wiener, first and second Zagreb, Schultz, Gutman and eccentric
connectivity indices of the intersection graph of $D_{2n}$ for $n=p^{2}$, $p$
is prime. We also find the metric dimension and the resolving polynomial of
the intersection graph of $D_{2p^{2}}$.

Keywords: Intersection graph of subgroups, Wiener index, Zagreb indices,
Schultz index, resolving polynomial of a graph.

##  1 Introduction

The notion of intersection graph of a finite group has been introduced by
Csákány and Pollák in 1969 [1]. For a finite group $G$, associate a graph
$\Gamma(G)$ with it in such away that the set of vertices of $\Gamma(G)$ is
the set of all proper non-trivial subgroups of $G$ and join two vertices if
their intersection is non-trivial. For more studies about intersection graphs
of subgroups, we refer the reader to see [9, 2, 3, 6, 7].

Suppose that $\Gamma$ is a simple graph, which is undirected and contains no
multiple edges or loops. We denote the set of vertices of $\Gamma$ by
$V(\Gamma)$ and the set of edges of $\Gamma$ by $E(\Gamma)$. We write $uv\in
E(\Gamma)$ if $u$ and $v$ form an edge in $\Gamma$. The size of the vertex-set
of $\Gamma$ is denoted by $|V(\Gamma)|$ and the number of edges of $\Gamma$ is
denoted by $|E(\Gamma)|$. The degree of a vertex $v$ in $\Gamma$, denoted by
$deg(v)$, is defined as the number of edges incident to $v$. The distance
between any pair of vertices $u$ and $v$ in $\Gamma$, denoted by $d(u,v)$, is
the shortest $u-v$ path in $\Gamma$. For a vertex $v$ in $\Gamma$, the
eccentricity of $v$, denoted by $ecc(v)$, is the largest distance between $v$
and any other vertex in $\Gamma$. The diameter of $\Gamma$, denoted as
$diam(\Gamma)$, is defined by $diam(\Gamma)=max\\{ecc(v):v\in V(\Gamma)\\}$. A
graph $\Gamma$ is called complete if every pair of vertices in $\Gamma$ are
adjacent. If $S\subseteq V(\Gamma)$ and no two elements of $S$ are adjacent,
then $S$ is called an independent set. The cardinality of the largest
independent set is called an independent number of the graph $\Gamma$. A graph
$\Gamma$ is called bipartite if the set $V(\Gamma)$ can be partitioned into
two disjoint independent sets such that each edge in $\Gamma$ has its ends in
different independent sets. A graph $\Gamma$ is called split if $V(\Gamma)$
can be partitioned into two different sets $U$ and $K$ such that $U$ is an
independent set and the subgraph induced by $K$ is a complete graph.

Let $W=\\{v_{1},v_{2},\cdots,v_{k}\\}\subseteq V(\Gamma)$ and let $v$ be any
vertex of $\Gamma$. The representation of $v$ with respect to $W$ is the
k-vector $r(v|W)=(d(v,v_{1}),d(v,v_{2}),$ $\cdots,d(v,v_{k}))$. If distinct
vertices have distinct representations with respect to $W$, then $W$ is called
a resolving set for $\Gamma$. A basis of $\Gamma$ is a minimum resolving set
for $\Gamma$ and the cardinality of a basis of $\Gamma$ is called the metric
dimension of $\Gamma$ and denoted by $\beta(\Gamma)$ [8]. Suppose $r_{i}$ is
the number of resolving sets for $\Gamma$ of cardinality $i$. Then the
resolving polynomial of a graph $\Gamma$ of order $n$, denoted by
$\beta(\Gamma,x)$, is defined as
$\beta(\Gamma,x)=\sum_{i=\beta(\Gamma)}^{n}r_{i}x^{i}$. The sequence
$(r_{\beta(\Gamma)},r_{\beta(\Gamma)+1},\cdots,r_{n})$ formed from the
coefficients of $\beta(\Gamma,x)$ is called the resolving sequence.

For a graph $\Gamma$, the Wiener index is defined by
$W(\Gamma)=\sum_{\\{u,v\\}\subseteq V(\Gamma)}d(u,v)$ [5]. The hyper-Wiener
index of $\Gamma$ is defined by
$WW(\Gamma)=\frac{1}{2}W(\Gamma)+\frac{1}{2}\sum_{\\{u,v\\}\subseteq
V(\Gamma)}(d(u,v))^{2}$ [10]. The Zagreb indices are defined by
$M_{1}(\Gamma)=\sum_{v\in V(\Gamma)}(deg(v))^{2}$ and
$M_{2}(\Gamma)=\sum_{uv\in E(\Gamma)}deg(u)deg(v)$ [13]. The Schultz index of
$\Gamma$, denoted by $MTI(\Gamma)$ is defined in [14] by
$MTI(\Gamma)=\sum_{\\{u,v\\}\subseteq V(\Gamma)}d(u,v)[deg(u)+deg(v)]$. In
[15, 11] the Gutman index has been defined by
$Gut(\Gamma)=\sum_{\\{u,v\\}\subseteq V(\Gamma)}d(u,v)[deg(u)\times deg(v)]$.
Sharma, Goswami and Madan defined the eccentric connectivity index of
$\Gamma$, denoted by $\xi^{c}(\Gamma)$, in [12] by $\xi^{c}(\Gamma)=\sum_{v\in
V(\Gamma)}deg(v)ecc(v)$.

For an integer $n\geq 3$, the dihedral group $D_{2n}$ of order $2n$ is defined
by

$D_{2n}=\langle r,s:r^{n}=s^{2}=1,srs=r^{-1}\rangle.$

In [6], Rajkumar and Devi studied the intersection graph of subgroups of some
non-abelian groups, especially the dihedral group $D_{2n}$, quaternion group
$Q_{n}$ and quasi-dihedral group $QD_{2^{\alpha}}$. They were only able to
obtain the clique number and degree of vertices. It seems difficult to study
most properties of the intersection graph of subgroups of these groups. In
this paper, the focus will be on the intersection graph of subgroups of the
dihedral group $D_{2n}$ for the case when $n=p^{2}$, $p$ is prime. It is clear
that when $n=p$, then the resulting intersection graph of subgroups is a null
graph, which is not of our interest. For $n=p^{2}$, the intersection graph
$\Gamma({D_{2p^{2}}})$ of the group $D_{2p^{2}}$ has $p^{2}+p+2$ vertices. We
leave the other possibilities for $n$ open and we might be able to work on
them in the future. So, all throughout this paper, the considered dihedral
group is of order $2p^{2}$, and by intersection graph we mean intersection
graph of subgroups.

This paper is organized as follows. In Section 2, some basic properties of the
intersection graph of $D_{2p^{2}}$ are presented. We see that the intersection
graph $\Gamma({D_{2p^{2}}})$ is split. In Section 3, we find some topological
indices of the intersection graph $\Gamma({D_{2p^{2}}})$ of $D_{2p^{2}}$ such
as the Wiener, hyper-Wiener and Zagreb indices. In Section 4, we find the
metric dimension and the resolving polynomial of the intersection graph
$\Gamma({D_{2p^{2}}})$.

##  2 Some properties of the intersection graph of $D_{2n}$

In [6], all proper non-trivial subgroups of the group $D_{2n}$ has been
classified as shown in the following lemma.

######  Lemma 2.1.

The proper non-trivial subgroups of $D_{2n}$ are:

  1. 1.

cyclic groups $H^{r}=\langle r^{\frac{n}{k}}\rangle$ of order $k$, where $k$
is a divisor of $n$ and $k\neq 1$,

  2. 2.

cyclic groups $H_{s}=\langle sr^{i}\rangle$ of order 2, where
$i=1,2,\cdots,n$, and

  3. 3.

dihedral groups $H_{s}^{r}=\langle r^{\frac{n}{k}},sr^{i}\rangle$ of order
$2k$, where $k$ is a divisor of $n$, $k\neq 1,n$ and
$i=1,2,\cdots,\frac{n}{k}$.

The total number of these proper subgroups is $\tau(n)+\sigma(n)-2$, where
$\tau(n)$ is the number of positive divisors of $n$ and $\sigma(n)$ is the sum
of positive divisors of $n$. We mentioned that we only focus on the case when
$n=p^{2}$, $p$ is prime. Recall that, for $n=p^{2}$, the intersection graph
$\Gamma({D_{2p^{2}}})$ of the group $D_{2p^{2}}$ has $p^{2}+p+2$ vertices. The
vertex set of $\Gamma({D_{2p^{2}}})$ is
$V(\Gamma({D_{2p^{2}}}))=\cup_{i=1}^{p}H_{i}\cup\\{H_{p}^{i}\\}\cup H_{1,p}$,
where

  1. 1.

$H_{i}=\\{\langle sr^{i+lp}\rangle;1\leq l\leq p\\}$,

  2. 2.

$H_{p}^{i}=\langle r^{p},sr^{i}\rangle$, and

  3. 3.

$H_{1,p}=\\{\langle r\rangle,\langle r^{p}\rangle\\}$

The following lemma is given in [6] to compute the degree of any vertex in
$\Gamma({D_{2n}})$. Since we only consider the case $n=p^{2}$, we restate it
as follows:

######  Theorem 2.2.

In the graph $\Gamma({D_{2p^{2}}})$,

$deg(v)=\left\\{\begin{tabular}[]{ll}$1$,&\mbox{ if }$v\in H_{i}$\\\
$2p+1$&\mbox{ if }$v=H_{p}^{i}$\\\ $p+1$,&\mbox{ if }$v\in H_{1,p}$\\\
\end{tabular}\right.$

where $i=1,2,...,p$.

The following theorem gives the exact number of edges in
$\Gamma({D_{2p^{2}}})$ which can be in the Section 3 to compute the second
Zagreb index.

######  Theorem 2.3.

In the graph $\Gamma({D_{2p^{2}}})$,
$|E(\Gamma(D_{2p^{2}}))|=\frac{1}{2}(3p^{2}+3p+2)$.

###### Proof.

It follows from Theorem 2.2 that there are $p^{2}$ vertices of degree 1, $p$
vertices of degree $2p+1$ and 2 vertices of degree $p+1$. Thus,
$|E(\Gamma(D_{2p^{2}}))|=\frac{1}{2}\sum_{v\in
V(\Gamma({D_{2p^{2}}}))}deg(v)=\frac{1}{2}(p^{2}\cdot
1+p\cdot(2p+1)+2\cdot(p+1))=\frac{1}{2}(3p^{2}+3p+2)$. ∎

######  Theorem 2.4.

Let $\Gamma=\Gamma({D_{2p^{2}}})$ be an intersection graph on $D_{2p^{2}}$.
Then $diam(\Gamma)=3$. In particular, $\Gamma$ is connected.

###### Proof.

Suppose $u$ and $v$ are two distinct vertices of $\Gamma({D_{2p^{2}}})$. If
$u$ and $v$ are adjacent, then $d(u,v)=1$. Otherwise, let $u\cap v=\\{e\\}$.
Then there are three possibilities: both $u$ and $v$ are in $H_{i}$, one of
them is in $H_{i}$ and the other is in $H_{1,p}$, or one of them is in $H_{i}$
and the other is in $H_{j}$ for $i\neq j$. For the first case, if $u,v\in
H_{i}$ for some $i$. There exists $z=H_{p}^{i}$ such that $uz,vz\in E(\Gamma)$
and then $d(u,v)=2$. If $u\in H_{i}$ and $v\in H_{1,p}$, then there exists
$z^{\prime}=H_{p}^{i}$ such that $uz^{\prime},vz^{\prime}\in E(\Gamma)$. Again
$z^{\prime}$ is adjacent to both $u$ and $v$, and then $d(u,v)=2$. Finally, if
$u\in H_{i}$ and $v\in H_{j}$ for $i\neq j$, then there exists $w=H_{p}^{i}$
and $w^{\prime}=H_{p}^{j}$ such that $uw,vw^{\prime}\in E(\Gamma)$. However,
$ww^{\prime}\in E(\Gamma)$, then the shortest path from $u$ to $v$ has length
3 and so $d(u,v)=3$. ∎

From Theorem 2.4, one can see that the maximum distance between any pair of
vertices in $\Gamma(D_{2p^{2}})$ is 3. In order to explore the exact distance
between any pair of vertices in $\Gamma(D_{2p^{2}})$ , we state the following
corollary which can be used in the next section to find some topological
indices of $\Gamma(D_{2p^{2}})$.

######  Corollary 2.5.

In the graph $\Gamma({D_{2p^{2}}})$,

$d(u,v)=\left\\{\begin{tabular}[]{ll}$1$,&\mbox{ if }$u,v\in
H_{1,p}\cup\\{H_{p}^{i}\\}$ \mbox{ or } $v\in H_{i},v=H_{p}^{i}$\\\
$2$,&\mbox{ if }$u,v\in H_{i}$ \mbox{ or } $u\in H_{j},v\in
H_{1,p}\cup\\{H_{p}^{i}\\},i\neq j$\\\ $3$,&\mbox{ if }$u\in H_{i},v\in
H_{j},i\neq j$\\\ \end{tabular}\right.$

where $i,j=1,2,...,p$.

######  Theorem 2.6.

Let $\Gamma=\Gamma({D_{2p^{2}}})$ be an intersection graph on $D_{2p^{2}}$.
Then for each $i$, $H_{i}$ forms an independent set.

###### Proof.

From Corollary 2.5, $d(u,v)=2$ for every distinct pairs of vertices $u,v\in
H_{i}$ and so $uv\notin E(\Gamma)$. Therefore, $H_{i}$ is an independent set
for each $i$. ∎

######  Corollary 2.7.

The independent number of the graph $\Gamma({D_{2p^{2}}})$ is $p^{2}+1$.

###### Proof.

From Theorem 2.6, there are $p$ independent sets of size $p$. Also, from
Corollary 2.5, one can see that none of the vertices of $H_{1,p}$ is adjacent
to vertices in $H_{i}$ for each $i$. So, in total the size of the largest
independent set is $p^{2}+1$. ∎

######  Theorem 2.8.

Let $H\subseteq V(\Gamma({D_{2p^{2}}}))$. Then the intersection graph
$\Gamma(H)$ is complete if and only if $H=\cup_{i=1}^{p}\\{H_{p}^{i}\\}\cup
H_{1,p}$.

###### Proof.

Suppose $H=\cup_{i=1}^{p}\\{H_{p}^{i}\\}\cup H_{1,p}$. By Corollary 2.5,
$d(u,v)=1$ for every distinct pairs of vertices $u,v\in H$. Then the graph
$\Gamma(H)$ is complete. The converse follows directly from Corollary 2.5. ∎

The complete graph in the previous theorem is the largest complete subgraph of
$\Gamma({D_{2n}})$. As a consequence, the clique number of $\Gamma({D_{2n}})$
is $p+2$ which coincides with Theorem 2.3 in [6].

######  Theorem 2.9.

Let $H\subseteq V(\Gamma({D_{2p^{2}}}))$. Then $\Gamma(H)=K_{1,p}$ if and only
if $H=H_{i}\cup\\{H_{p}^{i}\\}$ for some $i$.

###### Proof.

The proof follows from Theorems 2.6 and 2.8. ∎

As a consequence of the above theorem, we have the following corollary.

######  Corollary 2.10.

The graph $\Gamma({D_{2p^{2}}})$ is split.

######  Theorem 2.11.

In the graph $\Gamma({D_{2p^{2}}})$,

$ecc(v)=\left\\{\begin{tabular}[]{ll}$2$,&\mbox{ if }$v\in
H_{1,p}\cup\\{H_{p}^{i}\\}$\\\ $3$&\mbox{ if }$v\in H_{i}$\\\
\end{tabular}\right.$

where $i=1,2,...,p$.

###### Proof.

It follows from Corollary 2.5 that no vertex of $H_{1,p}\cup\\{H_{p}^{i}\\}$
is adjacent to any vertex of $H_{j}$, where $i,j=1,2,...,p$ and $i\neq j$.
Then the maximum distance between any vertex of $H_{1,p}\cup\\{H_{p}^{i}\\}$
and any other vertex in $H_{j}$, $i\neq j$, is 2. Thus, $ecc(v)=2$ for each
$v\in H_{1,p}\cup\\{H_{p}^{i}\\}$. Again, from Corollary 2.5, the maximum
distance between any vertex of $H_{i}$ and any other vertex of $H_{j}$, $i\neq
j$, is 3, so $ecc(v)=3$ for each $v\in H_{i}$. ∎

##  3 Some Topological Indices of intersection graph on $D_{2p^{2}}$

In this section, some topological indices, such as the Wiener index, Hyper-
Wiener index, Zagreb indices, the Schultz index, the Gutman index and the
eccentric connectivity index, of the intersection graph for the dihedral group
$D_{2n}$, where $n=p^{2}$, are computed.

######  Theorem 3.1.

Let $\Gamma=\Gamma({D_{2n}})$ be an intersection graph on $D_{2n}$. Then

$W(\Gamma)=\frac{1}{2}(3p^{4}+3p^{3}+5p^{2}+3p+2).$

###### Proof.

Let $u,v\in V(\Gamma)$. It follows from Corollary 2.5 that the number of
possibilities of $d(u,v)=1$ is $p^{2}+{{p+2}\choose{2}}$, the number of
possibilities of $d(u,v)=2$ is $p\cdot{{p}\choose{2}}+p\cdot p\cdot(p+1)$ and
the number of possibilities of $d(u,v)=3$ is
${{p}\choose{2}}{{p}\choose{1}}{{p}\choose{1}}$. Thus,
$W(\Gamma({D_{2n}}))=(p^{2}+\frac{1}{2}(p+1)(p+2))\cdot
1+(\frac{1}{2}(3p^{3}+p^{2}))\cdot 2+(\frac{1}{2}(p^{4}-p^{3}))\cdot
3=\frac{1}{2}(3p^{4}+3p^{3}+5p^{2}+3p+2)$. ∎

######  Theorem 3.2.

Let $\Gamma({D_{2n}})$ be an intersection graph on $D_{2n}$. Then

$WW(\Gamma({D_{2n}}))=\frac{1}{2}(6p^{4}+3p^{3}+6p^{2}+3p+2).$

###### Proof.

From Theorem 3.1 and Corollary 2.5, we can see that
$WW(\Gamma({D_{2n}}))=\frac{1}{2}\bigg{(}\frac{1}{2}(3p^{4}+3p^{3}+5p^{2}+3p+2)\bigg{)}+\frac{1}{2}\bigg{(}\bigg{(}p^{2}+\frac{1}{2}(p+1)(p+2)\bigg{)}\cdot
1^{2}+\bigg{(}\frac{1}{2}(3p^{3}+p^{2})\bigg{)}\cdot
2^{2}+\bigg{(}\frac{1}{2}(p^{4}-p^{3})\bigg{)}\cdot
3^{2}\bigg{)}=\frac{1}{2}(6p^{4}+3p^{3}+6p^{2}+3p+2)$. ∎

In the next two theorems, the first and second Zagreb indices for the
intersection graph $\Gamma({D_{2n}})$ are presented.

######  Theorem 3.3.

Let $\Gamma({D_{2n}})$ be an intersection graph on $D_{2n}$. Then

$M_{1}(\Gamma({D_{2n}}))=4p^{3}+7p^{2}+5p+2.$

###### Proof.

The proof is similar to the proof of Theorem 2.3. It follows from Theorem 2.2
that $M_{1}(\Gamma({D_{2n}}))=p^{2}\cdot
1^{2}+p\cdot(2p+1)^{2}+2\cdot(p+1)^{2}=4p^{3}+7p^{2}+5p+2$. ∎

######  Theorem 3.4.

Let $\Gamma({D_{2n}})$ be an intersection graph on $D_{2n}$. Then

$M_{2}(\Gamma({D_{2n}}))=2p^{4}+6p^{3}+\frac{13}{2}p^{2}+\frac{7}{2}p+1.$

###### Proof.

By Theorem 2.3, $\Gamma$ has $\frac{1}{2}(3p^{2}+3p+2)$ edges in which $p^{2}$
edges with one end-vertex of degree 1 and the other end-vertex of degree
$2p+1$, $\frac{p(p-1)}{2}$ edges where end-vertices have degree $2p+1$, $2p$
edges with one end-vertex of degree $2p+1$ and the other end-vertex of degree
$p+1$ and one edge where end-vertices have degree $p+1$. Thus,
$M_{2}(\Gamma({D_{2n}}))=p^{2}\cdot(1)(2p+1)+\frac{p(p-1)}{2}\cdot(2p+1)^{2}+2p\cdot(2p+1)(p+1)+1\cdot(p+1)^{2}=2p^{4}+6p^{3}+\frac{13}{2}p^{2}+\frac{7}{2}p+1$.
∎

######  Theorem 3.5.

Let $\Gamma({D_{2n}})$ be an intersection graph on $D_{2n}$. Then

$MTI(\Gamma({D_{2n}}))=7p^{4}+6p^{3}+5p^{2}+5p+2.$

###### Proof.

By Theorem 2.2 and Corollary 2.5, for $i,j=1,2,\cdots,p$,

$\displaystyle MTI(\Gamma({D_{2n}}))$ $\displaystyle=\bigg{(}\sum_{u\in
H_{i},v=H_{p}^{i}}d(u,v)[deg(u)+deg(v)]$
$\displaystyle+\sum_{u,v\in\\{H_{p}^{i}\\}}d(u,v)[deg(u)+deg(v)]$
$\displaystyle+\sum_{u,v\in H_{1,p}}d(u,v)[deg(u)+deg(v)]$
$\displaystyle+\sum_{u\in
H_{1,p},v\in\\{H_{p}^{i}\\}}d(u,v)[deg(u)+deg(v)]\bigg{)}$
$\displaystyle+\bigg{(}\sum_{u,v\in H_{i}}d(u,v)[deg(u)+deg(v)]$
$\displaystyle+\sum_{u\in H_{j},v=H_{p}^{i},i\neq j}d(u,v)[deg(u)+deg(v)]$
$\displaystyle+\sum_{u\in H_{i},v\in H_{1,p}}d(u,v)[deg(u)+deg(v)]\bigg{)}$
$\displaystyle+\bigg{(}\sum_{u\in H_{i},v\in H_{j},i\neq
j}d(u,v)[deg(u)+deg(v)]\bigg{)}$ $\displaystyle=\bigg{(}p^{2}\cdot
1\cdot[1+(p+1)]+{{p}\choose{2}}\cdot 1\cdot[(2p+1)+(2p+1)]$
$\displaystyle+1\cdot 1\cdot[(p+1)+(p+1)]$
$\displaystyle+{{2}\choose{1}}\cdot{{p}\choose{1}}\cdot
1\cdot[(p+1)+(2p+1)]\bigg{)}+\bigg{(}p\cdot{{p}\choose{2}}\cdot 2\cdot[1+1]$
$\displaystyle+p\cdot p\cdot(p-1)\cdot 2\cdot[1+(2p+1)]$ $\displaystyle+p\cdot
p\cdot 2\cdot
2\cdot[1+(p+1)]\bigg{)}+\bigg{(}{{p}\choose{1}}\cdot{{p}\choose{1}}\cdot{{p}\choose{2}}\cdot
3\cdot[1+1]\bigg{)}$ $\displaystyle=7p^{4}+6p^{3}+5p^{2}+5p+2.$

∎

######  Theorem 3.6.

Let $\Gamma({D_{2n}})$ be an intersection graph on $D_{2n}$. Then

$Gut(\Gamma({D_{2n}}))=\frac{1}{2}(15p^{4}+13p^{3}+15p^{2}+7p+2).$

###### Proof.

Again by Theorem 2.2 and Corollary 2.5, for $i,j=1,2,\cdots,p$,

$\displaystyle Gut(\Gamma({D_{2n}}))$ $\displaystyle=\bigg{(}\sum_{u\in
H_{i},v=H_{p}^{i}}d(u,v)[deg(u)\times deg(v)]$
$\displaystyle+\sum_{u,v\in\\{H_{p}^{i}\\}}d(u,v)[deg(u)\times deg(v)]$
$\displaystyle+\sum_{u,v\in H_{1,p}}d(u,v)[deg(u)\times deg(v)]$
$\displaystyle+\sum_{u\in H_{1,p},v\in\\{H_{p}^{i}\\}}d(u,v)[deg(u)\times
deg(v)]\bigg{)}$ $\displaystyle+\bigg{(}\sum_{u,v\in H_{i}}d(u,v)[deg(u)\times
deg(v)]$ $\displaystyle+\sum_{u\in H_{j},v=H_{p}^{i},i\neq
j}d(u,v)[deg(u)\times deg(v)]$ $\displaystyle+\sum_{u\in H_{i},v\in
H_{1,p}}d(u,v)[deg(u)\times deg(v)]\bigg{)}$ $\displaystyle+\bigg{(}\sum_{u\in
H_{i},v\in H_{j},i\neq j}d(u,v)[deg(u)\times deg(v)]\bigg{)}$
$\displaystyle=\bigg{(}p^{2}\cdot 1\cdot[1\times(p+1)]+{{p}\choose{2}}\cdot
1\cdot[(2p+1)\times(2p+1)]$ $\displaystyle+1\cdot 1\cdot[(p+1)\times(p+1)]$
$\displaystyle+{{2}\choose{1}}\cdot{{p}\choose{1}}\cdot
1\cdot[(p+1)\times(2p+1)]\bigg{)}$
$\displaystyle+\bigg{(}p\cdot{{p}\choose{2}}\cdot 2\cdot[1\times 1]+p\cdot
p\cdot(p-1)\cdot 2\cdot[1\times(2p+1)]$ $\displaystyle+p\cdot p\cdot 2\cdot
2\cdot[1\times(p+1)]\bigg{)}+\bigg{(}{{p}\choose{1}}\cdot{{p}\choose{1}}\cdot{{p}\choose{2}}\cdot
3\cdot[1\times 1]\bigg{)}$
$\displaystyle=\frac{1}{2}(15p^{4}+13p^{3}+15p^{2}+7p+2).$

∎

######  Theorem 3.7.

Let $\Gamma({D_{2n}})$ be an intersection graph on $D_{2n}$. Then

$\xi^{c}(\Gamma({D_{2n}}))=7p^{2}+6p+4.$

###### Proof.

By Theorems 2.2 and 2.11, for $i=1,2,\cdots,p$, we see that

$\displaystyle\xi^{c}(\Gamma({D_{2n}}))=$ $\displaystyle\sum_{v\in
H_{i}}deg(v)ecc(v)+\sum_{v\in\\{H_{p}^{i}\\}}deg(v)ecc(v)+\sum_{v\in
H_{1,p}}deg(v)ecc(v)$ $\displaystyle=\sum_{v\in H_{i}}1\times
3+\sum_{v\in\\{H_{p}^{i}\\}}(2p+1)\times 2+\sum_{v\in H_{1,p}}(p+1)\times 2$
$\displaystyle=p^{2}\times 1\times 3+p\times(2p+1)\times 2+2\times(p+1)\times
2$ $\displaystyle=7p^{2}+6p+4.$

∎

##  4 Metric dimension and resolving polynomial of intersection graph on
$D_{2p^{2}}$

For a vertex $u$ of a graph $\Gamma$, the set $N(u)=\\{v\in V(\Gamma):uv\in
E(\Gamma)\\}$ is called the open neighborhood of $u$ and the set
$N[u]=N(u)\cup\\{u\\}$ is called the closed neighborhood of $u$. If $u$ and
$v$ are two distinct vertices of $\Gamma$, then $u$ and $v$ are said to be
adjacent twins if $N[u]=N[v]$ and non-adjacent twins if $N(u)=N(v)$. Two
distinct vertices are called twins if they are adjacent or non-adjacent twins.
A subset $U\subseteq V(\Gamma)$ is called a twin-set in $\Gamma$ if every pair
of distinct vertices in $U$ are twins.

######  Lemma 4.1.

Let $\Gamma$ be a connected graph of order $n$ and $U\subseteq V(\Gamma)$ be a
twin set in $\Gamma$ with $|U|=m$. Then every resolving set for $\Gamma$
contains at least $m-1$ vertices of $U$.

######  Corollary 4.2.

[4] Let $\Gamma$ be a connected graph, $U$ resolves $\Gamma$ and $u$ and $v$
are twins. Then $u\in U$ or $v\in U$. In addition, if $u\in U$ and $v\notin
U$, then $(U\setminus\\{u\\})\cup\\{v\\}$ also resolves $\Gamma$.

######  Theorem 4.3.

Let $\Gamma(D_{2p^{2}})$ be an intersection graph on $D_{2p^{2}}$. Then

$\beta(\Gamma(D_{2p^{2}}))=p^{2}-p+1.$

###### Proof.

Let $W=\cup_{i=1}^{p}(H_{i}-\\{\langle sr^{i}\rangle\\})\cup\\{\langle
r^{p}\rangle\\}$. One can see that $W$ is a resolving set for
$\Gamma(D_{2p^{2}})$ of cardinality $p(p-1)+1$. Then
$\beta(\Gamma({D_{2p^{2}}}))\leq p^{2}-p+1$.
On the other hand, the sets $H_{i}$, for $i=1,2,\cdots,p$, and $H_{1,p}$ are
twin sets of cardinality $p$ and $2$, respectively. Then by Lemma 4.1, we see
that $\beta(\Gamma({D_{2p^{2}}}))\geq p^{2}-p+1$. ∎

The following is a useful property for finding a resolving polynomial of a
graph of order $n$.

######  Lemma 4.4.

If $\Gamma$ is a connected graph of order $n$, then $r_{n}=1$ and $r_{n-1}=n$.

######  Theorem 4.5.

Let $\Gamma=\Gamma(D_{2p^{2}})$ be an intersection graph on $D_{2p^{2}}$. Then
$\beta(\Gamma,x)=x^{p^{2}-p+1}\bigg{(}{{2}\choose{1}}{{p}\choose{p-1}}^{p}+\sum_{q=1}^{p}r_{p^{2}-p+1+q}x^{q}+\sum_{k=p+1}^{2p-1}r_{p^{2}-p+1+k}x^{k}+(p^{2}+p+1)x^{2p}+x^{2p+1}\bigg{)},$
where
$r_{p^{2}-p+1+q}={{p}\choose{q}}{{p}\choose{p-1}}^{p-q}{{2}\choose{1}}{{p}\choose{0}}+{{p}\choose{q-i}}{{p}\choose{p-1}}^{p-(q-i)}{{2}\choose{1}}{{p}\choose{i}}+{{p}\choose{q-1}}{{p}\choose{p-1}}^{p-(q-1)}{{2}\choose{2}}{{p}\choose{0}}\\\
+{{p}\choose{q-1-i}}{{p}\choose{p-1}}^{p-(q-1-i)}{{2}\choose{2}}{{p}\choose{i}}$,

$r_{p^{2}-p+1+k}={{p}\choose{k_{1}}}{{p}\choose{p-1}}^{p-k_{1}}{{2}\choose{1}}{{p}\choose{k_{2}}}+{{p}\choose{k_{2}}}{{p}\choose{p-1}}^{p-k_{2}}{{2}\choose{1}}{{p}\choose{k_{1}}}+\\\
{{p}\choose{k_{1}-1}}{{p}\choose{p-1}}^{p-(k_{1}-1)}{{2}\choose{2}}{{p}\choose{k_{2}}}+{{p}\choose{k_{1}}}{{p}\choose{p-1}}^{p-k_{1}}{{2}\choose{2}}{{p}\choose{k_{2}-1}}+{{p}\choose{k_{2}-1}}{{p}\choose{p-1}}^{p-(k_{2}-1)}{{2}\choose{2}}{{p}\choose{k_{1}}}\\\
+{{p}\choose{k_{2}}}{{p}\choose{p-1}}^{p-k_{2}}{{2}\choose{2}}{{p}\choose{k_{1}-1}}$,

$i\leq q,k_{1}+k_{2}=k,k_{1}\neq k_{2},k_{1}-1\neq k_{2},k_{1}\neq k_{2}-1$
and $1\leq k_{j}\leq p$ for $j=1,2$.

###### Proof.

By Theorem 4.3, $\beta(\Gamma)=p^{2}-p+1$. It is required to find the
resolving sequence
$(r_{\beta(\Gamma)},r_{\beta(\Gamma)+1},\cdots,r_{\beta(\Gamma)+2p+1})$ of
length $2p+2$.
To find $r_{\beta(\Gamma)}$. For the reason that $H_{i}(1\leq i\leq p)$ and
$H_{1,p}$ are twin sets, then by Corollary 4.2 and the principal of
multiplication, we see that there are

$\underbrace{{{p}\choose{p-1}}{{p}\choose{p-1}}\cdots{{p}\choose{p-1}}}_{p-times}{{2}\choose{1}}=2p^{p}$

possibilities of resolving sets of cardinality $\beta(\Gamma)$, that is,
$r_{\beta(\Gamma)}=2p^{p}$.
For $1\leq l\leq 2p-1$, we aim to find $r_{\beta(\Gamma)+l}$.
First, we try to find $r_{\beta(\Gamma)+q}$, where $1\leq q\leq p$. Suppose
$u_{1},u_{2},\cdots,u_{q}$ be $q$ distinct vertices of $\Gamma$ that do not
belong to any resolving set of cardinality $\beta(\Gamma)+q-1$. Then there are
four possibilities to consider:

$u_{1},u_{2},\cdots,u_{q}\in\cup_{j=1}^{p}H_{j}$;
$u_{1},u_{2},\cdots,u_{q}\in\cup_{j=1}^{p}H_{j}\cup\\{H_{p}^{j}\\}$;
$u_{1},u_{2},\cdots,u_{q}\in\cup_{j=1}^{p}H_{j}\cup H_{1,p}$ or
$u_{1},u_{2},\cdots,u_{q}\in\cup_{j=1}^{p}H_{j}\cup\\{H_{p}^{j}\\}\cup
H_{1,p}$.

Altogether, by principals of addition and multiplication, there are
${{p}\choose{q}}{{p}\choose{p-1}}^{p-q}{{2}\choose{1}}{{p}\choose{0}}+{{p}\choose{q-i}}{{p}\choose{p-1}}^{p-(q-i)}{{2}\choose{1}}{{p}\choose{i}}+{{p}\choose{q-1}}{{p}\choose{p-1}}^{p-(q-1)}{{2}\choose{2}}{{p}\choose{0}}+\\\
{{p}\choose{q-1-i}}{{p}\choose{p-1}}^{p-(q-1-i)}{{2}\choose{2}}{{p}\choose{i}}$
possibilities of resolving sets of cardinality $\beta(\Gamma)+q$, where $1\leq
q\leq p$ and $i\leq q$.
Second, to find $r_{\beta(\Gamma)+k}$, where $p+1\leq k\leq 2p-1$. Take the
set of vertices $v_{1},v_{2},\cdots,v_{k}$ in $\Gamma$ that do not belong to
any resolving set of cardinality $\beta(\Gamma)+k-1$. Since $k>p$, then we
assume that $k=k_{1}+k_{2}$ such that $k_{1}\neq k_{2},k_{1}-1\neq k_{2}$ and
$k_{1}\neq k_{2}-1$, where $1\leq k_{j}\leq p$ and $j=1,2$. Then there are the
following possibilities:

$k_{1}$ vertices of the set $\\{v_{1},v_{2},\cdots,v_{k}\\}$ are in
$\cup_{i}^{p}H_{i}$ and $k_{2}$ vertices of the set
$\\{v_{1},v_{2},\cdots,v_{k}\\}$ are in $\cup_{i}^{p}\\{H_{p}^{i}\\}$,

$k_{2}$ vertices of the set $\\{v_{1},v_{2},\cdots,v_{k}\\}$ are in
$\cup_{i}^{p}H_{i}$ and $k_{1}$ vertices of the set
$\\{v_{1},v_{2},\cdots,v_{k}\\}$ are in $\cup_{i}^{p}\\{H_{p}^{i}\\}$,

$k_{1}$ vertices of the set $\\{v_{1},v_{2},\cdots,v_{k}\\}$ are in
$\cup_{i}^{p}H_{i}\cup H_{1,p}$ and $k_{2}$ vertices of the set
$\\{v_{1},v_{2},\cdots,v_{k}\\}$ are in $\cup_{i}^{p}\\{H_{p}^{i}\\}$,

$k_{1}$ vertices of the set $\\{v_{1},v_{2},\cdots,v_{k}\\}$ are in
$\cup_{i}^{p}H_{i}$ and $k_{2}$ vertices of the set
$\\{v_{1},v_{2},\cdots,v_{k}\\}$ are in $\cup_{i}^{p}\\{H_{p}^{i}\\}\cup
H_{1,p}$,

$k_{2}$ vertices of the set $\\{v_{1},v_{2},\cdots,v_{k}\\}$ are in
$\cup_{i}^{p}H_{i}\cup H_{1,p}$ and $k_{1}$ vertices of the set
$\\{v_{1},v_{2},\cdots,v_{k}\\}$ are in $\cup_{i}^{p}\\{H_{p}^{i}\\}$ or

$k_{2}$ vertices of the set $\\{v_{1},v_{2},\cdots,v_{k}\\}$ are in
$\cup_{i}^{p}H_{i}$ and $k_{1}$ vertices of the set
$\\{v_{1},v_{2},\cdots,v_{k}\\}$ are in $\cup_{i}^{p}\\{H_{p}^{i}\\}\cup
H_{1,p}$.
Again, by the principal of addition and multiplication, there are

${{p}\choose{k_{1}}}{{p}\choose{p-1}}^{p-k_{1}}{{2}\choose{1}}{{p}\choose{k_{2}}}+{{p}\choose{k_{2}}}{{p}\choose{p-1}}^{p-k_{2}}{{2}\choose{1}}{{p}\choose{k_{1}}}+{{p}\choose{k_{1}-1}}{{p}\choose{p-1}}^{p-(k_{1}-1)}{{2}\choose{2}}{{p}\choose{k_{2}}}+{{p}\choose{k_{1}}}{{p}\choose{p-1}}^{p-k_{1}}{{2}\choose{2}}{{p}\choose{k_{2}-1}}+{{p}\choose{k_{2}-1}}{{p}\choose{p-1}}^{p-(k_{2}-1)}{{2}\choose{2}}{{p}\choose{k_{1}}}+{{p}\choose{k_{2}}}{{p}\choose{p-1}}^{p-k_{2}}{{2}\choose{2}}{{p}\choose{k_{1}-1}}$
possible resolving sets of cardinality $\beta(\Gamma)+k$, where $p<k\leq
2p-1$.

By Lemma 4.4, $r_{\beta(\Gamma)+2p}=p^{2}+p+1$ and $r_{\beta(\Gamma)+2p+1}=1$.
∎

In the following remark, some additional possibilities of
$r_{\beta(\Gamma)+k}$, where $p<k\leq 2p-1$, are given.

######  Remark 4.6.

In Theorem 4.5, we have the following additional possibilities:

  1. 1.

if $k_{1}=k_{2}$, then $r_{\beta(\Gamma)+k}=\\\
{{p}\choose{k_{1}}}{{p}\choose{p-1}}^{p-k_{1}}{{2}\choose{1}}{{p}\choose{k_{2}}}+{{p}\choose{k_{1}-1}}{{p}\choose{p-1}}^{p-(k_{1}-1)}{{2}\choose{2}}{{p}\choose{k_{2}}}+{{p}\choose{k_{1}}}{{p}\choose{p-1}}^{p-k_{1}}{{2}\choose{2}}{{p}\choose{k_{2}-1}}$,

  2. 2.

if $k_{1}-1=k_{2}$, then $r_{\beta(\Gamma)+k}=\\\
{{p}\choose{k_{1}}}{{p}\choose{p-1}}^{p-k_{1}}{{2}\choose{1}}{{p}\choose{k_{2}}}+{{p}\choose{k_{2}}}{{p}\choose{p-1}}^{p-k_{2}}{{2}\choose{1}}{{p}\choose{k_{1}}}+{{p}\choose{k_{1}-1}}{{p}\choose{p-1}}^{p-(k_{1}-1)}{{2}\choose{2}}{{p}\choose{k_{2}}}+{{p}\choose{k_{1}}}{{p}\choose{p-1}}^{p-k_{1}}{{2}\choose{2}}{{p}\choose{k_{2}-1}}+{{p}\choose{k_{2}-1}}{{p}\choose{p-1}}^{p-(k_{2}-1)}{{2}\choose{2}}{{p}\choose{k_{1}}}$,
and

  3. 3.

if $k_{1}=k_{2}-1$, then $r_{\beta(\Gamma)+k}=\\\
{{p}\choose{k_{1}}}{{p}\choose{p-1}}^{p-k_{1}}{{2}\choose{1}}{{p}\choose{k_{2}}}+{{p}\choose{k_{2}}}{{p}\choose{p-1}}^{p-k_{2}}{{2}\choose{1}}{{p}\choose{k_{1}}}+{{p}\choose{k_{1}-1}}{{p}\choose{p-1}}^{p-(k_{1}-1)}{{2}\choose{2}}{{p}\choose{k_{2}}}+{{p}\choose{k_{1}}}{{p}\choose{p-1}}^{p-k_{1}}{{2}\choose{2}}{{p}\choose{k_{2}-1}}+{{p}\choose{k_{2}}}{{p}\choose{p-1}}^{p-k_{2}}{{2}\choose{2}}{{p}\choose{k_{1}-1}}$.

## References

  * [1] B. Csákány and G. Pollák, The graph of subgroups of a finite group (Russian), Czechoslovak Math. J., 19(1969), 241–247.
  * [2] Ivy Chakrabarty, Shamik Ghosh, T.K. Mukherjee and M.K. Sen, Intersection graphs of ideals of rings, Discrete mathematics, 309(17) (2009), 5381–5392.
  * [3] Rulin Shen, Intersection graphs of subgroups of finite groups, Czechoslovak mathematical journal, 60(4) (2010), 945–950.
  * [4] C. Hernando, M. Mora, I. M. Pelayo, C. Seera and D. R. Wood, Extremal graph theory for metric dimension and diameter, The Elec. J. of Combin., 17 (2010), Research–Paper.
  * [5] H. Wiener, Structural determination of the paraffin boiling points, J. Am. Chem. Soc., 69(1947), 17–20.
  * [6] R. Rajkumara and P. Devib, Intersection graph of subgroups of some non-abelian groups, Malaya Journal of Mathematik, 4(2)(2016), 238–242.
  * [7] R. Rajkumar and P. Devi, Intersection graphs of cyclic subgroups of groups, Electronic Notes in Discrete Mathematics, 53 (2016), 15–24.
  * [8] G. Chartrand, L. Eroh, M.A. Johnson and O.R. Oellermann, Resolvability in graphs and the metric dimension of a graph, Disc. Appl. Math., 105(2000), 99–113.
  * [9] Bohdan Zelinka, Intersection graphs of finite abelian groups, Czechoslovak Mathematical Journal, 25(2)(1975), 171–174.
  * [10] D.J. Klein, I. Lukovits and I. Gutman, On the definition of the hyper-Wiener index for cycle-containing structures, J. Chem. Inf. Comput. Sci., 35(1995), 50–52.
  * [11] I. Gutman, W. Yan, Y.-N. Yeh and B.-Y. Yang, Generalized Wiener indices of zigzagging pentachains, J. Math. Chem., 42(2007), 103–117.
  * [12] V. Sharma, R. Goswami and A.K. Madan, Eccentric connectivity index: A novel highly discriminating topological descriptor for structure-property and structure-activity studies, J. Chem. Inf. Comput. Sci., 37(1997), 273–282.
  * [13] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals, Total ${\pi}$-electron energy of alternant hydrocarbons, Chem. Phys. Lett., 17(1972), 535–538.
  * [14] H.P. Schultz, Topological organic chemistry 1. Graph theory and topological indices of Alkanes, J. Chem. Inf. Comput. Sci., 29(1989), 227–228.
  * [15] I. Gutman, Selected properties of the Schultz molecular topological index, J. Chem. Inf. Comput. Sci., 34(1994), 1087–1089.