# Aspects of entanglement in non-local field theories with fractional
Laplacian

(Pratim Roy ††thanks: E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
Harish-Chandra Research Institute, Homi Bhabha National Institute (HBNI),
Chhatnag Road, Jhunsi, Allahabad, India 211019)

##  1 Introduction

The physics of long range systems in general, and entanglement in particular
have recently been active topics of research. This is primarily because long
range models often exhibit novel physical characteristics such as power law
fall-off in correlations [1], rather than exponential. This has significant
implications for the propagation of correlations throughout the system, which
affects properties such as different entanglement measures. In the present
paper, a mixed state entanglement measure known as logarithmic negativity
along with entanglement entropy is calculated numerically for a class of
simple non-local theories with a fractional Laplacian.

It may be noted that the majority of recent research on entanglement
concentrates on entanglement entropy, which is a valid measure of correlation
between a system and its complement in pure state. However, it is not a
suitable measure of entanglement for mixed states (a simple example being a
finite temperature state in free scalar field theory). For instance,
entanglement entropy in thermal states receives both classical and quantum
contributions and is thus not effective at capturing correlations in these
situations. Naturally, the problem of quantifying entanglement in general
mixed states has received much attention, and several quantities such as the
entanglement of purification, entanglement of formation, entanglement of
distillation and entanglement negativity etc. were put forward (an overview
may be found in [2]). The main criterion for an entanglement measure to be a
valid one depends on whether it is a monotone under local quantum operations
and classical correlations (LOCC). Mathematically, a quantity $E$ that is a
faithful measure of entanglement must have the following property,

$E(\rho)\geq\sum_{i}p_{i}E(\rho_{i}),$ (1)

which states that the entanglement of a state $\rho$ must (on average) be
greater than the entanglement of the total entanglement of states $\rho_{i}$,
which are obtained with probability $p_{i}$ when a LOCC operation is performed
on $\rho$. A quantity which satisfies the above condition is said to be a
monotone under LOCC. It was proved in [3] that entanglement negativity
satisfies condition (1). Recently, logarithmic negativity has been
investigated in various physical systems [4, 5, 6, 7] and has also been
calculated for scalar and conformal field theories in various scenarios
(including quantum quench) in [8, 9, 10].

A crucial point investigated in this paper is the effect of long range
interactions on entanglement. It is known in the literature [11, 12, 13] that
long-range interactions lead to substantial changes in entanglement entropy,
namely, the behaviour of entanglement entropy changes from area law to a
volume law. It is therefore interesting to investigate what effect inclusion
of long-range interactions has on the logarithmic negativity. A candidate for
a physical system which has long-range interactions is a scalar field theory
with a fractional Laplacian [14]. The physics of field theories with
fractional Laplacian was also recently investigated from the point of view of
entanglement entropy in [12] (see also [15]), where it was found that the non-
locality of this class of theories is responsible for the modification of the
behaviour of entanglement entropy.

Aside from entanglement entropy, long-range quantum systems have recently
become interesting from the point of view of experiments involving cold atoms
[16]. Entanglement of such systems has also been studied [17], along with
entanglement [18] and the quench dynamics of entanglement [19]. Studies on
entanglement entropy in Lifshitz type of theories have been done in [20, 21].
One of the factors that make the quench in long-range systems interesting is
the lack of the Lieb-Robinson velocity [1, 22]. The Lieb-Robinson velocity,
which can be related to causality, governs the spread of a perturbation in a
short-range system. Specifically, there is a causal region, outside which the
perturbations drop off exponentially. Physically, this is equivalent to the
presence of a velocity (the Lieb-Robinson velocity), which defines a cone-like
causal region, in effect acting as the speed of light in these theories.
Recently, there have also been a number of works studying the structure of
light cones and the Lieb-Robinson velocity in long range systems (see, for
example, [23, 24, 25]).

The present paper is organised as follows. In section 2, the theoretical set-
up of the model is described, along with the method used to calculate
entanglement numerically. In section 3, the results for the calculation of
logarithmic negativity in the static case are given for both the critical and
the non-critical harmonic chains and the effect of the long range interaction
is discussed. Subsequently, in section 4, the phenomenon of quantum quench is
described, along with the type of quench considered in this paper. The
quasiparticle model of understanding post-quench evolution of the entanglement
is also briefly introduced. Finally, the numerical results concerning the
temporal evolution of logarithmic negativity are presented, with the lack of
revivals in long range regimes emphasised. At the end, section 5 describes
similar results in entanglement entropy, with confirmation from the
quasiparticle model alluded to earlier. The paper concludes with a summary of
results and pointing out some future directions. Since there are indications
of scrambling present in the results for logarithmic negativity and
entanglement entropy, the calculation of the out-of-time-ordered correlators
(OTOC) to probe this aspect is presented in the appendix.

##  2 Non-local field theories with fractional Laplacian

###  2.1 The Model

Some classes of non-local field theories considered in this paper have been
studied from the point of view of entanglement entropy in [11]. A slightly
modified form of non-local scalar field theories with long-range interaction
have also been investigated in [15] from the point of view of a quantum
quench. In the present paper, logarithmic negativity will be calculated for
non-local scalar field theories with fractional Laplacian, which is a non-
local operator (see, for example, [26]).

The class of theories under consideration are described by the Hamiltonian
[15, 26],

$H=\frac{1}{2}\int
dx\left[\pi^{2}+\left(-\phi(-\Delta)^{\alpha/2}\phi\right)+m^{\alpha}\phi^{2}\right].$
(2)

It may be observed that the above Hamiltonian generalises the Klein-Gordon
scalar field theory by inclusion of the exponent, $\alpha$ of the Laplacian.
The parameter $\alpha$ can take value greater than zero, with $\alpha=2$
leading to the standard Klein-Gordon theory. The Hamiltonian can be
discretised in the following form,

$H=aH_{0}=\sum_{i=1}^{L}\frac{1}{2}\pi_{i}^{2}+\sum_{i,j=1}^{L}\frac{1}{2}\phi_{i}V_{ij}\phi_{j}.$
(3)

Here, $a$ is the lattice spacing, which will henceforth be set to unity. For
convenience of computations, the coupled system of oscillators obtained in eq.
(3) can easily be decoupled by Fourier transformation to assume the form,

$H=\sum_{k=1}^{L}\omega_{k}(a^{\dagger}_{k}a_{k}+\frac{1}{2}),$ (4)

where the dispersion relation is given by,

$\omega_{k}=\sqrt{m^{\alpha}+\left(4\sin^{2}(\frac{\pi
k}{L})\right)^{\alpha/2}}.$ (5)

###  2.2 Correlator method of calculating entanglement

The subsequent sections of the paper are concerned with calculating
entanglement in field theories in both static and quantum quench (time
dependent) scenarios. The correlator method is a standard method used for
calculating entanglement measures such as entanglement entropy and logarithmic
negativity (see, for example [27]). Since this is discussed extensively in the
literature, only a brief description is given here. For generality, the
relevant expressions for the time dependent case will be given, which may
readily be reduced to the corresponding quantities for the static scenario in
the appropriate limit. To calculate entanglement measures, consider the time
dependent correlators,

$\displaystyle Q_{ij}(t)$
$\displaystyle\equiv\langle{\psi}|\phi_{i}\phi_{j}|\psi\rangle=\frac{1}{2L}\sum_{i=1}^{L}Q_{k}\cos\left[\frac{2\pi(i-j)k}{L}\right],$
(6) $\displaystyle P_{ij}(t)$
$\displaystyle\equiv\langle{\psi}|\pi_{i}\pi_{j}|\psi\rangle=\frac{1}{2L}\sum_{i=1}^{L}P_{k}\cos\left[\frac{2\pi(i-j)k}{L}\right],$
(7) $\displaystyle R_{ij}(t)$
$\displaystyle\equiv\langle{\psi}|\frac{1}{2}\\{\phi_{i},\pi_{j}\\}|\psi\rangle=\frac{1}{2L}\sum_{i=1}^{L}R_{k}\cos\left[\frac{2\pi(i-j)k}{L}\right],$
(8)

and the “correlation matrix” $\Gamma_{A\cup B}$ and the symplectic matrix $J$
which are written as,

$\Gamma_{A\cup B}=\begin{bmatrix}Q_{ij}(t)&R_{ij}(t)\\\
R_{ji}(t)&P_{ij}(t)\end{bmatrix},~{}~{}~{}~{}J=\begin{bmatrix}0&I_{l\times
l}\\\ -I_{l\times l}&0\end{bmatrix}.$

The indices $(i,j)$ of the correlators runs over the regions included in the
subsystem (whether adjacent or disjoint). For the quench protocol described
above, the 2-point correlators are given by,

$\displaystyle Q_{k}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{4\omega_{1k}\omega_{2k}^{2}}\left(\omega_{1k}^{2}+\omega_{2k}^{2}-(\omega_{1k}^{2}-\omega_{2k}^{2})\cos(2\omega_{2}t)\right),$
(9) $\displaystyle P_{k}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{4\omega_{1k}}\left(\omega_{1k}^{2}+\omega_{2k}^{2}+(\omega_{1k}^{2}-\omega_{2k}^{2})\cos(2\omega_{2k}t)\right),$
(10) $\displaystyle R_{k}$ $\displaystyle\equiv$
$\displaystyle\frac{\omega_{1k}^{2}-\omega_{2k}^{2}}{4\omega_{1k}\omega_{2k}}\sin(2\omega_{2k}t).$
(11)

Here, $\omega_{1k}$ and $\omega_{2k}$ are the pre and post-quench dispersion
relations. In the quench scenario considered in this paper, the mass is
abruptly changed. For generality, the expressions for the 2-point correlators
are recorded above. The corresponding expressions for the static situation may
be obtained by putting $\omega_{1k}=\omega_{2k}$ in the above expressions.

It may be shown that the trace of the reduced density matrix
$\textrm{Tr}\rho_{A}$ (where $A$ is the subsystem under consideration) can be
related to eigenvalues ($\pm\lambda$) of the matrix $iJ.\Gamma$ by the
relation,

$\textrm{Tr}\,\rho^{n}=\prod_{i\,=\,1}^{\ell}\left[\bigg{(}\lambda_{i}+\frac{1}{2}\bigg{)}^{n}-\bigg{(}\lambda_{i}-\frac{1}{2}\bigg{)}^{n}\,\right]^{-1}.$
(12)

The entanglement entropy is then given by,

$S_{A}(t)=\sum_{i=1}^{\ell}(\lambda_{i}+\frac{1}{2})\log(\lambda_{i}+\frac{1}{2})-(\lambda_{i}-\frac{1}{2})\log(\lambda_{i}-\frac{1}{2}),$
(13)

where the size of the subsystem concerned is given by $\ell$. To calculate the
logarithmic negativity, consider a region $(A\cup B)$ of the system and
partial transpose with respect to $B$. In the correlation matrix $\Gamma$, the
partial transpose merely inverts the signs of the matrix elements
corresponding to momenta included in $B$. So the partial transpose of the
matrix $\Gamma$ is given by,

$\Gamma_{A\cup
B}^{T_{2}}=\begin{pmatrix}\mathbb{I}_{\ell}&\boldsymbol{0}_{\ell}\\\
\boldsymbol{0}_{\ell}&\mathbb{R}_{B}\end{pmatrix}\cdot\Gamma_{A\cup
B}\cdot\begin{pmatrix}\mathbb{I}_{\ell}&\boldsymbol{0}_{\ell}\\\
\boldsymbol{0}_{\ell}&\mathbb{R}_{B}\end{pmatrix}.$ (14)

Finally the trace norm of the partially transposed density matrix $\rho_{A\cup
B}^{T_{2}}$ which is needed for calculation of negativity is given as,

$||\rho^{T_{2}}||=\prod_{i\,=\,1}^{\ell}\Bigg{[}\,\bigg{|}\chi_{i}+\frac{1}{2}\bigg{|}-\bigg{|}\chi_{i}-\frac{1}{2}\bigg{|}\,\Bigg{]}^{-1}=\prod_{i\,=\,1}^{\ell}\textrm{max}\bigg{[}1,\frac{1}{2\chi_{i}}\bigg{]}\,,$
(15)

where $\chi\equiv\chi(t)$ is the set of eigenvalues of the matrix
$iJ.\Gamma_{A}^{T_{2}}$. Finally, the logarithmic negativity of the subsystems
$A$ and $B$ may be calculated from eqn.

$E_{LN}(A,B)\equiv\lvert\lvert\rho_{A\cup B}^{T_{2}}\rvert\rvert,$ (16)

by substituting the expression of $||\rho_{A}^{T_{2}}||$.

The mutual information similarly may be obtained simply by calculating,

$I(A,B)=S(A)+S(B)-S(A\cup B).$ (17)

##  3 Static scenario: Results for the logarithmic negativity

The simplest physical scenario to investigate the effect of the long range
interactions (controlled by the parameter $\alpha$) on logarithmic negativity
is the static situation. This would serve as a warm-up to understand better
the behaviour of logarithmic negativity in more complex scenarios (such as the
quantum quench considered in subsequent sections). Since the regime $\alpha>2$
corresponds to short range models (with $\alpha=2$ corresponding to the Klein-
Gordon case), the range of $\alpha$ is confined to $0<\alpha<2$ in the
remainder of this section. The analysis is done for the critical ($m\approx
0$)111In practice, a small value of $m=10^{-5}$ is chosen to ensure numerical
stability. and the non-critical cases, similar to [7], which investigated
similar issues for the Klein-Gordon field. As with the rest of the present
work, the results are presented for a finite size system, i.e, for a system in
which the ratio $\ell/L$ is non-zero and finite.

As mentioned earlier, the logarithmic negativity is calculated by means of the
correlator method, with the pre and post-quench masses set equal in the
general (time-dependent) expressions for the 2-point correlators in eq. (9).
In Fig. 1, for the critical case the results are presented, focusing on the
fall-off the entanglement as the separation between the subsystems is
increased.

|
---|---
(a) | (b)
Figure 1: Critical: (a) $\log E_{LN}$ vs $n_{d}$ for different values of the
parameter $\alpha$. (b) $\log E_{LN}$ as a function of $\log n_{d}$. In all
cases, the black lines denote a linear fit. The total number of sites is
$L=1250$ and the subsystem size is $n_{A}=n_{B}=50$ with $m=10^{-5}$.

In Fig. 1, it is of interest to check regions of exponential and/or power law
fall-off in the negativity. Consequently, for Fig. 1(a), a linear fit is done
at large separation ($n_{d}>60$ in this particular case) and at small
separation ($n_{d}<2.5$) for Fig. 1(b). From Fig. 1, it is clear that, the
fall-off is power-law at short distances and changes to an exponential fall-
off at larger distances. It is also clear especially from Fig. 1(a) that the
linear approximation (corresponding to exponential decay) appears to break
down for increasing values of $n_{d}$ as $\alpha$ increases, i.e., the fall-
off of $E_{LN}$ assumes an exponential form at increasingly smaller
separations of the subsystems when the the long range interaction increases.
It may be noted that the slope of the linear fit in Fig. 1(b) depends on the
parameter $\alpha$.222For the critical Klen-Gordon field and for critical spin
chain, the power-law exponent has been determined to be $1/3$, which is the
same as the prefactor to the entanglement entropy [7], [28].

For the non-critical case, it is also desirable to study the variation of
$E_{LN}$ as a function of the separation $n_{d}$ between the subsystems, the
results of which are shown in Fig. 2. Comparing Figs. 1 and 2, it is clear
that the deviation from exponential fall-off behaviour is more pronounced for
the non-critical case, i.e, the power law fall-off persists for larger
distances.

|
---|---
(a) | (b)
Figure 2: Non-Critical: (a) $\log E_{LN}$ vs $n_{d}$ for different values of
the parameter $\alpha$. (b) $\log E_{LN}$ as a function of $\log n_{d}$. In
all cases, the black lines denote linear fit. The total number of sites is
$L=1250$ and the subsystem size is $n_{A}=n_{B}=50$ with $m=1.0$.

It is also interesting to plot the logarithmic negativity with respect to
subsystem size, for the critical and non-critical cases (Fig. 3). From the Fig
3, it is clear that the critical case can be distinguished from the non-
critical by the observation that $E_{LN}$ saturates rapidly for the non-
critical case compared to the critical case.

|
---|---
(a) | (b)
Figure 3: (a) Critical: $E_{LN}$ vs $\ell$ for $m=10^{-5}$. (b) Non-critical:
$E_{LN}$ vs $\ell$ for $m=1.0$. In both cases, total number of sites $N=1250$
and $n_{A}=n_{B}=50$.

##  4 Quantum quench scenario

With the results of the previous section which detail the general properties
of logarithmic negativity in a static scenario, it is suitable to analyse the
temporal evolution of logarithmic negativity in a time dependent scenario. A
quantum quench is defined as a physical situation in which a parameter in the
Hamiltonian undergoes a change with respect to time. In the specific protocol
for the quench considered in this paper, the mass parameter in the Hamiltonian
of the system (eq. (2)) is considered to change suddenly. To gain physical
understanding of the behaviour of the system post-quench, it is useful to
recall a heuristic model of a quantum quench that exists in the literature.

###  4.1 The quasiparticle picture for quantum quench

It is well known that the post-quench behaviour of a measure of entanglement
may be understood in qualitative (and in some cases quantitative) terms by a
simple semiclassical model of the quench [29]. This model has been applied
successfully to study quenches in various systems like conformal field theory
and many-body systems [30]. Although it is difficult to apply this model to
complicated entanglement measures like negativity, it is sufficient to
describe measures like entanglement entropy well, which would serve to provide
some indication of the physics in other scenarios also.

Considering the energy difference between the initial and the post-quench
state, the quench results in generation of quasiparticle excitations in pairs,
which then propagate through the system and generate entanglement. However, to
contribute to entanglement, the quasiparticle pair should be generated at the
same point in the system. Specifically, entanglement between two regions (say,
$A$ and $B$) is generated by one member of the quasiparticle pair arriving at
$A$, while the other member arrives at $B$ simulteneously.

It has been proposed in [29] that the entanglement is proportional to the
number of quasiparticles, resulting in the relation (for a subsystem of size
$\ell)$,

$S_{\ell}(t)\propto
2t\int_{2|v|t<\ell}dkv(k)s(k)+\ell\int_{2|v|t>\ell}dks(k).$ (18)

Above, $v(k)$ refers to the velocity of propagation of the quasiparticles
$s(k)$ indicates the production rate of quasiparticles and the entropy
density. According to [29], $s(k)$ refers to the contribution of
quasiparticles with momenta $\pm k$ to the thermodyanamic entropy of the
Generalized Gibbs Ensemble (GGE) of the steady state. It can be observed from
this equation that the existence of a maximum velocity of excitations
($v_{M}$) ensures a light-cone like spreading of correlations. It should be
noted that the above equation is valid for the continuum limit.

For systems with finite size, the above equation is modified to obtain [31],

$S_{\ell}(t)=\int_{\\{\frac{2v(k)t}{L}\\}<\frac{\ell}{L}}\frac{dk}{2\pi}s(k)L\\{\frac{2v(k)t}{L}\\}+\ell\int_{\frac{\ell}{L}\leq\\{\frac{2v(k)t}{L}\\}<1-\frac{\ell}{L}}\frac{dk}{2\pi}s(k)+\\\
\int_{1-\frac{\ell}{L}\leq\\{\frac{2v(k)t}{L}\\}}\frac{dk}{2\pi}s(k)L\left(1-\\{\frac{2v(k)t}{L}\\}\right).$
(19)

In the above equation, $\\{x\\}$ indicates fractional part of $x$. A feature
of the above equation is that “revivals” in entanglement are accounted for,
which also occur in numerical results for finite systems (for example, in
[32]).

At this point, it is worthwhile writing down the exact expressions for the
specific class of models which occur in this paper. Firstly, the velocity of
propagation of the excitations is given by,

$v(k)=\frac{d\omega_{k}}{dk}=\frac{2^{-2+a}a\cot(\pi k/L)(\sin(\pi
k/L))^{a}}{\sqrt{m_{2}^{a}+(4\sin(\pi k/L)^{2})^{a/2}}},$ (20)

where $m_{2}$ represents the post-quench mass. The GGE density matrix can be
constructed from the creation/annihilation operators as,

$\rho_{\textrm{GGE}}=\frac{e^{-\sum_{k}\lambda_{k}\hat{n}_{k}}}{Z},$ (21)

where $Z=\textrm{Tr}e^{-\sum_{k}\lambda_{k}\hat{n}_{k}}$. Note that
$\lambda_{k}$ denotes Lagrange multiplier, and they are evaluated from the
constraint that
$\langle\hat{n}_{k}\rangle=\langle\psi_{0}|n_{k}|\psi_{0}\rangle$ in the
initial state must equal the value according to GGE,

$n_{k}=\langle\hat{n}_{k}\rangle=\frac{1}{e^{\lambda_{k}}-1}.$ (22)

The thermodynamic entropy according to GGE is,

$S_{\textrm{GGE}}=\sum_{k}s(k)=-\textrm{Tr}\left(\rho_{\textrm{GGE}}\log\rho_{\textrm{GGE}}\right)=\sum_{k}\left[(n_{k}+1)\log(n_{k}+1)-n_{k}\log
n_{k}\right]$ (23)

In case of, for example, the massive to massless quench, the occupation number
$n_{k}$ can be easily calculated to be,

$n_{k}=\frac{1}{4}\left(\frac{\omega_{2k}}{\omega_{1}k}+\frac{\omega_{1k}}{\omega_{2k}}\right)-\frac{1}{2}.$
(24)

Here, $\omega_{1k}$ and $\omega_{2k}$ are pre and post-quench dispersion
relations respectively.

The 2-point correlators for these models have also been studied in [33], for
both massless and massive post-quench Hamiltonians. It was established that
for the critical quench, the ballistic velocity of propagation of the
quasiparticles showed no maximum, corresponding to absence of Lieb-Robinson
bound for all values of $\alpha$. For massive post-quench Hamiltonian, the
velocity of propagation showed a maximum for $\alpha>1.0$ and was unbounded
for $\alpha<1$, with the former case corresponding to a generalised Lieb-
Robinson bound.

|
---|---
(a) | (b)
|
(c) | (d)
Figure 4: Density plot for $|Q_{ij}|$ against $t$ and $|i-j|$ reveal the
change in shape of the light cone and the range of interactions in the model
is varied. Clockwise from the top, the values for $a$ considered are
$a=0.2,0.9,1.1,1.6$. In each case, the total number of sites is $L=1250$ and
the quench protocol is the sudden change of $m=4$ to $m=2$.

Fig. 4 depicts the spread of information in a finite-sized system and it may
be observed that the shape of the “light cone” undergoes a marked change as
$a$ is increased. From the preceding description of the quasiparticle model
for quenches, it is natural to expect that this would influence the temporal
behaviour of the entanglment post-quench.

###  4.2 Quantum quench scenario: Numerical results for logarithmic negativity

In this subsection, numerical results for the logarithmic negativity are
presented, considering a quench from $m=1$ to $m=0$. As mentioned earlier, the
correlator method is used, with the relevant expressions for the individual
correlators given by eqn. (9).

|
---|---
(a) | (b)
|
(c) | (d)
Figure 5: $E_{LN}$ vs $t$ for different values of the $\alpha$ and the
separation between subsystems $n_{d}$. For (a), (b), (c) and (d), $n_{d}$
takes values $0$, $2$, $4$ and $6$ respectively. The inset(s) examine the
limit $\alpha\rightarrow 1$ in more detail. In each case, $L=1250$ and
$n_{A}=n_{B}=50$, with the mass parameter quenched from $m=1$ to $m=0$.

It is instructive to note how the qualitative behaviour of logarithmic
negativity changes from $0<\alpha<1$ to $\alpha>1$ in Fig. (5). To emphasise
the robustness of the results, several different cases, corresponding to
different separations are examined and confirmed to reach the same conclusion.
As can be observed, (partial) revivals are observed in each case as the limit
$\alpha\rightarrow 1$ is crossed. Recalling the content of the previous
section, this corresponds to the limit of the ballistic velocity of
propagation of quasiparticles acquiring a bound.

The inset of each figure examines more closely the limit $\alpha=1.0$, at
which it appears that qualitative changes to appear. Of course, it is
reasonable to ask whether as $\alpha\rightarrow 1$ is crossed, the revivals
are pushed out to $t=\infty$. It is difficult to confirm this entirely
conclusively in a numerical study, but this issue is partly addressed in the
subsequent section.

Figure 6: $E_{LN}$ vs $t$ for different values of the $\alpha$ with the
separation between subsystems $n_{d}=10$. Here $L=1250$ and $n_{A}=n_{B}=50$,
with the mass parameter quenched from $m=4$ to $m=2$.

It is also relevant to check whether the post-quench Hamiltonian being
massless affects the results. From Fig. 6 where the mass parameter is quenched
from $m=4$ to $m=2$, this would appear to not be the case since revivals occur
in this case also. It is clear that the revivals appear for $\alpha>1$, with
the logarithmic negativity behaving in a qualitatively different manner for
lower values of $\alpha$. From the results in this section, it is clear that
revivals are present in logarithmic negativity in quantum quenches from
$\alpha\approx 1$ onwards, thus establishing this as a general feature.

##  5 Investigating revivals in entanglement entropy

Based on the results in the previous section, it seems that gradual appearance
of revivals as $\alpha$ is increased is a general feature in logarithmic
negativity. As a check, it is desirable to confirm the general nature of
revivals in another measure of entanglement, namely, entanglement entropy.
Although entanglement entropy is a much simpler measure of entanglement, it
would be interesting if it showed any notable/similar charactertistics in long
range theories. This has the advantage that predictions can be checked against
the quasiparticle model (described in section 4.1).333To the author’s
knowledge, there is no proposal in the literature for calculating a
complicated measure such as logarithmic negativity via the quasiparticle
model. The results are given in Fig. 7, which shows that a dip in entanglement
occurs at $t=2v_{k}^{max}/L$ for $\alpha>1$. This is expected from the
quasiparticle model and is also observed in [31], which is the main
motivations for this section. As in the logarithmic negativity, no revivals
are observed for approximately $\alpha<1$. The results show good agreement
with the predictions from the quasiparticle picture, denoted by the dashed
lines in Fig. 7

It has been pointed out in [31] that the lack of dips points to scrambling,
which leads to loss of information about initial correlations being dispersed.
It is also noted in [34] that disappearance of peaks in mutual information is
related to scrambling. Physically, the same should be true for logarithmic
negativity. But, due to the appearance of multiple peaks (for example, Fig.
6), it is difficult to discriminate peak height. Hence, it is more convenient
to investigate the dips in entanglement entropy. More precisely, height of the
dips is given by [31],

$\delta S=\frac{S_{\ell}(\infty)-S_{\ell}(t_{R})}{\ell},\quad\text{where}\quad
S_{\ell}(\infty)=\lim_{t\to\infty}\lim_{L\to\infty}S_{\ell}(t).$ (25)

It is interesting to note that how fast the dip is suppressed as the ratio
$\ell/L$ is increased and the parameter $\alpha$ is decreased. This is
encapsulated in Fig. 8. The plot also reveals that the fall-off eventually
assumes an exponential form for large $\ell/L$, which is consistent with
results in [31]. It may be observed that the only effect of the parameter
$\alpha$ seems to decrease the magnitude of the dip and there is no
significant effect on the rate at which the dip disappears with increasing
system size $L$.

Figure 7: $S_{\ell}/\ell$ vs $t$ for different values of the $\alpha$. Here
$L=1250$ and $n_{A}=50$, with the mass parameter quenched from $m=4.0$ to
$m=2.0$. Here, the dotted lines correspond to the predictions from the
quasiparticle model. Figure 8: The first dip in the entanglement entropy is
plotted against system size $L$. The solid lines indicate a linear fit and the
crosses indicate the predictions from the quasiparticle model.

##  6 Conclusions

In this paper, aspects of entanglement in non-local field theories have been
examined. First, the investigation is carried out in a static setting,
involving two subsystems with a separation between them. The main conclusions
from this investigation are that for long-ranged models, the logarithmic
negativity falls off more gradually with subsystem separation. Secondly, time
dependence is introduced by a simple model for an instantaneous quantum
quench. The variation of the negativity with respect to time is studied, with
the focus being mainly on the long time behaviour to check if the negativity
in long-range models showed any indication of revival. The main result of
these studies may be summarised as follows: long-range models (for
$0<\alpha<1$) exhibit some novel characteristics which are distinct from
results obtained so far in the literature. One of the most interesting
features of long-range models is that no signs of revival of entanglement
negativity are found, at least in the time interval considered.

To reinforce the results about revivals in the logarithmic negativity, the
same is investigated in the context of entanglement entropy, and dips in the
entanglement entropy (comparable to revivals in logarithmic negativity) are
noted. The numerical results show excellent agreement with predictions from
the quasiparticle picture. This establishes the feature of revivals/dips
disappearing gradually in long range models. As the literature suggests that
this signals scrambling, the out-of-ordered correlators (OTOC), is calculated
in the appendix, which shows that fast scrambling is not present in this class
of models.

This work may be extended in various directions. For example, there also exist
in the literature various other quench protocols [35, 36, 37] which have been
extensively investigated from the point of view of entanglement entropy. Thus
negativity might therefore also be investigated in similar settings to gain
further insight into mixed state entanglement.

## Appendix: Calculating the out-of-time-ordered correlator

Considering the indications of scrambling for a specific regime of $\alpha$ in
non-local field theories, it is desirable to further investigate the issue
through calculating another quantity which has been used in the literature,
namely, out-of-time-ordered correlators (OTOC) [38]. Recently, the OTOC has
seen a resurgence in the context of the AdS/CFT duality, where it has been
used by Kitaev [39] for one-dimensional systems. It was proved that for the
Sachdev-Ye-Kitaev (SYK) model [40] that the thermal OTOC exhibits exponential
growth, which indicates chaos. This is because it is possible to relate the
decay rate of the OTOC to the Lyapunov exponent in the semiclassical limit.

It may be noted that the growth of the OTOC has been used to probe scrambling
and chaos in numerous few [41, 42] and many-body systems [43, 44]. A procedure
of calculating OTOCs in quantum mechanical systems has also been formulated in
[45]. In this appendix, the same formulation is adapted for the system under
consideration, i.e, a long range harmonic chain consisting of coupled
oscillators. Firstly, the microcanonical (ground state) OTOC for the harmonic
chain (eq. 2) is defined as follows,

$c_{ij}(t)=-\langle{0}|[\phi_{i}(t),\pi_{j}]^{2}|{0}\rangle,$ (26)

where the expectation value is calculated according to the ground state of the
system. Inserting the resolution of unity, $\sum_{m}|m\rangle\langle m|=1$ in
the above equation,

$c_{ij}(t)=\sum_{m}(b_{ij}^{m})^{*}(t)b_{ij}^{m}(t),\quad
b_{ij}^{m}(t)=-i\langle{0}|[\phi_{i}(t),\pi_{j}]^{2}|{m}\rangle.$ (27)

As in subsection 2.1, it is convenient to make a discrete Fourier
transformation to decouple the oscillators,

$\phi_{i}(t)=\sum_{k=1}^{L}O_{ik}q_{k}(t),\quad\pi_{j}=\sum_{k=1}^{L}O_{jk}p_{k}.$
(28)

Taking into account the fact that the oscillators are decoupled in Fourier
space,

$b^{m}_{ij}(t)=-i\sum_{k,k^{\prime}=1}^{L}O_{ik}O_{jk^{\prime}}\langle{0}|[q_{k}(t),p_{k^{\prime}}|{m}\rangle\delta_{kk^{\prime}}.$
(29)

Another insertion of the resolution of unity yields,

$\langle{0}|[q_{k}(t),p_{k}]|{m}\rangle=-\frac{i}{2}\sum_{p}q_{0p}q_{pm}\left(E_{pm}e^{iE_{0p}t}-E_{0p}e^{iE_{pm}t}\right),$
(30)

where $x_{mn}=\langle{m}|x|n\rangle$, calculated from the pre-quench state of
the system. Evaluating the right hand side of the above equation,

$\langle{0}|[q_{k}(t),p_{k}]|{m}\rangle=\frac{\omega_{2k}}{\omega_{1k}}\cos(\omega_{2k}t).$
(31)

For the final expression for $b_{m}(t)$, the expression for the matrix $O$
needs to be inserted, which yields,

$b^{0}_{ij}(t)\equiv
b_{ij}(t)=-i\sum_{k=1}^{L}\frac{\omega_{2k}}{\omega_{1k}}\cos(\omega_{2k}t)\cos\left[\frac{2\pi(i-j)k}{L}\right]$
(32)

From the above expression, it is easy to obtain the OTOC for the harmonic
chain.

With the above expressions in hand, it is possible to plot the the
microcanonical OTOC against time, as shown in Fig. 9. It may be observed that
the microcanonical OTOC shows oscillates extremely rapidly in a small time
interval. Nevertheless, the general feature appears to be that the OTOC shows
“revivals” at $t=2v_{k}^{max}/L$ as $\alpha>1.0$, similar to the case for
logarithmic negativity and entanglement entropy. It is also clear that there
seems to be no exponential growth of the OTOC, indicating the lack of fast
scrambling. This is physically not surprising, since it is usually the case
that fast scrambling is present in non-integrable models.

|
---|---
(a) | (b)
|
(c) | (d)
Figure 9: $c(t)\equiv c_{ij}(t)$ vs $t$ for different values of the $\alpha$
and $n=2000$ and $|i-j|=25$. For (a), (b), (c) and (d), $\alpha$ takes values
$0.6$, $0.9$, $0.95$ and $1.1$ respectively. The mass parameter is quenched
from $m=4$ to $m=2$.

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# General position sets, colinear sets, and Sierpiński product graphs

Jing Tian a,b Sandi Klavžar b,c,d

a School of Science, Zhejiang University of Science and Technology,
Hangzhou, Zhejiang 310023, PR China
<EMAIL_ADDRESS>
b Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia
cFaculty of Mathematics and Physics, University of Ljubljana, Slovenia
d Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia
<EMAIL_ADDRESS>

###### Abstract

Let $G\otimes_{f}H$ denote the Sierpiński product of $G$ and $H$ with respect
to the function $f$. The Sierpiński general position number ${\rm gp}{{}_{\rm
S}}(G,H)$ is introduced as the cardinality of a largest general position set
in $G\otimes_{f}H$ over all possible functions $f$. Similarly, the lower
Sierpiński general position number $\underline{{\rm gp}}{{}_{\rm S}}(G,H)$ is
the corresponding smallest cardinality. The concept of vertex-colinear sets is
introduced. Bounds for the general position number in terms of extremal
vertex-colinear sets, and bounds for the (lower) Sierpiński general position
number are proved. The extremal graphs are investigated. Formulas for the
(lower) Sierpiński general position number of the Sierpiński products with
$K_{2}$ as the first factor are deduced. It is proved that if $m,n\geq 2$,
then ${\rm gp}{{}_{\rm S}}(K_{m},K_{n})=m(n-1)$ and that if $n\geq 2m-2$, then
$\underline{{\rm gp}}{{}_{\rm S}}(K_{m},K_{n})=m(n-m+1)$.

Keywords: general position set; colinear set; Sierpiński product of graphs;
Sierpiński general position number

AMS Subj. Class. (2020): 05C12, 05C76

##  1 Introduction

General position sets were introduced to graph theory in [16], but Chandran
S.V. and Parthasarathy in [3] earlier studied the concept of geodetic
irredundant sets which turned out to be an equivalent concept to general
position sets. Moreover, yet much earlier Körner [14] studied the general
position sets of hypercubes (in a completely different context thought).

The problem of finding a largest general position set in a graph, that is, the
general position number of the graph, is NP-hard [16]. In [1], general
position sets of a graph were characterized. Afterwards, the problem of
determining the general position number of (classes of) graphs received a wide
attention, see [7, 19, 24, 26]. General position sets were generalized to
$d$-position sets [10] and to Steiner general position sets [8]. The edge
version of general position sets were investigated in [12, 17, 22], while for
the monophonic version see [21].

A lot of attention has been gives to the general position sets in Cartesian
product graphs, see [11, 13, 23]. In this paper we are interested in general
position sets in the recently introduced, attractive graph operation called
the Sierpiński product. The operation was introduced by Kovič, Pisanski,
Zemljič, and Žitnik in [15] with the idea to generalize the Sierpiński graphs
[9]. The investigation of the latter graphs up to 2017 is summarized in the
survey [6], some of the recent research of Sierpiński graphs can be found, for
instance, in [18, 20, 25], see also the references therein. The Sierpiński
product was further investigated in [4, 5], where the (upper) Sierpiński
domination number and the (upper) Sierpiński metric dimension were
respectively introduced. Following this line of investigation, in this paper
we introduce the Sierpiński general position number ${\rm gp}{{}_{\rm
S}}(G,H)$, and the lower Sierpiński general position number $\underline{{\rm
gp}}{{}_{\rm S}}(G,H)$, of graphs $G$ and $H$.

In the rest of the paper we proceed as follows. In the next section we give
definitions and recall known results needed. In Section 3 we introduce a new
concept into the theory of graph general position sets, the vertex-colinear
sets. This concept is useful while studying the general position sets in
Sierpiński product graphs, but we believe to be also of independent interest.
In particular, we prove bounds for the general position number of a graph in
terms of extremal vertex-colinear sets and show their usefulness for graphs
with bridges. In Section 4, we present bounds for the (lower) Sierpiński
general position number for general graphs and give formulas for these numbers
for the Sierpiński product graphs with $K_{2}$ as the first factor. In Section
5, we consider the Sierpiński product of complete graphs
$K_{m}\otimes_{f}K_{n}$. We determine their Sierpiński general position
number, more precisely, if $m,n\geq 2$, then ${\rm gp}{{}_{\rm
S}}(K_{m},K_{n})=m(n-1)$. For the lower Sierpiński general position number we
prove that $\underline{{\rm gp}}{{}_{\rm S}}(K_{m},K_{n})=m(n-m+1)$ provided
that $n\geq 2m-2$. The assumption $n\geq 2m-2$ is required as demonstrated by
our last result which asserts that $\underline{{\rm gp}}{{}_{\rm
S}}(K_{6},K_{9})=25$.

##  2 Preliminaries

In this section we define concepts, notation, and results needed, and
introduce the (lower) Sierpiński general position number. We begin by basic
definitions, and follow by introducing the general position sets and the
Sierpiński product graphs.

For a positive integer $k$ we set $[k]=\\{1,\ldots,k\\}$. Unless stated
otherwise, all graphs $G=(V(G),E(G))$ in the paper are connected. The order
$n(G)$ of $G$ is equal $|V(G)|$. The degree of a vertex $u$, $\deg_{G}(u)$, is
the number of adjacent vertices of $u$ in $G$. Vertices of degree one are
called leaves. The number of leaves of $G$ will be denoted by $\ell(G)$. For a
graph $G$, let $v_{k}(G)$ denote the number of vertices of $G$ of degree $k$,
cf. [2]. An edge $e$ of $G$ is a bridge if $G-e$ is disconnected. A vertex of
$G$ is simplicial if its neighbourhood induces a complete subgraph. If
$S\subseteq V(G)$, then the subgraph of $G$ induced by $S$ is denoted by
$G[S]$. The distance $d_{G}(u,v)$ between vertices $u$ and $v$ of $G$ is the
number of edges on a shortest $u,v$-path. The interval between vertices $u$
and $v$ is

$I_{G}[u,v]=\\{w:\ d_{G}(u,v)=d_{G}(u,w)+d_{G}(w,v)\\}\,.$

A subgraph $H$ of $G$ is isometric if for each pair of vertices $u,v\in V(H)$
we have $d_{H}(u,v)=d_{G}(u,v)$ and is convex if whenever $u,v\in V(H)$ and
$P$ is a shortest $u,v$-path in $G$, then $P$ lies completely in $H$.

Let $X\subseteq V(G)$. Then $X$ is a general position set of $G$ if as soon as
$P$ is a shortest $u,v$-path, we have $V(P)\cap X=\\{u,v\\}$. The cardinality
of a largest general position set of $G$ is the general position number, ${\rm
gp}(G)$, of $G$. A general position set $X$ of cardinality ${\rm gp}(G)$ is
refereed to as a ${\rm gp}$-set of $G$. Clearly, if $G$ has at least two
vertices, then ${\rm gp}(G)\geq 2$.

Let $G$ and $H$ be graphs and let $f\colon V(G)\rightarrow V(H)$ be a
function. The Sierpiński product of $G$ and $H$ (with respect to $f$) is the
graph $G\otimes_{f}H$ with vertices

  * •

$V(G\otimes_{f}H)=V(G)\times V(H)$,

and with edges

  * •

$(g,h)(g,h^{\prime})$, where $g\in V(G)$ and $hh^{\prime}\in E(H)$, and

  * •

$(g,f(g^{\prime}))(g^{\prime},f(g))$, where $gg^{\prime}\in E(G)$.

The second type of the just defined edges will be called the _connecting
edges_ of $G\otimes_{f}H$. Note that for each vertex $g\in V(G)$, the subgraph
of $G\otimes_{f}H$ induced by the set of vertices $\\{(g,h):\ h\in V(H)\\}$,
is isomorphic to $H$; it will be denoted by $gH$.

Let $G$ and $H$ be connected graphs and denote by $H^{G}$ the set of functions
from $V(G)$ to $V(H)$. We introduce the Sierpiński general position number,
${\rm gp}{{}_{\rm S}}(G,H)$, as the cardinality of a largest general position
set in $G\otimes_{f}H$ over all possible functions $f\in H^{G}$ and the lower
Sierpiński general position number, $\underline{{\rm gp}}{{}_{\rm S}}(G,H)$,
as the corresponding smallest cardinality. That is,

${\rm gp}{{}_{\rm S}}(G,H)=\max_{f\in H^{G}}\\{{\rm
gp}(G\otimes_{f}H)\\}\qquad{\rm and}\qquad\underline{{\rm gp}}{{}_{\rm
S}}(G,H)=\min_{f\in H^{G}}\\{{\rm gp}(G\otimes_{f}H)\\}\,.$

At the end of the section, we will call up several known results needed later
on. For the first one, known as Isometric Cover Lemma, we need the following
definition. A set of subgraphs $\\{H_{1},\ldots,H_{k}\\}$ is an isometric
cover of $G$ if each $H_{i}$, $i\in[k]$, is isometric in $G$ and
$\bigcup_{i=1}^{k}V(H_{i})=V(G)$.

###### Theorem 2.1

[16, Theorem 3.1] If $\\{H_{1},\ldots,H_{k}\\}$ is an isometric cover of $G$,
then

${\rm gp}(G)\leq\sum_{i=1}^{k}{\rm gp}(H_{i})\,.$

###### Theorem 2.2

[16, Theorem 3.6] If $S$ is the set of simplicial vertices of a block graph
$G$, then ${\rm gp}(G)=|S|$.

If $G$ is a connected graph, $S\subseteq V(G)$, and ${\cal
P}=\\{S_{1},\ldots,S_{t}\\}$ a partition of $S$, then ${\cal P}$ is _distance-
constant_ if for any $i,j\in[t]$, $i\neq j$, the distance $d_{G}(x,y)$, where
$x\in S_{i}$ and $y\in S_{j}$, is independent of the selection of $x$ and $y$.
A distance-constant partition ${\cal P}$ is in-transitive if
$d_{G}(S_{i},S_{k})\neq d_{G}(S_{i},S_{j})+d_{G}(S_{j},S_{k})$ holds for
$i,j,k\in[p]$. The following characterization of general position sets will be
used either implicitly or explicitly in the rest of the paper.

###### Theorem 2.3

[1, Theorem 3.1] Let $G$ be a connected graph. Then $S\subseteq V(G)$ is a
general position set if and only if the components of $G[S]$ are complete
subgraphs, the vertices of which form an in-transitive, distance-constant
partition of $S$.

The last retrieved result is about the metric structure of Sierpiński product
graphs.

###### Theorem 2.4

[5, Theorem 4.1] If $G$ and $H$ are connected graphs, $f\in H^{G}$, and $g\in
V(G)$, then $gH$ is a convex subgraph of $G\otimes_{f}H$.

##  3 Colinear sets

In this section we introduce a new concept into the theory of graph general
position sets, the vertex-colinear sets. We believe that the concept is
interesting in its own right. Moreover, from our aspect it turns out to be
extremely useful for studying the general position sets in Sierpiński product
graphs.

Let $G$ be a connected graph and $u\in V(G)$. A set $S\subseteq V(G)$ is a
$u$-colinear set if $S$ is a general position set such that $u\notin S$ and
$y\notin I_{G}[x,u]$ for any $x,y\in S$. We are interested in $u$-colinear
sets of largest cardinality and hence set

$\xi_{G}(u)=\max\\{|S|:\ S\ {\rm is\ a}\ u\mbox{-}{\rm colinear\ set}\\}\,.$

Further, let

$\displaystyle\xi_{\rm m}(G)$ $\displaystyle=\min\\{\xi_{G}(u):\ u\in
V(G)\\}\,,$ $\displaystyle\xi_{\rm M}(G)$ $\displaystyle=\max\\{\xi_{G}(u):\
u\in V(G)\\}\,.$

To illustrate these concepts, we state the following easy result.

###### Proposition 3.1

If $T$ is a tree with $n(T)\geq 3$ and $u\in V(T)$, then
$\xi_{T}(u)=\ell(T)-1$ if $u$ is a leaf, and $\xi_{T}(u)=\ell(T)$, otherwise.
In particular, $\xi_{\rm m}(T)=\ell(T)-1$ and $\xi_{\rm M}(T)=\ell(T)$.

Proof. Assume first that $u$ is a leaf and let $S$ be a $u$-colinear set. Let
$u^{\prime}$ be the support vertex adjacent to $u$. In the following arguments
we use the the consequence of Theorem 2.2 asserting that the general position
number of a tree is the number of its leaves. If $\deg_{T}(u^{\prime})>2$,
then ${\rm gp}(T-u)=\ell(T)-1$, and then the leaves of $T-u$ form a largest
$u$-colinear set. Assume next that $\deg_{T}(u^{\prime})=2$. Then $T-u$
contains $\ell(T)$ leaves (one of them being $u^{\prime}$). However, if
$u^{\prime}$ lies in a $u$-colinear set, then no other vertex lies in such a
set. Hence we can again conclude that $\xi_{T}(u)=\ell(T)-1$. Since $n(T)\geq
3$, there exists a vertex $u$ which is not a leaf. Then the leaves of $T$ form
a $u$-colinear set and hence $\xi_{T}(u)=\ell(T)$ and so also $\xi_{\rm
M}(T)=\ell(T)$. $\square$

The above invariants fulfil the following chain of inequalities involving the
general position number. This in part demonstrate that our definitions are
meaningful, further reasons for their introduction will be given later on.

###### Theorem 3.2

If $G$ is a connected graph of order at least $2$ and $u\in V(G)$, then

$\xi_{\rm m}(G)\leq\xi_{G}(u)\leq\xi_{\rm M}(G)\leq{\rm gp}(G)\leq 2\xi_{\rm
m}(G)\,.$

Proof. The left three inequalities follow directly from definitions. To prove
the last inequality, let $u$ be a vertex of $G$ such that $\xi_{G}(u)=\xi_{\rm
m}(G)$ and let $X$ be an arbitrary gp-set of $G$. We distinguish two cases.

Assume first that $u\in X$. Since $X$ is a general position set of $G$, we
infer that $X\setminus\\{u\\}$ is a $u$-colinear set. Then it follows that
$\xi_{G}(u)\geq|X|-1$, hence $|X|\leq\xi_{\rm m}(G)+1$. Since the order of $G$
is at least two, $\xi_{\rm m}(G)\geq 1$ and thus we have $|X|\leq\xi_{\rm
m}(G)+\xi_{\rm m}(G)\leq 2\xi_{\rm m}(G)$.

Assume second that $u\notin X$. In this case we partition $X$ into two sets
$X_{1}$ and $X_{2}$ as follows. For every $x\in X$, we put each vertex from
$(I_{G}[x,u]\setminus\\{x\\})\cap X$ into $X_{2}$. This defines $X_{2}$, and
then $X_{1}=X\setminus X_{2}$. We claim that each of $X_{1}$ and $X_{2}$ is a
$u$-colinear set. By the definition, this clearly holds for $X_{1}$. Consider
now two vertices $x,x^{\prime}\in X_{2}$ and suppose by way of contradiction
that there exists a shortest $x,u$-path $P_{x,u}$ that contains $x^{\prime}$.
Then $d_{G}(x,u)=d_{G}(x,x^{\prime})+d_{G}(x^{\prime},u)$. Since $x\in X_{2}$,
there exists a vertex $x^{\prime\prime}\in X$ and a shortest
$x^{\prime\prime},u$-path $P_{x^{\prime\prime},u}$. We are going to show that
$x^{\prime\prime}$, $x$, and $x^{\prime}$ lie on a common shortest path. If
not, then
$d_{G}(x^{\prime\prime},x^{\prime})<d_{G}(x^{\prime\prime},x)+d_{G}(x,x^{\prime})$,
but then

$\displaystyle d_{G}(x^{\prime\prime},u)$
$\displaystyle=d_{G}(x^{\prime\prime},x)+d_{G}(x,u)$
$\displaystyle=d_{G}(x^{\prime\prime},x)+d_{G}(x,x^{\prime})+d_{G}(x^{\prime},u)$
$\displaystyle>d_{G}(x^{\prime\prime},x^{\prime})+d_{G}(x^{\prime},u)$
$\displaystyle=d_{G}(x^{\prime\prime},u)\,,$

which is not possible. We have thus seen that also $X_{2}$ is a $u$-colinear
set. Therefore,

$\xi_{G}(u)\geq\max\\{|X_{1}|,|X_{2}|\\}\geq\frac{1}{2}|X|=\frac{1}{2}{\rm
gp}(G)\,,$

hence ${\rm gp}(G)\leq 2\xi_{G}(u)=2\xi_{\rm m}(G)$. $\square$

We now give some examples which demonstrate that in the inequality chain of
Theorem 3.2 the inequalities can be sharp or not. If $n\geq 2$, then $\xi_{\rm
m}(K_{n,n})=\xi_{\rm M}(K_{n,n})={\rm gp}(K_{n,n})$, hence for these graphs
the left three equalities hold. On the other hand, $\xi_{\rm M}(G)$ can be
smaller than ${\rm gp}(G)$. For instance, $\xi_{\rm M}(K_{n})=n-1={\rm
gp}(K_{n})-1$ for $n\geq 2$. In addition, if $P$ is the Petersen graph, then
$\xi_{\rm m}(P)=\xi_{\rm M}(P)=5$ while ${\rm gp}(P)=6$.

Consider next the cycle chain graphs $C_{2k}^{\ell}$, $k,\ell\geq 2$, where
$C_{2k}^{\ell}$ consists of $\ell$ cycles $C_{2k}$ sharing a vertex, such that
a middle cycle shares its diametral vertices with its two neighboring cycles.
See Fig. 1 where $C_{6}^{5}$ is shown. Consider the vertex $u$ of
$C_{2k}^{\ell}$ as shown in Fig. 1. Then $\xi_{C_{2k}^{\ell}}(u)=2$ and hence
$\xi_{\rm m}(C_{2k}^{\ell})=2$. On the other hand, for the vertex $v$ we have
$\xi_{C_{2k}^{\ell}}(v)=4$ which in turn implies that $\xi_{\rm
M}(C_{2k}^{\ell})=4$. Thus $\xi_{\rm m}(G)$ can be strictly smaller than
$\xi_{\rm M}(G)$. Moreover, the vertex $v$ of $C_{2k}^{\ell}$ also
demonstrates that in Theorem 3.2, $\xi_{G}(u)$ can be equal to $2\xi_{\rm
m}(G)$.

$u$$v$ Figure 1: The graph $C_{6}^{5}$.

We conclude this section by demonstrating how colinear sets can be used while
determining the general position number of graphs containing bridges.

###### Proposition 3.3

Let $e=u_{1}u_{2}$ be a bridge of a connected graph $G$, and $G_{1}$, $G_{2}$
the two components of $G-e$, where $u_{i}\in V(G_{i})$, $i\in[2]$. Then ${\rm
gp}(G)\geq\xi_{G_{1}}(u_{1})+\xi_{G_{2}}(u_{2})$. Moreover, the equality holds
in block graphs containing at least one bridge.

Proof. Let $X_{1}$ be a $u_{1}$-colinear set of $G_{1}$ with cardinality
$\xi_{G_{1}}(u_{1})$, and let $X_{2}$ be an analogous $u_{2}$-colinear set of
$G_{2}$ with cardinality $\xi_{G_{2}}(u_{2})$. Then it is straightforward to
check that $X_{1}\cup X_{2}$ is a general position set of $G$, hence ${\rm
gp}(G)\geq\xi_{G_{1}}(u_{1})+\xi_{G_{2}}(u_{2})$.

Consider the block graph $G$ with bridges, and let $e=u_{1}u_{2}$ be an
arbitrary bridge of $G$. Let $G_{1}$ and $G_{2}$ be the two components of
$G-e$, where $u_{1}\in V(G_{1})$ and $u_{2}\in V(G_{2})$. From the above, we
have ${\rm gp}(G)\geq\xi_{G_{1}}(u_{1})+\xi_{G_{2}}(u_{2})$. Let $S$ be the
set of simplicial vertices of $G$ and set $S_{1}=S\cap V(G_{1})$ and
$S_{2}=S\cap V(G_{2})$. Since the vertices from $S_{i}$, $i\in[2]$, are
simplicial vertices in $G_{i}$, we infer that $S_{i}$ is a $u_{i}$-colinear
set of $G_{i}$. Clearly, $|S_{1}|+|S_{2}|=|S|$ and by Theorem 2.2 we know that
${\rm gp}(G)=|S|$. Putting all these fact together we have

${\rm
gp}(G)\geq\xi_{G_{1}}(u_{1})+\xi_{G_{2}}(u_{2})\geq|S_{1}|+|S_{2}|=|S|={\rm
gp}(G)\,,$

which proves that ${\rm gp}(G)=\xi_{G_{1}}(u_{1})+\xi_{G_{2}}(u_{2})$.
$\square$

Next, we give two examples of non-block graphs showing that the inequality of
Proposition 3.3 can be sharp or not. Consider the graph $H$ as shown in Fig.
2. It is straightforward to see that ${\rm gp}(H)=6$ and
$\xi_{H_{1}}(u_{1})+\xi_{H_{2}}(u_{2})=3+2=5$, hence ${\rm gp}(G)$ can be
larger than $\xi_{G_{1}}(u_{1})+\xi_{G_{2}}(u_{2})$. On the other hand, to see
that ${\rm gp}(G)$ can be equal to $\xi_{G_{1}}(u_{1})+\xi_{G_{2}}(u_{2})$ on
non-block graphs with bridges, consider the graph $H^{\prime}$ as shown in
Fig. 2. We have ${\rm gp}(H^{\prime})=6$ and
$\xi_{H^{\prime}_{1}}(u_{1})+\xi_{H^{\prime}_{2}}(u_{2})=3+3=6$.

$u_{1}$$u_{2}$$H$$u_{1}$$u_{2}$$H^{\prime}$ Figure 2: The graphs $H$ and
$H^{\prime}$.

##  4 Sierpiński products of arbitrary graphs

In this section we bound the (lower) Sierpiński general position number for
general graphs. Then we give formulas for the (lower) Sierpiński general
position number for the Sierpiński product of graphs with $K_{2}$ as the first
factor. We begin with the following simple, but useful lemma.

###### Lemma 4.1

Let $X$ be a general position set of $K_{2}\otimes_{f}H$, where $V(K_{2})=[2]$
and $f(2)=u\in V(H)$. If $|X\cap V(1H)|\geq 2$ and $(1,u)\in X$, then $X\cap
2H=\emptyset$.

Proof. Let $(1,v)$ be a vertex of $X$ different from $(1,u)$. Then every
vertex from $2H$, say $(2,x)$, is an endvertex of a shortest
$(1,v),(2,x)$-path passing through $(1,u)$, hence $(2,x)\notin X$. $\square$

###### Theorem 4.2

If $G$ and $H$ are two connected graphs of order at least $2$, then

${\rm gp}(H)\leq\underline{{\rm gp}}{{}_{\rm S}}(G,H)\leq{\rm gp}{{}_{\rm
S}}(G,H)\leq n(G){\rm gp}(H).$

Moreover, ${\rm gp}{{}_{\rm S}}(G,H)=n(G){\rm gp}(H)$ if and only if ${\rm
gp}(H)=\xi_{\rm M}(H)$.

Proof. Let $G\otimes_{f}H$ be an arbitrary Sierpiński product of graphs $G$
and $H$. By Theorem 2.4, the product $G\otimes_{f}H$ contains $n(G)$ convex
subgraphs $gH$, $g\in V(G)$. Since a convex subgraph is an isometric subgraph,
these subgraphs form an isometric cover. Therefore, by Theorem 2.1, we have
${\rm gp}{{}_{\rm S}}(G,H)\leq n(G){\rm gp}(H)$, hence the right most
inequality holds. The convexity of the subgraphs $gH$ also implies the left
inequality, while the middle inequality is obvious.

Assume now that ${\rm gp}{{}_{\rm S}}(G,H)=n(G){\rm gp}(H)$. Consider an
arbitrary function $f\in H^{G}$. Using once more the convexity of the
subgraphs $gH$, there exists a gp-set $X$ of $G\otimes_{f}H$, such that
$|V(gH)\cap X|={\rm gp}(H)$ for each $g\in V(G)$. Consider a fixed subgraph
$gH$ and let $gg^{\prime}\in E(G)$. Since $|V(gH)\cap X|={\rm gp}(H)$ and
${\rm gp}{{}_{\rm S}}(G,H)=n(G){\rm gp}(H)$, Lemma 4.1 implies that
$(g,f(g^{\prime}))\notin X$. Setting $u=(g,f(g^{\prime}))$ we obtain, having
in mind Lemma 4.1 again, that $\xi_{gH}(u)={\rm gp}(H)$ which in turn implies
that $\xi_{\rm M}(H)={\rm gp}(H)$.

Assume next that $\xi_{\rm M}(H)={\rm gp}(H)$. Let $u$ be a vertex of $H$ such
that $\xi_{H}(u)=\xi_{\rm M}(H)$ and let $f\in H^{G}$ be the identity function
$f(g)=u$ for any $g\in V(G)$. Each $gH$ has at least $\xi_{\rm M}(H)$ vertices
from some gp-set of the graph $G\otimes_{f}H$, hence ${\rm
gp}(G\otimes_{f}H)\geq n(G)\xi_{\rm M}(H)=n(G){\rm gp}(H)$. Thus we have ${\rm
gp}{{}_{\rm S}}(G,H)\geq n(G){\rm gp}(H)$. Since $gH$ is convex in
$G\otimes_{f}H$, Theorem 2.1 gives ${\rm gp}(G\otimes_{f}H)\leq n(G){\rm
gp}(H)$ and then ${\rm gp}{{}_{\rm S}}(G,H)\leq n(G){\rm gp}(H)$. From the
above, we conclude that ${\rm gp}{{}_{\rm S}}(G,H)=n(G){\rm gp}(H)$. $\square$

###### Corollary 4.3

If $G$ is a connected graph with $n(G)\geq 2$ and $T$ is a tree with $n(T)\geq
3$, then ${\rm gp}{{}_{\rm S}}(G,T)=n(G)\ell(T)$.

Proof. Since $n(T)\geq 3$, Proposition 3.1 gives ${\rm gp}(T)=\ell(T)=\xi_{\rm
M}(T)$. Hence Theorem 4.2 yields the conclusion. $\square$

###### Theorem 4.4

If $H$ is a connected graph with $n(H)\geq 2$, then the following assertions
hold.

(i) $\underline{{\rm gp}}{{}_{\rm S}}(K_{2},H)=2\xi_{\rm m}(H)$.

(ii) ${\rm gp}{{}_{\rm S}}(K_{2},H)=2\xi_{\rm M}(H)$.

Proof. Set $V(K_{2})=[2]$ for the rest of the proof.

(i) Let $f\in H^{K_{2}}$ be an arbitrary function, say $f(1)=u$ and $f(2)=v$.
By Proposition 3.3 we have ${\rm
gp}(K_{2}\otimes_{f}H)\geq\xi_{H}(v)+\xi_{H}(u)\geq\xi_{\rm m}(H)+\xi_{\rm
m}(H)$. It follows that $\underline{{\rm gp}}{{}_{\rm S}}(K_{2},H)\geq
2\xi_{\rm m}(H)$.

To prove the reverse inequality, let $u$ be a vertex of $H$ such that
$\xi_{H}(u)=\xi_{\rm m}(H)$. Define $f\in H^{K_{2}}$ by $f(1)=f(2)=u$ and let
$X$ be a gp-set of $K_{2}\otimes_{f}H$. If $V(1H)\cap X=\emptyset$ or
$V(2H)\cap X=\emptyset$, then ${\rm gp}(K_{2}\otimes_{f}H)\leq{\rm gp}(H)$
holds and then Theorem 3.2 gives ${\rm gp}(K_{2}\otimes_{f}H)\leq 2\xi_{\rm
m}(H)$. Assume second that $V(1H)\cap X\neq\emptyset$ and $V(2H)\cap
X\neq\emptyset$. Then we claim that $|V(1H)\cap X|\leq\xi_{H}(u)$ and
$|V(2H)\cap X|\leq\xi_{H}(u)$. By establishing this claim, the ${\rm
gp}(K_{2}\otimes_{f}H)\leq 2\xi_{\rm m}(H)$ follows. Suppose on the contrary
that this is not the case and assume without loss of generality that
$|V(1H)\cap X|\geq\xi_{H}(u)+1$. Then there exist two vertices
$(1,h),(1,h^{\prime})\in V(1H)\cap X$, where $h\neq h^{\prime}$, such that the
vertices $(1,h)$, $(1,h^{\prime})$, and $(2,u)$ lie on a common shortest path.
If $(2,h^{\prime\prime})$ is an arbitrary vertex from $V(2H)\cap X$, then the
vertices $(1,h)$, $(1,h^{\prime})$, and $(2,h^{\prime\prime})$ lie on a common
shortest path. It is impossible because we have assumed that $V(2H)\cap
X\neq\emptyset$. This contradiction proves the claim.

(ii) Let $f\in H^{K_{2}}$ be an arbitrary function, say $f(1)=u$ and $f(2)=v$
and let $X$ be an arbitrary gp-set of $K_{2}\otimes_{f}H$. We claim that ${\rm
gp}(K_{2}\otimes_{f}H)\leq 2\xi_{\rm M}(H)$. Suppose on the contrary that
${\rm gp}(K_{2}\otimes_{f}H)\geq 2\xi_{\rm M}(H)+1$. Since the order of $H$ at
least $2$, Theorem 3.2 implies that $2\xi_{\rm M}(H)\geq{\rm gp}(H)\geq 2$.
Then ${\rm gp}(K_{2}\otimes_{f}H)\geq{\rm gp}(H)+1\geq 3$ and thus $|X\cap
V(1H)|\geq 2$ or $|X\cap V(2H)|\geq 2$.

Assume, without loss of generality, that $|X\cap V(1H)|\geq 2$. If $(1,v)\in
X$, then Lemma 4.1 implies that $X\cap V(2H)=\emptyset$ and then, using
Theorem 3.2 once more, ${\rm gp}(K_{2}\otimes_{f}H)\leq{\rm gp}(H)\leq
2\xi_{\rm m}(H)\leq 2\xi_{\rm M}(H)$, a contradiction. Hence assume in the
rest that $(1,v)\notin X$. Since ${\rm gp}(K_{2}\otimes_{f}H)\geq{\rm
gp}(H)+1$, we get $|X\cap V(2H)|\geq 1$. But as argued in (i), this is not
possible. This contradiction implies that $|X\cap
V(2H)|\leq\xi_{H}(u)\leq\xi_{\rm M}(H)$. By symmetry we also have that if
$|X\cap V(2H)|\geq 2$, then $|X\cap V(1H)|\leq\xi_{H}(v)\leq\xi_{\rm M}(H)$
holds. We can thus conclude that ${\rm gp}{{}_{\rm S}}(K_{2},H)\leq 2\xi_{\rm
M}(H)$.

To complete the argument we need to demonstrate that ${\rm gp}{{}_{\rm
S}}(K_{2},H)\geq 2\xi_{\rm M}(H)$. Let $x$ be a vertex of $H$ such that
$\xi_{H}(x)=\xi_{\rm M}(H)$ and let the function $f\in H^{K_{2}}$ be defined
by $f(1)=f(2)=x$. By Proposition 3.3, it follows that ${\rm
gp}(K_{2}\otimes_{f}H)\geq 2\xi_{H}(x)=2\xi_{\rm M}(H)$. Then we have ${\rm
gp}{{}_{\rm S}}(K_{2},H)\geq 2\xi_{\rm M}(H)$, and we are done. $\square$

##  5 Sierpiński products of complete graphs

In this section we consider the Sierpiński product of complete graphs
$K_{m}\otimes_{f}K_{n}$. In the first main result we determine their
Sierpiński general position number. For the lower Sierpiński general position
number we prove that $\underline{{\rm gp}}{{}_{\rm S}}(K_{m},K_{n})=m(n-m+1)$
provided that $n\geq 2m-2$. The latter assumption is required as demonstrated
by our last result which asserts that $\underline{{\rm gp}}{{}_{\rm
S}}(K_{6},K_{9})=25$.

Throughout this section we will assume that $V(K_{m})=[m]$ and
$V(K_{n})=\\{x_{i}:\ i\in[n]\\}$.

###### Theorem 5.1

If $m,n\geq 2$, then

${\rm gp}{{}_{\rm S}}(K_{m},K_{n})=m(n-1)\,.$

Proof. Let $f\in K_{n}^{K_{m}}$ be defined by $f(i)=x_{1}$ for $i\in[m]$. Let
$X=\\{(i,x_{j}):\ i\in[m],\ 2\leq j\leq n\\}$. Since $f$ is an identify
function, for any two vertices $(i,x_{j}),(i^{\prime},x_{j}^{\prime})\in X$,
where $i\neq i^{\prime}$, we have
$d_{K_{m}\otimes_{f}K_{n}}((i,x_{j}),(i^{\prime},x_{j}^{\prime}))=3$. By
Theorem 2.3, $X$ is a general position set, hence ${\rm
gp}(K_{m}\otimes_{f}K_{n})\geq m(n-1)$. It follows that ${\rm gp}{{}_{\rm
S}}(K_{m},K_{n})\geq{\rm gp}(K_{m}\otimes_{f}K_{n})\geq m(n-1)$.

To prove equality, suppose on the contrary that ${\rm gp}{{}_{\rm
S}}(K_{m},K_{n})\geq m(n-1)+1$. It means that there exists a function
$f^{\prime}\in K_{n}^{K_{m}}$ and an index $i\in[m]$, such that $V(iK_{n})$
completely lies in some general position set $S$ of
$K_{m}\otimes_{f^{\prime}}K_{n}$. Since $m,n\geq 3$, there exists a vertex
$(i^{\prime},x_{j})\in S$, where $i\neq i^{\prime}$. Assume that
$f^{\prime}(i^{\prime})=x_{p}$ and let $q\in[n]$ be such that $q\neq p$. Then
the vertices $(i,x_{q})\in S$, $(i,x_{p})\in S$, and $(i^{\prime},x_{j})\in S$
lie on a common shortest path (of length $2$) of
$K_{m}\otimes_{f^{\prime}}K_{n}$, a contradiction. This contradiction implies
that ${\rm gp}(K_{m}\otimes_{f^{\prime}}K_{n})\leq m(n-1)$, hence we have
${\rm gp}{{}_{\rm S}}(K_{m},K_{n})=m(n-1)$. $\square$

In the rest we focus on the lower Sierpiński general position number.

###### Lemma 5.2

Let $m,n\geq 2$ and let $f\in K_{n}^{K_{m}}$. Then the following hold.

(i) If $m\leq n$, then ${\rm gp}(K_{m}\otimes_{f}K_{n})\geq
v_{n-1}(K_{m}\otimes_{f}K_{n})\geq(n-m+1)m$. In particular,

$\underline{{\rm gp}}{{}_{\rm S}}(K_{m},K_{n})\geq(n-m+1)m\,.$

(ii) If $m\geq n$, then ${\rm
gp}(K_{m}\otimes_{f}K_{n})\geq\max\\{2(n-1),m\\}$. In particular,

$\underline{{\rm gp}}{{}_{\rm S}}(K_{m},K_{n})\geq\max\\{2(n-1),m\\}\,.$

Proof. Let $f\in K_{n}^{K_{m}}$, and set $Z=K_{m}\otimes_{f}K_{n}$ for the
rest of the proof.

(i) Note that a vertex of $Z$ is of degree $n-1$ if and only if it is not
incident to a connecting edge. For each $i\in[m]$, let $X_{i}$ be the set of
vertices from $V(Z)\cap V(iK_{n})$ of degree $n-1$. (Note that it is possible
that $X_{i}=\emptyset$.) Then $X_{i}$ clearly induces a complete subgraph.
Moreover, if $i\neq j$, and if $(i,x_{k})\in X_{i}$ and $(j,x_{\ell})\in
X_{j}$, then $d_{Z}((i,x_{k}),(j,x_{\ell}))=3$. Theorem 2.3 implies that
$\bigcup_{i=1}^{m}X_{i}$ is a general position set. This in turn yields ${\rm
gp}(K_{m}\otimes_{f}K_{n})\geq v_{n-1}(K_{m}\otimes_{f}K_{n})$ because each
vertex not in $\bigcup_{i=1}^{m}X_{i}$ is of degree at least $n$. Moreover,
since each $iK_{n}$ contains at most $m-1$ connecting edges, we have
$|X_{i}|\geq n-m+1$, so we also have
$v_{n-1}(K_{m}\otimes_{f}K_{n})\geq(n-m+1)m$.

The above argument holds true for any function $f\in K_{n}^{K_{m}}$, hence we
have $\underline{{\rm gp}}{{}_{\rm S}}(K_{m},K_{n})\geq(n-m+1)m$.

(ii) Assume first that $f$ is an identity function, without loss of generality
let $f(i)=x_{1}$ for $i\in[m]$. By Theorem 2.3 we then get that
$V(Z)\setminus\\{(1,x_{1}),\ldots,(m,x_{1})\\}$ is a general position set,
hence in this case ${\rm gp}(Z)\geq 2(n-1)$. Assume second that $f$ is not an
identity function and assume without loss of generality that $f(1)=x_{1}$ and
$f(2)=x_{2}$. Now we claim that $S=V(1K_{n})\cup
V(2K_{n})\setminus\\{(1,x_{2}),(2,x_{1})\\}$ is a general position set. To
prove this claim it suffices to show (again in view of Theorem 2.3) that if
$(1,x_{i}),(2,x_{j})\in S$, then $d_{Z}((1,x_{i}),(2,x_{j}))=3$. Suppose on
the contrary that $d_{Z}((1,x_{i}),(2,x_{j}))=2$. Then there exits a vertex
$(p,x_{k})$, where $k\neq 1,2$, such that $(1,x_{i})(p,x_{k})\in E(Z)$ and
$(2,x_{j})(p,x_{k})\in E(Z)$. But this means that $f(1)=x_{k}$ and
$f(2)=x_{k}$, a contradiction since we have assumed that $f(1)\neq f(2)$. This
contradiction proves the claim. We conclude that $S$ is a general position set
of $Z$ and as $|S|=2(n-1)$ we have proved that ${\rm gp}(Z)\geq 2(n-1)$.

To prove that ${\rm gp}(Z)\geq m$, we may assume without loss of generality
that the function $f$ is non-decreasing with respect to the indices of
$x_{i}$s. That is, let $k_{1}\geq\cdots\geq k_{n}\geq 0$, where
$k_{1}+\cdots+k_{n}=m$, and set $k_{0}=0$. Then for $i\in[n]$,

$f(j)=x_{i}\quad{\rm for}\quad
j\in\\{k_{1}+\cdots+k_{i-1}+1,\ldots,k_{1}+\cdots+k_{i}\\}\,.$

For $i\in[n]$, set

$X_{i}=\big{\\{}(j,x_{i}):\
j\in\\{k_{1}+\cdots+k_{i-1}+1,\ldots,k_{1}+\cdots+k_{i}\\}\big{\\}}\,.$

Note that for some $i$ we can have $X_{i}=\emptyset$, but in any case
$\sum_{i=1}^{n}|X_{i}|=m$.

We claim that $X=\bigcup_{i=1}^{n}X_{i}$ is a general position set of $Z$.
Note first that $X_{i}$, $i\in[n]$, induces a complete graph $K_{k_{i}}$ of
$Z$. Therefore, in view of Theorem 2.3, to prove that $X$ is a general
position set, it suffices to demonstrate that if $(j,x_{i})\in X_{i}$ and
$(j^{\prime},x_{i^{\prime}})\in X_{i^{\prime}}$, where $i\neq i^{\prime}$,
then $d_{Z}((j,x_{i}),(j^{\prime},x_{i^{\prime}}))=3$. Since $(j,x_{i})$ and
$(j^{\prime},x_{i^{\prime}})$ are not adjacent, suppose on the contrary that
$d_{Z}((j,x_{i}),(j^{\prime},x_{i^{\prime}}))=2$. Then there exists a vertex
$(p,x_{q})$, where $p\neq j,j^{\prime}$, such that $(j,x_{i})(p,x_{q})\in
E(Z)$ and $(j^{\prime},x_{i^{\prime}})(p,x_{q})\in E(Z)$. Then it follows that
$f(j)=x_{q}$ and $f(j^{\prime})=x_{q}$, contradicting our assumption. This
contradiction proves the claim, hence ${\rm gp}(Z)\geq m$.

Again, the above arguments hold true for any function $f\in K_{n}^{K_{m}}$,
hence we have $\underline{{\rm gp}}{{}_{\rm
S}}(K_{m},K_{n})\geq\max\\{2(n-1),m\\}$. $\square$

For the lower Sierpiński general position number of two complete graphs we
have the following result, where we need to assume that the second factor is
relatively large with respect to the first factor.

###### Theorem 5.3

If $m\geq 2$ and $n\geq 2m-2$, then

$\underline{{\rm gp}}{{}_{\rm S}}(K_{m},K_{n})=m(n-m+1)\,.$

Proof. Using Lemma 5.2(i) once more we have $\underline{{\rm gp}}{{}_{\rm
S}}(K_{m},K_{n})\geq m(n-m+1)$.

To prove that $\underline{{\rm gp}}{{}_{\rm S}}(K_{m},K_{n})=m(n-m+1)$,
consider the function $f\in K_{n}^{K_{m}}$ defined by $f(i)=x_{i}$ for
$i\in[m]$. Setting $G=K_{m}\otimes_{f}K_{n}$ we are going to show that ${\rm
gp}(G)\leq m(n-m+1)$.

Suppose on the contrary that ${\rm gp}(G)\geq m(n-m+1)+1$ and let $R$ be a gp-
set of $G$. Then it follows that there exists an index $i\in[m]$ such that
$|V(iK_{n})\cap R|\geq n-m+2\geq 2$. Assume, without loss of generality, that
$|V(1K_{n})\cap R|$ is as large as possible.

We claim that if $(1,x_{j})\in V(1K_{n})\cap R$, then $V(jK_{n})\cap
R=\emptyset$, $j\in[m]$. Indeed, since $|V(1K_{n})\cap R|\geq 2$, there exists
a vertex $(1,x_{j^{\prime}})\in R$, where $j^{\prime}\neq j$ and
$j^{\prime}\in[m]$. Then the vertices $(1,x_{j^{\prime}})$, $(1,x_{j})$,
$(j,x_{1})$, and $(j,x_{p})\in V(jK_{n})$ form a shortest path, hence we
conclude that $(j,x_{p})\notin R$ for $p\in[n]$.

As we have assumed that ${\rm gp}(G)\geq m(n-m+1)+1$, we can write
$|V(1K_{n})\cap R|=n-m+1+k$, where $k\geq 1$ (and $k\leq m-1$). Then
$|V(sK_{n})\cap R|\leq n-m+1+k$, where $2\leq s\leq m$. By the above, $G$
contains at least $k$ copies $iK_{n}$ such that $V(iK_{n})\cap R=\emptyset$.
Then we have

$\displaystyle|R|$ $\displaystyle\leq(n-m+1+k)+(m-k-1)(n-m+1+k)$
$\displaystyle=m(n-m+1)+k(2m-n-1-k)$ $\displaystyle<m(n-m+1)+1.$

Here the last inequality holds because $n\geq 2m-2$ and $k\geq 1$ which
implies that $2m-n-1-k\leq 2-1-k=1-k\leq 0$. This contradiction implies that
${\rm gp}(G)\leq m(n-m+1)$. We conclude that $\underline{{\rm gp}}{{}_{\rm
S}}(K_{m},K_{n})=m(n-m+1)$. $\square$

In Theorem 5.3 we have assumed that $n\geq 2m-2$. The following result
explains why this assumption cannot be avoided in general.

###### Proposition 5.4

$\underline{{\rm gp}}{{}_{\rm S}}(K_{6},K_{9})=25$.

Proof. By Lemma 5.2(i), we have $\underline{{\rm gp}}{{}_{\rm
S}}(K_{6},K_{9})\geq 24$. To prove that $\underline{{\rm gp}}{{}_{\rm
S}}(K_{6},K_{9})=25$, we consider two cases depending on the function $f\in
K_{9}^{K_{6}}$.

Case 1. $f$ is injective.
In this case we may assume, without loss of generality, that $f(i)=x_{i}$ for
$i\in[6]$. The obtained graph $K_{6}\otimes_{f}K_{9}$ is shown in Fig. 3,
where the edges of complete subgraphs $iK_{9}$, $i\in[6]$, and the edges of
the factors $K_{6}$ and $K_{9}$ are not drawn for a clearer picture.

$1K_{9}$$1$$2$$3$$4$$5$$6$$x_{1}$$x_{2}$$x_{3}$$x_{4}$$x_{5}$$x_{6}$$x_{7}$$x_{8}$$x_{9}$$2K_{9}$$3K_{9}$$4K_{9}$$5K_{9}$$6K_{9}$
Figure 3: The Sierpiński product graph $K_{6}\otimes_{f}K_{9}$, where
$f(i)=x_{i}$ for $i\in[6]$

We can check that the set consisting of black vertices from Fig. 3 is a
general position set of cardinality $25$ of $K_{6}\otimes_{f}K_{9}$. Hence
${\rm gp}(K_{6}\otimes_{f}K_{9})\geq 25$. To prove the reverse inequality,
suppose there exists a general position set $X$ of $K_{6}\otimes_{f}K_{9}$
with $|X|\geq 26$. Let $X_{i}=X\cap V(iK_{9})$, $i\in[6]$. We may assume
without loss of generality that $|X_{1}|\geq|X_{i}|$ for
$i\in\\{2,\ldots,6\\}$. Clearly, $|X_{1}|\geq 5$. If $|X_{1}|=5$, then for
some $i\in\\{2,\ldots,6\\}$ we have $X_{i}=\emptyset$. This implies that
$|X|\leq 25$. Hence we must have $|X_{1}|=4+k$, where $2\leq k\leq 5$. Then
there are $k$ copies $iK_{9}$ with no vertex from $X$. It follows that
$|X|\leq(4+k)+(6-1-k)(4+k)\leq 25$. We can conclude that ${\rm
gp}(K_{6}\otimes_{f}K_{9})\leq 25$ and thus ${\rm
gp}(K_{6}\otimes_{f}K_{9})=25$.

Case 2. $f$ is not injective.
In this case we may assume, without loss of generality, that $f(1)=f(2)$. Then
in each of the copies $iK_{9}$, $i\in\\{3,4,5,6\\}$, the graph
$K_{6}\otimes_{f}K_{9}$ contains at least five vertices of degree $8$. Since
in each of $1K_{9}$ and $2K_{9}$ there are at least four such vertices, we can
see (using the argument from the proof of Lemma 5.2(i)) that ${\rm
gp}(K_{6}\otimes_{f}K_{9})\geq 8+20=28$. $\square$

##  6 Concluding remarks

In Proposition 3.3 we have bounded from below the general position number of
graphs with bridges. It would be interesting to characterize the graphs that
attain the equality in the proved bound.

In view of Theorem 4.2 it would be interesting to characterize the graphs $G$
with ${\rm gp}(G)=\xi_{\rm M}(G)$. Such graphs are, for instance, grid graphs
$P_{n}\,\square\,P_{m}$, $n,m\geq 3$, for which we have ${\rm
gp}(P_{n}\,\square\,P_{m})=4=\xi_{\rm M}(P_{n}\,\square\,P_{m})$. Similarly,
in view of Theorem 4.4 it would be interesting to study the graphs $G$ with
$\xi_{\rm m}(G)=\xi_{\rm M}(G)$.

In Theorem 5.3 we have proved that $\underline{{\rm gp}}{{}_{\rm
S}}(K_{m},K_{n})=m(n-m+1)$ if $m\geq 2$ and $n\geq 2m-2$. It remains as an
open problem to determine $\underline{{\rm gp}}{{}_{\rm S}}(K_{m},K_{n})$ for
the cases when $n<2m-2$. In Proposition 5.4 the particular case of
$\underline{{\rm gp}}{{}_{\rm S}}(K_{6},K_{9})$ has been solved.

## Acknowledgements

This work was supported by the Slovenian Research Agency ARIS (research core
funding P1-0297 and projects N1-0285, N1-0218).

## Declaration of interests

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Data availability

Our manuscript has no associated data.

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