# Modeling the Galactic Neutron Star Population for Use in Continuous
Gravitational Wave Searches

Brendan T. Reed<EMAIL_ADDRESS>Department of Astronomy, Indiana University,
Bloomington, IN 47405, USA  Center for Exploration of Energy and Matter and
Department of Physics, Indiana University, Bloomington, IN 47405, USA  Alex
Deibel<EMAIL_ADDRESS>Department of Astronomy, Indiana University,
Bloomington, IN 47405, USA  C. J. Horowitz<EMAIL_ADDRESS>Center for
Exploration of Energy and Matter and Department of Physics, Indiana
University, Bloomington, IN 47405, USA

###### Abstract

Searches for continuous gravitational waves from unknown Galactic neutron
stars provide limits on the shapes of neutron stars. A rotating neutron star
will produce gravitational waves if asymmetric deformations exist in its
structure that are characterized by the star’s ellipticity. In this study, we
use a simple model of the spatial and spin distribution of Galactic neutron
stars to estimate the total number of neutron stars probed, using
gravitational waves, to a given upper limit on the ellipticity. This may help
optimize future searches with improved sensitivity. The improved sensitivity
of third-generation gravitational wave detectors may increase the number of
neutron stars probed, to a given ellipticity, by factors of 100 to 1000.

##  1 Introduction

Within an order of magnitude, the age of the Milky Way is $\sim
10^{10}\,\mathrm{yrs}$ and has a Galactic supernovae rate of $\sim 1$ per
century (Diehl et al., 2006). We can therefore estimate that $N_{0}\sim
10^{8}$ neutron stars (NS) have been born in our galaxy to-date. Of that
number, a relatively small fraction are known through electromagnetic searches
– a few thousand mostly radio pulsars – see, for example, the ATNF Pulsar
Database (Manchester et al., 2005; Hobbs et al., 2020). Gravitational waves
(GW) may be a means to discover some of the remaining unknown NSs and study
the distribution of their shapes.

Any rotating NS with asymmetric deformations will produce continuous
gravitational waves (CGWs) via quadrupole radiation (Zimmermann & Szedenits,
1979; Lasky, 2015) and the observed background of CGWs from GW detectors may
reveal unknown NSs (Caride et al., 2019; Abbott et al., 2019a). A rotating NS
radiates CGWs with strain amplitude $h_{0}$ according to

$\displaystyle h_{0}=\frac{4\pi^{2}G}{c^{4}}\frac{I_{zz}f_{\rm
GW}^{2}}{d}\epsilon\ ,$ (1)

where $d$ is the distance to the source and the gravitational wave frequency
is $f_{\rm GW}=2\nu$ for a NS rotating with spin frequency $\nu$ (Riles, 2017;
Abbott et al., 2019a). This relation is notable in that it is linearly
dependent on the NS’s ellipticity,

$\displaystyle\epsilon=\sqrt{\frac{8\pi}{15}}\frac{Q_{22}}{I_{zz}}=\frac{I_{xx}-I_{yy}}{I_{zz}},$
(2)

defined here in either terms of the quadrupole moment $Q_{22}$ or fractional
difference in principle moments of inertia (Owen, 2005; Riles, 2017).

We expect a distribution of ellipticities across Galactic NSs. The maximum
allowed ellipticity may be limited by the breaking strain of the NS crust to
$\epsilon\lesssim{\rm few}\times 10^{-6}$ (Ushomirsky et al., 2000; Horowitz &
Kadau, 2009; Gittins et al., 2020). In order for a NS to support a larger
$\epsilon$, there may need to be an exotic solid phase in the core, such as
crystalline quark matter (Owen, 2005; Johnson-McDaniel & Owen, 2013).
Constraining the ellipticity further, observations of millisecond pulsars
(MSPs) suggest that the NS ellipticity reaches a minimum near $\epsilon\approx
10^{-9}$ (Woan et al., 2018).

CGWs from Galactic NS are expected to be $\sim 10^{-4}$ times lower in
amplitude than the GW signal from the binary mergers of compact objects
(Riles, 2017). There have been many CGW searches from known pulsars, see for
example (Abbott et al., 2019b). Furthermore, one can gain sensitivity to weak
CGW signals by integrating for a long time. For known pulsars, however, radio
or X-ray spin-down luminosity place an observational limit on the power in GW
radiation. Alternatively, there are a number of all sky searches for CGWs from
unknown NSs (Abbott et al., 2019a; Steltner et al., 2021; Dergachev & Papa,
2020, 2021; Dergachev & Alessandra Papa, 2021). An unknown NS could be a
strong CGW source with an unconstrained spin-down power. However, to-date no
CGW signals have been detected.

The physics implications of negative CGW searches are presently unclear, but
we can use these result to infer some interesting limits on the NS population.
Because the CGW strain amplitude depends inversely on the distance to the NS
($h_{0}\sim 1/d$), the lack of detected CGWs constrains the NS population
within that distance from Earth. Of course, the strain amplitude also depends
on the spin-frequency of the NS producing the GWs as $f_{\mathrm{G}W}=2\nu$
and the ellipticity of the NS ($h_{0}\sim f^{2}_{\mathrm{G}W}\epsilon$).
Assuming a spatial distribution and spin distribution for Galactic NSs near
Earth, we can therefore infer limits on the ellipticities of the NS population
producing CGWs within a distance $d$ from Earth. Doing so would allow us to
estimate _how many NSs are actually being probed by a given CGW search._

In this paper, we develop a very simple population model of the spatial and
spin distribution of Galactic NSs. We then use this model to estimate the
number of Galactic NSs probed by recent CGW searches. We detail our
distribution choices in Sec. 2. These distributions can be used to help
optimize future searches and to infer the physics implications of search
results. In Sec. 3 we use our NS distribution, along with search limits on
$h_{0}$, to infer a distribution of upper limits on NS ellipticities. Lastly,
we discuss the implications and conclusions of our findings and possible
future studies to improve these limits in Sec. 4. We also discuss our
assumptions and their impact on the results; for example, the time-dependence
of the CGW source, the effects of binary pairs of NS, and the NS birth history
in the Milky Way.

##  2 Modeling Neutron Star Distributions

In this section, we develop a simple model for the distribution of NSs in the
galaxy to provide a simple first estimate of the number of NSs probed to a
variety of ellipticities via CGW data. We explain our calculations of the
maximum distance from earth that is probed at a given frequency in subsection
2.1. We then give our assumptions and calculations for the spatial
distribution of NSs in subsection 2.2 and detail our choice of spin-frequency
distribution in subsection 2.3. Finally, we show the calculation of the
unknown NS population in subsection 2.4.

###  2.1 Gravitational Wave Strain Data

Equation 2 gives the definition of $\epsilon$, characterized by an asymmetric
deformation on the surface of a NS. This asymmetry will cause a rapidly
rotating NS to emit CGWs with a strain amplitude given by Equation 1. The
strain amplitude $h_{0}$ is sensitive to the frequency of the GW signal and
has a complicated behavior.

Figure 1: Strain amplitude versus frequency used in this work. The three
colored lines represent the 95% confidence level obtained for each of the
three pipelines used in Abbott et al. (2019a). The black line is the data that
we use in the calculation of $d(f_{\mathrm{G}W},\epsilon)$, obtained by taking
the smallest value for $h_{0}$ at each frequency from among the three
pipelines.

We use data from Abbott et al. (2019a) which presented a detailed analysis of
their CGW search limits on $h_{0}$ as a function of $f_{\rm GW}$ at 95%
confidence. Within this data are the constraints from the three pipelines
SkyHough (Krishnan et al., 2004), Frequency-Hough (Astone et al., 2014), and
TDFstat (Jaranowski et al., 1998) which have different sensitivities in the
range of frequencies considered by Abbott et al. (2019a). Because there exists
some overlap in the strain among the three pipelines, we define a grid of
20-1922 Hz and take the smallest of the three’s $h_{0}$ at each frequency to
use in our calculations. We plot this and the data from the three pipelines in
Figure 1.

By simple inversion of Equation 1, we can solve for the maximum distance from
Earth to which a NS with frequency $f_{\rm GW}$ and ellipticity $\epsilon$ has
been excluded. Explicitly,

$\displaystyle d(f_{\rm
GW},\epsilon)=\frac{4\pi^{2}G}{c^{4}}\frac{I_{zz}f^{2}_{\rm
GW}\epsilon}{h_{0}(f_{\rm GW})}\,.$ (3)

The distance can be easily obtained by fixing a desired value for $\epsilon$,
using a canonical value for $I_{zz}=10^{45}\,\mathrm{g\,cm^{2}}$, and then
choosing a desired $f_{\mathrm{GW}}$ value. For illustration, we plot the
distance versus $f_{\rm GW}$ for various $\epsilon$ values in Figure 2. Note
that current GW interferometers are insensitive below 20 Hz.

Because the maximum distance in Figure 2 goes as $d\sim
f_{\mathrm{GW}}^{2}\epsilon$, the greatest distances probed are for
$f_{\mathrm{GW}}\gtrsim 1,000\,\mathrm{Hz}$ and large ellipticity
$\epsilon\sim 10^{-5}$. In particular, NSs with $\epsilon\sim 10^{-5}$ are
excluded to around $d\approx 20\,\mathrm{kpc}$ (likely the entire Galactic
disk) for $f_{\mathrm{GW}}\gtrsim 1,000\,\mathrm{Hz}$ and to $d\approx
500\,\mathrm{pc}$ for $f_{\mathrm{GW}}=100\,\mathrm{Hz}$. For NSs with
$\epsilon\sim 10^{-6}$, closer to the breaking strain of the crust, they are
excluded to approximately $d\approx 2\,\mathrm{kpc}$ for
$f_{\mathrm{GW}}\gtrsim 1,000\,\mathrm{Hz}$ and around $d\approx
50\,\mathrm{pc}$ for $f_{\mathrm{GW}}=100\,\mathrm{Hz}$. For NSs with
$\epsilon\sim 10^{-9}$ they are excluded up to $d\approx 2\,\mathrm{pc}$ for
$f_{\mathrm{GW}}\gtrsim 1,000\,\mathrm{Hz}$ and $d<<1\,\mathrm{pc}$ for
$f_{\mathrm{GW}}=100\,\mathrm{Hz}$.

Figure 2: Maximum distance from Earth probed at a given GW frequency for a
range of possible NS ellipticities. The curves shown here are the result of
using $h_{0}(f_{\rm GW})$ tabulated in Figure 1 which is then substituted into
Equation 3 for different $\epsilon$.

###  2.2 Spatial Distribution

The NSs that fall within the range of the GW detector as defined by Equation 3
reside in the Galactic disk. The spatial distribution of Galactic NSs is
believed to approximately follow an exponential distribution in the vertical
direction above the disk and a Gaussian-like distribution in the radial
direction (Binney & Merrifield, 1998; Faucher-Giguère & Loeb, 2010; Taani et
al., 2012). We shall adopt the following equation for the 3D-density of
neutron stars in the Galaxy

$\displaystyle\rho_{c}(r_{c},z)=\frac{N_{0}}{4\pi\sigma_{r}^{2}z_{0}}\exp\left[-\frac{r_{c}^{2}}{2\sigma_{r}^{2}}\right]\exp\left[-\frac{|z|}{z_{0}}\right],$
(4)

where $r_{c}$ is the cylindrical radius from the Galactic Center, $\sigma_{r}$
is a radius parameter, $N_{0}$ is the total number of NSs, and $z_{0}$ is the
disk thickness. For $\sigma_{r}$ we adopt a value of 5 kpc as in Faucher-
Giguère & Loeb (2010) and we use $N_{0}=10^{8}$ as discussed in section 1. For
normal stars, $z_{0}$ is often in the range of 0.5-1.0 kpc (Binney &
Merrifield, 1998; Binney & Tremaine, 2008). However, NSs may follow a
different distribution due to supernova kicks. Therefore, to probe a wider
range of models for the distribution, we choose to vary $z_{0}$ for the values
in Table 1.

Parameter | Symbol | Adopted Value(s)
---|---|---
Radius Parameter | $\sigma_{r}$ | 5 kpc
Disk Thickness | $z_{0}$ | 0.1, 2.0, 4.0 kpc
Distance to GC | $R_{e}$ | 8.25 kpc
Normalization | $N_{0}$ |  $10^{8}$ stars
Table 1: Adopted parameter values for the population distributions used in
this study. Figure 3: Left: 1D probability density distribution at three
different values for the scale height, see Equation 8. Right: Cumulative
distribution function of NS, calculated by integrating the 1D density from 0
to $d$. Inset shows the spread in models’ behavior at low values of $d$.

We then perform a coordinate transformation from the cylindrical $(r_{c},z)$
to the average 3D distance from Earth, $d$. This can be done by first
transforming $r_{c}$ to be centered on the Earth via
$\vec{r_{c}}\longrightarrow\vec{r}+\vec{R_{e}}$ where
$R_{e}=8.25\,\textrm{kpc}$ is the distance from the Galactic Center to Earth
(Gravity Collaboration et al., 2019). Plugging this substitution into Equation
4 and integrating over the angular direction gives us

$\rho^{\prime}(r,z)=\frac{N_{0}{\rm
e}^{-\frac{|z|}{z_{0}}}}{2\sigma_{r}^{2}z_{0}}I_{0}\left(\frac{R_{e}r}{\sigma_{r}^{2}}\right)\exp\left[-\frac{(r^{2}+R_{e}^{2})}{2\sigma_{r}^{2}}\right]$
(5)

where $I_{0}$ is the modified Bessel function. We note that this distribution
is normalized to $N_{0}$,

$\displaystyle\int_{-\infty}^{\infty}dz\int_{0}^{\infty}rdr\rho^{\prime}(r,z)=N_{0}$
(6)

Now, using $d$ for the 3D distance from Earth, $d=\sqrt{r^{2}+z^{2}}$, we can
then arrive at an average 1D density $\rho(d)$ by performing the following
integral

$\rho(d)=\int_{-\infty}^{\infty}dz\int_{0}^{\infty}rdr\rho^{\prime}(r,z)\delta(\sqrt{r^{2}+z^{2}}-d)\,.$
(7)

Integrating over the radial coordinate first, we then recast $z$ in terms of a
scaled variable $x=z/d$. With this, we have arrived at the probability density
distribution

$\displaystyle\rho(d)=$
$\displaystyle\frac{N_{0}d^{2}}{\sigma_{r}^{2}z_{0}}\int_{0}^{1}\exp\left[-\frac{xd}{z_{0}}\right]I_{0}\left[\frac{R_{e}d\sqrt{1-x^{2}}}{\sigma_{r}^{2}}\right]$
$\displaystyle\times\exp\left[-\frac{R_{e}^{2}+d^{2}(1-x^{2})}{2\sigma_{r}^{2}}\right]dx\,,$
(8)

which gives the likelihood that a NS is a distance $d$ from Earth. We plot the
probability density distribution within $30\,\mathrm{kpc}$ of Earth in the
left panel of Figure 3. Finally, we integrate Equation 8 to arrive at the
cumulative distribution function $N(d)$ at a distance $d$, defined here as

$\displaystyle N(d)=\int_{0}^{d}\rho(y)dy$ (9)

We plot $N(d)$ in the right panel of Figure 3.

###  2.3 Spin-frequency Distribution

A neutron star spinning with frequency $\nu$ emits CGWs with frequency $f_{\rm
GW}=2\nu$ according to Equation 1. The observed frequency of many MSPs is
believed to be due to spin-up in a NS’s low-mass X-ray binary phase (e.g.
Radhakrishnan & Srinivasan, 1982; Wijnands & van der Klis, 1998; Papitto et
al., 2013). Over the lifetime of this phase, asymmetric electron-capture
reaction layers on the accreting neutron star may lead to an asymmetric
deformation (Bildsten, 1998; Ushomirsky et al., 2000) because of an asymmetry
in the temperature distribution of the NS’s magnetic field. The observed
distribution of $\nu$ largely depends on the spin evolution in this phase
(Bhattacharyya, 2021). After this phase concludes, the spinning NS continues
to emit CGWs which affects both the time evolution of $\epsilon$ and $\nu$.

As a simple starting point, we assume that the true distribution of Galactic
NS spin-frequencies is the same as the observed spin-frequency distribution of
2811 pulsars from the ATNF Pulsar Database (Manchester et al., 2005; Hobbs et
al., 2020). In the left panel of Figure 4, we show a histogram of the $f_{GW}$
expected from pulsars in the database. Within the sample, there are 489
pulsars with a spin frequency above $10$ Hz that produce
$f_{\mathrm{GW}}>20\,\mathrm{Hz}$ and fall within GW detector sensitivity. We
note the maximally rotating pulsar in the catalog is PSR J1748-2446ad (Hessels
et al., 2006) and has $\nu\approx 716$ Hz ($f_{\mathrm{GW}}\approx
1432\,\mathrm{Hz}$), which is the fastest rotating pulsar yet observed.

Using a Kernel-Density Estimator (KDE) from Virtanen et al. (2020) (package
documentation in SciPy Community (2021)) we calculate a probability
distribution function (PDF) of $f_{\rm GW}$ from the pulsar distribution,
$\Phi$. This method uses a Gaussian with Scott’s rule (Scott, 2015) of
$n^{-1/5}$ and is convenient as it produces a continuous function of
$f_{\mathrm{G}W}$ which makes solving integrals with it much easier. We then
normalize $\Phi$ to unity

$\displaystyle 1=\int_{-\infty}^{\infty}\Phi(\log f)d(\log f)$ (10)

giving us a normalized PDF of $f_{\mathrm{GW}}$ We note that $\Phi(f)$ is
continuous, but drops off quickly after $f_{\mathrm{G}W}\gtrsim 1430$ Hz
because there are no observed pulsars with spin frequencies above
$\nu=716\,\mathrm{Hz}$. Above 2000 Hz, we set $\Phi(f>2000~{}\rm{Hz})=0$ since
the data from subsection 2.1 does not go above $f_{\mathrm{G}W}=2000$ Hz. We
show the PDF in the right panel of Figure 4. Note that the KDE was fit in
$\log_{10}(f_{\rm GW})$ to achieve higher accuracy at low values of $f_{\rm
GW}$. Henceforth, all $\log_{10}$ shall be shortened to just $\log$.

###  2.4 Unknown NS Population Estimate

Ground-based GW detectors are insensitive for $f_{\rm GW}<20$ Hz and based on
our catalog, $\gtrsim 83\%$ of known pulsars are therefore spinning too slowly
to be detectable via CGWs. As a first-order estimate we can say that CGW
searches are at most sensitive to the remaining $17\%$, that is, approximately
$\sim 17$ million NSs throughout the Galaxy,

$\displaystyle 0.17\approx\int_{\mathrm{\log}(20Hz)}^{\infty}\Phi(\log
f)d(\log f).$ (11)

In practice, ground-based GW detector sensitivity rapidly declines below
$f_{\mathrm{GW}}<100\,\rm{Hz}$, so it may be difficult to detect slowly
spinning stars.

Figure 4:  Left: Histogram of the distribution of $f_{\rm GW}$ from pulsars in
the ATNF pulsar database. Here, we convert the normal spin-frequency $\nu$ of
the NS’s rotation to the GW frequency via $f_{\rm GW}=2\nu$. Right: Calculated
probability density function of the histogram on the left, normalized to
unity. The 20 Hz limit of GW detector sensitivity is shown on both by the
vertical red line and arrow to indicate direction of limit.

We can now estimate the number of unknown NSs probed at a given $\epsilon$,
defined as $N_{\star}$. We define our grid of $\epsilon$ values to range from
$\log(\epsilon)=[-9,-5]$. Then, we perform the following integral to calculate
$N_{\star}$

$\displaystyle
N_{\star}(\epsilon)=\int_{\mathrm{\log}(f_{1})}^{\mathrm{\log}(f_{2})}N(d(\log
f,\epsilon))\times$ (12) $\displaystyle\Phi(\log f)\mathrm{d}(\log f)$

where $f_{1}=20\,\mathrm{Hz}$ is the minimum sensitivity of GW detectors and
$f_{2}$ is the spin-down limited frequency that a NS could produce a
detectable signal in GWs. This value is dependent on the particular search and
will mostly affect the number of fast-spinning highly elliptical NSs. In
Abbott et al. (2019a), the maximum spin-down considered in their search was
$|\dot{f}|=1\times 10^{-8}\,\mathrm{Hz\ s^{-1}}$ for the SkyHough and TDFStat
pipelines and $|\dot{f}|=2\times 10^{-9}\,\mathrm{Hz\ s^{-1}}$ for Frequency-
Hough. It should be noted that the limit on $|\dot{f}|$ for Frequency-Hough
quoted here is the value considered in the range of 512-1024 Hz, whereas in
the range of 20-512 Hz $|\dot{f}|=1\times 10^{-8}\,\mathrm{Hz\ s^{-1}}$. For a
NS with ellipticity $\epsilon$

$\displaystyle
f_{\mathrm{max}}=225\,\rm{Hz}\,\Big{(}\frac{|\dot{\it{f}}|^{1/5}}{\epsilon^{2/5}}\Big{)}$
(13)

where $|\dot{f}|$ is in units of $[\mathrm{Hz\ s^{-1}}]$ (Abbott et al.,
2019b). Using the strain data from Abbott et al. (2019a), we will adopt
$f_{2}=f_{\rm max}$ for $|\dot{f}|=2\times 10^{-9}\,\mathrm{Hz\ s^{-1}}$ for
simplicity since the overall search is limited by the smallest maximum value
of $|\dot{f}|$.

##  3 Results

###  3.1 Neutron Star Estimates

Figure 5: Predicted number of NSs probed by GW detectors to a given NS
ellipticity. We show the predictions of Equation 12 for $z_{0}=0.1,2.0,$ and
$4.0\,\textrm{kpc}$. Also plotted are the limits on known NS ellipticities
derived from Abbott et al. (2017) in light blue.

In Figure 5 are model populations from solving Equation 12 for the chosen
values of $z_{0}$ and these show the number of NSs probed to a given
ellipticity. We also record the characteristic values for each of the models
in Figure 5 in Table 2. We find that between $\approx 3\textrm{--}5.3\times
10^{5}$ stars of the $1.7\times 10^{7}$ NSs with
$f_{\mathrm{GW}}>20\,\mathrm{Hz}$ have been probed to $\epsilon\sim 10^{-5}$—
corresponding to between $\approx 1.7\textrm{--}3.1\%$ of the model
population. By contrast, only between $\approx 1\textrm{--}5.3\times 10^{4}$
stars are probed to $\epsilon\sim 10^{-6}$, corresponding to only between
$\approx 0.06\textrm{--}0.31\%$ of the visible population. For comparison, we
show the limits on $\epsilon$ from the analysis of Abbott et al. (2017) in
Figure 5 using CGW searches of known pulsars at known $f_{\mathrm{GW}}$. This
data probes significantly fewer NSs at ellipticities above $\epsilon\gtrsim
10^{-7}$ than using our method. This is because most of the NS in the galaxy
are unknown. We discuss this further in section 4.

Table 2: Estimates for number of NSs probed at the value of ellipticity $\epsilon$ for different values of disk thickness $z_{0}$. $\log_{10}(\epsilon)$ |  $z_{0}=0.1$ kpc |  $z_{0}=2.0$ kpc |  $z_{0}=4.0$ kpc
---|---|---|---
-5.00 | 5.3$\times 10^{5}$ | 4.1$\times 10^{5}$ | 3.0$\times 10^{5}$
-5.25 | 3.8$\times 10^{5}$ | 2.7$\times 10^{5}$ | 1.8$\times 10^{5}$
-5.50 | 2.1$\times 10^{5}$ | 1.2$\times 10^{5}$ | 7.6$\times 10^{4}$
-5.75 | 1.2$\times 10^{5}$ | 5.4$\times 10^{4}$ | 3.2$\times 10^{4}$
-6.00 | 5.3$\times 10^{4}$ | 1.8$\times 10^{4}$ | 1.0$\times 10^{4}$
-6.25 | 2.0$\times 10^{4}$ | 4.8$\times 10^{3}$ | 2.6$\times 10^{3}$
-6.50 | 6.5$\times 10^{3}$ | 1.0$\times 10^{3}$ | 540
-6.75 | 1.8$\times 10^{3}$ | 190 | 99
-7.00 | 430 | 35 | 18
-7.25 | 95 | 6 | 3
-7.50 | 19 | 1 | 1
-7.75 | 4 | 0 | 0
-8.00 | 1 | 0 | 0
-8.25 | 0 | 0 | 0
-8.50 | 0 | 0 | 0
-8.75 | 0 | 0 | 0
-9.00 | 0 | 0 | 0

###  3.2 Effects of Improved Strain Sensitivity

The strain amplitude $h_{0}$ used in this study is limited by the sensitivity
of GW interferometers and the parameters of the search. We can see the effects
of improving the sensitivity of $h_{0}$ directly in Equation 3 in that the
distance we can be sensitive to will increase with decreasing $h_{0}$. This
will then increase our estimate for the total number of NSs probed at a given
ellipticity. As an example of this effect, we test what would happen to
$N_{\star}$ should a new search reduce $h_{0}$ in either the high-frequency
($f_{\rm GW}\geq 1000\rm{Hz}$) or low-frequency ($f_{\rm GW}\leq 100\rm{Hz}$)
regimes by a factor of two.

We present the predicted values of $N_{\star}$ for improved detector
sensitivity in Figure 6 for a disk model which has $z_{0}=2.0\,\mathrm{kpc}$
and we also tabulate characteristic values in Table 3. In this figure, we show
the original prediction for $N_{\star}$ in Figure 5 along with the number of
new NSs probed, $\Delta N_{\star}$, when $h_{0}$ is decreased by a factor of
two in the high or low frequency regimes. We find that improving the high-
frequency regime by a factor of two has a much larger effect on the number
estimates compared to improving the low-frequency by a factor of two. This is
due to the larger number of MS-pulsars from our catalog compared to those with
$f_{\mathrm{G}W}\lesssim 100$ Hz and because one is sensitive to greater
distances at higher frequencies. In this simple example, we see that lowering
the value of $h_{0}$ by a factor of two in the high frequency regime can add
nearly three times as many new NSs as the current estimates for
$\epsilon\lesssim 10^{-6}$.

While improved sensitivity in the high-frequency regime will increase
$N_{\star}$, it’s also worth examining the sensitivity of future third-
generation GW detectors — for example the _Einstein Telescope_ (Punturo et
al., 2010) and _Cosmic Explorer_ (Dwyer et al., 2015). This generation of
detectors at present is estimated to be a factor of ten times more sensitive
than present detectors. To explore this possibility we revisit the model of
$N_{\star}$ with $z_{0}=2\,\mathrm{kpc}$, but with $h_{0}$ reduced by a factor
of ten at all frequencies.

The resulting improvement to $N_{\star}$ – which we define as $\Delta N_{3g}$
– is shown in Figure 6. We see that there is an increase in $N_{\star}$ for
all $\epsilon$ by at least an order-of-magnitude and for $\epsilon\sim
10^{-7}$ the improvement is approximately three orders-of-magnitude. While for
the original study we were well below the observational limit of $\sim 17$
million NSs (probing $\lesssim 3\,\%$), with the sensitivity of third
generation GW detectors this limit is much closer to being reached (probing
$\lesssim 40\,\%$), see Table 3. We note that this assumes the same search
parameters as in Abbott et al. (2019a). An improved search could further
improve these limits as well.

Table 3: Estimates for number of new NSs probed with $z_{0}=2.0$ kpc when the strain amplitude $h_{0}$ sensitivity in different frequency regimes is increased by a factor of 2. The rightmost column is the result of decreasing the strain at all frequencies by a factor of 10. $\log_{10}(\epsilon)$ |  $\Delta N_{\star}$($\leq 100$ Hz) |  $\Delta N_{\star}$($\geq 1000$ Hz) | $\Delta N_{3g}$
---|---|---|---
-5.00 | 390 | 0 | 4.1$\times 10^{6}$
-5.25 | 74 | 0 | 5.5$\times 10^{6}$
-5.50 | 14 | 0 | 6.4$\times 10^{6}$
-5.75 | 3 | 0 | 6.2$\times 10^{6}$
-6.00 | 0 | 920 | 4.2$\times 10^{6}$
-6.25 | 0 | 6.9$\times 10^{3}$ | 1.9$\times 10^{6}$
-6.50 | 0 | 2.2$\times 10^{3}$ | 5.6$\times 10^{5}$
-6.75 | 0 | 450 | 1.3$\times 10^{5}$
-7.00 | 0 | 84 | 2.8$\times 10^{4}$
-7.25 | 0 | 15 | 5.5$\times 10^{3}$
-7.50 | 0 | 3 | 1.0$\times 10^{3}$
-7.75 | 0 | 1 | 190
-8.00 | 0 | 0 | 35
-8.25 | 0 | 0 | 6
-8.50 | 0 | 0 | 1
-8.75 | 0 | 0 | 0
-9.00 | 0 | 0 | 0
Figure 6: Number of additional NSs probed to a given ellipticity if the the
strain sensitivity is improved using models with $z_{0}=2\,\mathrm{kpc}$. The
solid yellow line is the total number of new NSs when the strain amplitude is
improved by a factor of two in the high frequency regime [$\geq
1000\,\rm{Hz}$] and the dashed black line the effect of improving the low
frequency regime [$\leq 500\,\rm{Hz}$] by a factor of two. The blue dot-dash
line is the $N_{\star}$ for a 10 times better strain sensitivity potentially
achievable with the third generation of GW detectors. The dotted red line
represents the original data for $z_{0}=2$ kpc from Figure 5.

###  3.3 Alternative Searches

We present now an example of our methodology using new data for the strain
sensitivity. Here, we follow the same process for both the determination of
$\Phi(f)$ and $N(d)$ as described in section 2. For our strain sensitivity, we
use the results of Steltner et al. (2021) for frequencies 20 Hz - 500 Hz,
Dergachev & Papa (2020) for frequencies 500 Hz - 1700 Hz, and Dergachev & Papa
(2021) for frequencies 1700 Hz - 2000 Hz. We show this data in Figure 7. These
latter two searches were intended to search for low ellipticity NSs. As such,
the maximum spin-down allowed for a given ellipticity is considerably lower,
$|\dot{f}|=2.5\times 10^{-12}\,\mathrm{Hz\ s^{-1}}$. The search performed by
Steltner et al. (2021) did consider a higher spin-down, $|\dot{f}|=2.6\times
10^{-9}\,\mathrm{Hz\,s^{-1}}$. For this reason, we have now split up the
integral in Equation 12 with one integral being
$f_{1}=20\,\mathrm{Hz}\,\&\,f_{2}=f_{\mathrm{max}}(|\dot{f}|=2.6$$\times
10^{-9}$$\,\mathrm{Hzs}^{-1})$ and the second being
$f_{1}=500\,\mathrm{Hz}\,\&\,f_{2}=f_{\mathrm{max}}(|\dot{f}|=2.5$$\times
10^{-12}$$\,\mathrm{Hzs}^{-1})$. This ensures that we do not hinder the
breadth of the search of Steltner et al. (2021) with the lower $|\dot{f}|$
value.

Indeed, our results show that using this improved data does yield more NSs
probed at small $\epsilon$, see Figure 8. This can largely be attributed to
the overall decrease in $h_{0}$ at $f_{\rm GW}\geq 500$ Hz when compared to
the strain from Abbott et al. (2019a). However, at $\epsilon\gtrsim 10^{-7}$
this data set probes significantly fewer NSs than for Abbott et al. (2019a).
This is expected since NSs with these high $\epsilon$ were not prioritized in
the study of $h_{0}(f_{\mathrm{GW}})$ cited above. We note that the low-
frequency regime has been considered by Dergachev & Alessandra Papa (2021)
since this work was submitted.

Figure 7: Strain sensitivity used in this study. We show in the blue the
strain data described in subsection 3.3 and the data used in subsection 3.1 in
orange. In addition to the noise being much less throughout, the blue data
also has a smaller $h_{0}$ for the majority of frequencies. Figure 8: Models
for $N_{\star}$ with $z_{0}=2$ kpc using strain data from Abbott et al.
(2019a) (light) or Steltner et al. (2021), Dergachev & Papa (2020), and
Dergachev & Papa (2020) (dark).

##  4 Discussion

Prior searches for CGWs from known pulsars involve searching a well-defined
number of NSs near an expected $f_{\mathrm{GW}}$ for each source (Abbott et
al., 2017). Limits on $\epsilon$ in Figure 5 show the number of unknown NSs
where GW (assuming a source with a given $\epsilon$) have been searched for
and not found. We see that the models begin to result in similar estimates
near $N_{\star}\sim 10^{6}$ above $\epsilon\gtrsim 10^{-6}$. This may be the
maximum $\epsilon$ allowed by the NS’s crust (Ushomirsky et al., 2000;
Horowitz & Kadau, 2009; Gittins et al., 2020), which is only slightly
disfavored by our results. We predict that only $\gtrsim 10^{5}$, or 0.1%, of
Galactic NSs have been probed above $\epsilon=10^{-5.5}$. This puts a limit on
about one in ten million NSs may have such an ellipticity or we would have
detected a signal in gravitational waves.

The largest ellipticity in our tested range $\epsilon=10^{-5}$, though heavily
disfavored from studies of the breaking strain of a NS’s crust, cannot be
ruled out entirely using current CGW data. From our results, we only rule out
this ellipticity for $\approx 1.6\%$ of all Galactic NSs. This may be somewhat
unrealistic, for example, a millisecond pulsar would produce a very large
strain amplitude in CGW signal with such a large ellipticity.

The theoretical upper limit on $\epsilon$ of $\sim\textrm{few }\times 10^{-6}$
has likewise not been ruled out. In fact, our results suggest that $\lesssim
0.1\%$ of all Galactic NSs have been probed at this ellipticity for all values
of the disk thickness. Therefore there is great need to continue searching for
CGWs arising from NSs with ellipticities near this value. With further studies
of NS ellipticities and CGW searches, this limit may become more apparent.

Our methodology used in this study attempted to keep things as simple as
possible. Several additional complications to the study could be introduced to
further constrain $N_{\star}$. Firstly, our choice of Equation 3 as the
distribution of Galactic NSs is a simple model which largely follows the star
formation pattern in the Galactic disk. In reality, NSs may have a much
different distribution, in part resulting from large transverse velocity kicks
during their birth. We have attempted to mitigate this by picking different
values for $z_{0}$ which either condense ($z_{0}=0.1\,\mathrm{kpc}$) or expand
($z_{0}=4.0\,\mathrm{kpc}$) the density distribution of NSs as seen from
Earth.

Most NSs are expected to be born with a transverse space-velocity from a
supernovae kick (Shklovskii, 1970). Analysis of NS orbits suggests that fewer
than $\lesssim 20\,\%$ are retained in the disk and a greater fraction remain
in bound orbits in the Galactic Halo (Sartore et al., 2010). Furthermore, some
NSs have a sufficient space-velocity to escape the Galactic potential entirely
(Arzoumanian et al., 2002; Katsuda et al., 2018; Nakamura et al., 2019). As a
result, kicked NSs leaving the disk will spread the density distribution in
Equation 4 to larger $z_{0}$ than is typical for other stellar populations.
Additionally, our calculated estimate for the total number of NSs probed in
Table 2 could be reduced by more than factor of two depending on the real
distribution of supernovae kick velocities. A clear next step with this type
of estimate would be to self-consistently include an empirical density
distribution of NSs that can account for a kicked NS population.

Additionally, we have chosen to neglect CGWs arising from NSs in binaries
because of the added complication it would cause on the GW signal and on the
search parameters. However, in future work, it would be useful if the GW
search treated binary NSs and isolated NSs separately. The newly developed
BinarySkyHough (Covas & Sintes, 2019) pipeline is much better equipped to
search for CGWs in binaries than its predecessor SkyHough (Krishnan et al.,
2004) which was used in Abbott et al. (2019a). By better constraining the
values of $\epsilon$, this can also further improve the search parameter
computation time.

We find that the disk thickness parameter $z_{0}$ from Equation 4 has a
significant impact on the estimated number of nearby NS. These nearby sources
of CGWs would be vital in constraining ellipticities $\lesssim 10^{-7}$. In
the thin disk approximation ($z_{0}\longrightarrow 0$), Equation 3 then goes
like $\rho(d)\approx d$ for small values of $d$. However, for other values of
$z_{0}$ where $d\ll z_{0}$, the distribution instead goes like $\rho(d)\approx
d^{2}$. This has significant effects on nearby number estimates as any stars
lying above the plane of the disk are then condensed, thereby increasing the
total number of stars estimated. This is easily seen in the right-hand figure
of Figure 3, whose effects on the estimated numbers of NSs seen in Figure 5
and in Table 2. Figure 5 is the result of Equation 12 for the values of
$z_{0}$ used in Table 1. We see that $N_{\star}$ is very sensitive to $z_{0}$
for small values of $\epsilon$, decreasing for increasing $z_{0}$. Better
determination of the disk thickness of the Galactic NS population is important
for constraining $\epsilon\lesssim 10^{-5.5}$, where an order of magnitude
difference exists between the models used here.

We explore the implications of a new CGW search with improved $h_{0}(f_{\rm
GW})$ sensitivity using the current generation of GW detectors. Our results
show that improving sensitivity in the high-frequency regime ($f_{\rm GW}\geq
1000\,\mathrm{Hz}$) can have the greatest impact on the search for CGWs. From
Figure 6, we can see that the improvements in $h_{0}$ sensitivity can have
much higher returns on the total number of new NSs probed. At
$\epsilon\lesssim 10^{-6}$, for example, this results in finding approximately
three times $N_{\star}$ new NSs.

Interestingly, this is not true for very high values of $\epsilon$. We see
that the high frequency regime has a turnoff point at $\epsilon\sim 10^{-6}$
which occurs for two reasons. First, the original search already probed a
significant fraction of visible NSs in the disk for $\epsilon>10^{-6}$, and so
fewer new NSs would become visible. Secondly, for $\epsilon>10^{-5.5}$,
$f_{\mathrm{max}}<1000\,\mathrm{Hz}$ and so the improvement is no longer
limiting the contribution of $h_{0}(f_{\mathrm{GW}})$ to the integral in
Equation 12. From these two points, we see that the strain sensitivity at high
frequency has a significant impact on searches for NSs with moderate
ellipticity. Conversely, improving the low-frequency regime ($f_{\rm GW}\leq
100\,\mathrm{Hz}$) certainly increases the number probed, it is approximately
four orders of magnitude smaller in effect than improving the high-frequency
regime for moderate ellipticity. This is because there are a much larger
number of MS-pulsars with spin frequencies in excess of $\nu>50$ Hz, as
discussed in subsection 2.3.

Third generation detectors may dramatically increase the number of NSs probed.
Given the blue dash-dot curve in Figure 6 we can see that improving $h_{0}$ by
a factor of ten increases $N_{\star}$ by more than a factor of 100-1000 times.
We note that this assumes for Equation 13 $|\dot{f}|=2\times
10^{-9}\,\mathrm{Hz\,s^{-1}}$ which may not be the true limit considered when
searches with these instruments take place. Despite this, however, just the
improvements to $h_{0}$ we estimate will probe almost 40% of all the NSs in
the Galaxy at large $\epsilon$. In addition, improvements in search techniques
and computer resources may further increase the number of NSs probed.

We conclude our discussion with the analysis of subsection 3.3. The data used
here is comprised of several additional analyses of the data from Abbott et
al. (2019a), however now with improved strain sensitivity. Both searches use
different techniques and have different goals for performing their respective
searches. For example, Dergachev & Papa (2020) were primarily interested in
finding low $\epsilon$ neutron stars which allows for a smaller maximum value
for $|\dot{f}|$. If one sets $f_{\mathrm{max}}=2000$ Hz, then for
$\epsilon=10^{-8}$ $|\dot{f}|\approx 5.55\times
10^{-12}\,\mathrm{Hz\,s^{-1}}$. This allows for more restrictive limits on
$h_{0}$, as seen in Figure 7. While the value of $|\dot{f}|=2.5\times
10^{-12}\,\mathrm{Hz\,s^{-1}}$ is consistent with pulsar data, this constraint
on the analysis has two effects on the overall results. First, it reduces the
search parameter space considerably and therefore allows for a better
determination of $h_{0}(f_{\mathrm{GW}})$, as seen in Figure 7. As we have
shown in subsection 3.2, reducing the strain amplitude does increase
$N_{\star}$.

However, the second, and most important effect for this work, is that it
limits the amount of detectable NSs. For example, taking $|\dot{f}|=2\times
10^{-9}\,\mathrm{Hz\ s^{-1}}$ as we did in subsection 3.1, for
$\epsilon=10^{-6}$ this means $f_{\rm max}\approx 1029\,\mathrm{Hz}$. Using
$|\dot{f}|=2.5\times 10^{-12}\,\mathrm{Hz\ s^{-1}}$ instead, $f_{\rm
max}\approx 220\,\mathrm{Hz}$. This means that this data set may be
inefficient when looking for highly elliptical neutron stars because none of
the MSPs are being probed. Interestingly though, the improved data set does
probe more NSs at $\epsilon>10^{-5.75}$. This is because the values of
$f_{\mathrm{max}}$ for both sets exclude the highest frequencies from the
search, meaning only the low frequency sources contribute to $N_{\star}$.
Since we see in Figure 7 that the strain used in this analysis is smaller than
from Abbott et al. (2019a), slightly more NSs are probed.

Note that there are two possible approaches when selecting the optimal choice
for $|\dot{f}|$ because the ellipticity distribution of NSs is unknown. Should
a future CGW search occur with the intent of probing the highest ellipticities
near $\epsilon\sim 10^{-6}$ one should consider using a higher $|\dot{f}|$
limit. Possibly the most promising value of $|\dot{f}|$ is slightly higher
than considered by Abbott et al. (2019a), $5.55\times 10^{-8}\,\mathrm{Hz}\
s^{-1}$. This value will ensure that $f_{\mathrm{max}}=2000$ Hz for
$\epsilon=10^{-6}$ so that any fast-spinning NSs won’t be excluded. On the
other hand, if the intent is to find lower ellipticity NSs – for instance,
near $\epsilon\sim 10^{-9}$ – one should consider a deeper search with a lower
$|\dot{f}|$ limit.

##  5 Conclusion

We have detailed estimates on the total number of NSs probed with
gravitational wave detectors. In doing so, we have shown that continuous
gravitational wave searches suggest that fewer than about one in ten thousand
NSs have an ellipticity $\gtrsim 10^{-6}$. Additionally, we have shown that
the disk thickness strongly affects the number counts of nearby neutron stars
while leaving more distant stars largely unaffected. We have explored the
effects of improving strain amplitude sensitivity at higher frequencies which
can increase the amount of NSs probed to a given ellipticity. These estimates
are important for setting upper limits on the ellipticity of a NS as well as
detecting radio quiet neutron stars that may be nearby, yet unobserved.
Finally, we discuss the impact of third-generation detectors and find that
they may probe 100-1000 times more NSs than have presently been probed.

We would like to thank the anonymous referee for their helpful comments and
kind words. We would also like to extend our gratitude to M. Papa for useful
comments and for providing the data used in subsection 3.3. We also thank N.
Andersson, F. Gittins, V. Dergachev, F. De Lillo, and P. Covas for their
helpful comments. This material is based upon work supported by the U.S.
Department of Energy Office of Science, Office of Nuclear Physics under Awards
DE-FG02-87ER40365 (Indiana University) and Number DE-SC0008808 (NUCLEI SciDAC
Collaboration).

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# The generalized 3-(edge) connectivity of total graphs ***Supported by NSFC
No. 11561056 and 11661066.

Yinkui Li1,2
1Center for Applied Mathematics Tianjin University
Tianjin 300071, P.R. China
2College of mathematics and statistics, Qinghai Nationalities University,
Xining, Qinghai 810000, P.R. China
<EMAIL_ADDRESS>
###### Abstract

The generalized $k$-connectivity $\kappa_{k}(G)$ of a graph $G$, introduced by
Hager in 1985, is a natural generalization of the concept of connectivity
$\kappa(G)$, which is just for $k=2$. Total graph is generalized line graph
and a large graph which obtained by incidence relation between vertices and
edges of original graph. T. Hamada and T. Nonaka et al., in [32] determined
the connectivity of the total graph $T(G)$ for a graph $G$. In this paper, we
determine the generalized $3$-(edge)-connectivity of some total graphs and
give the bounds on the generalized $3$-(edge)-connectivity for total graph.
Keywords: generalized connectivity, generalized edge connectivity, total
graph.
AMS subject classification 2010: 05C05, 05C40, 05C70, 05C75.

##  1 Introduction

All graphs considered in this paper are undirected, finite and simple. We
refer to the book [1] for graph theoretical notation and terminology not
described here. For a graph $G$, we by $V(G)$, $E(G)$, $L(G)$, $T(G)$ denote
the set of vertices, the set of edges, the line graph and the total graph of
$G$, respectively. The following we state the motivations and our results of
this paper.

Connectivity and edge-connectivity are two of the most basic concepts of
graph-theoretic subjects, both in a combinatorial sense and an algorithmic
sense. As we know, Menger’s theorem is the most important basic result and
fundamental theorem of connectivity. Based on this theorem, the ‘path’ version
of _connectivity_ of a graph $G$ is defined as
$\kappa(G)=\min\\{\kappa_{G}(x,y)\,|\,x,y\in V(G),x\neq y\\}$, where
$\kappa_{G}(x,y)$ is the maximum number of internally disjoint paths
connecting two distinct vertices $x$ and $y$ in $G$. Similarly, the _edge-
connectivity_ of graph $G$ is defined as
$\lambda(G)=\min\\{\lambda_{G}(x,y)\,|\,x,y\in V(G),x\neq y\\}$, where
$\lambda_{G}(x,y)$ is the maximum number of edge-disjoint paths connecting $x$
and $y$.

Although there are many elegant and powerful results on these two parameters
in graph theory, they also have their defects on measuring connection of a
graph. So people want some generalizations of both connectivity and edge-
connectivity.

The generalized connectivity of a graph $G$, introduced by Hager [5], is a
natural and nice generalization of the ‘path’ version definition of
connectivity. For a graph $G=(V,E)$ and a set $S\subseteq V$ of at least two
vertices, _an $S$-Steiner tree_ or _a Steiner tree connecting $S$_ (or simply,
_an $S$-tree_) is a subgraph $T=(V^{\prime},E^{\prime})$ of $G$ that is a tree
with $S\subseteq V^{\prime}$. Two Steiner trees $T$ and $T^{\prime}$
connecting $S$ are said to be _internally disjoint_ if $E(T)\cap
E(T^{\prime})=\varnothing$ and $V(T)\cap V(T^{\prime})=S$. For $S\subseteq
V(G)$ and $|S|\geq 2$, the _generalized local connectivity_ $\kappa(S)$ is the
maximum number of internally disjoint Steiner trees connecting $S$ in $G$.
Note that when $|S|=2$ a minimal Steiner tree connecting $S$ is just a path
connecting the two vertices of $S$. For an integer $k$ with $2\leq k\leq n$,
_generalized $k$-connectivity_ (or _$k$ -tree-connectivity_) is defined as
$\kappa_{k}(G)=\min\\{\kappa(S)\,|\,S\subseteq V(G),|S|=k\\}$. Clearly, when
$|S|=2$, $\kappa_{2}(G)$ is the connectivity $\kappa(G)$ of $G$, that is,
$\kappa_{2}(G)=\kappa(G)$. As a natural counterpart of the generalized
connectivity, in [25] X.Li and Y.Mao introduced the concept of generalized
edge-connectivity, which is a generalization of the ‘path’ version definition
of edge-connectivity. For $S\subseteq V(G)$ and $|S|\geq 2$, the _generalized
local edge-connectivity_ $\lambda(S)$ is the maximum number of edge-disjoint
Steiner trees connecting $S$ in $G$. For an integer $k$ with $2\leq k\leq n$,
the _generalized $k$-edge-connectivity_ $\lambda_{k}(G)$ of $G$ is then
defined as $\lambda_{k}(G)=\min\\{\lambda(S)\,|\,S\subseteq V(G)\ and\
|S|=k\\}$. It is also clear that $\lambda_{2}(G)=\lambda(G)$. Results on the
generalized edge-connectivity can be found in [16, 25].

Recently, book[16], written by X. Li and Y. Mao, has been published, where
authors bring together the known results, conjectures, and open problems on
generalized connectivity and edge generalized connectivity. And thus many
researchers pay more attention to the study for this topic. See [8, 17, 22,
24, 23, 30].

_The total graph_ $T(G)$ of $G$ is graph with the vertex set is $V(G)\cup
E(G)$ and two vertices $x,y$ of $T(G)$ are adjacent if one of the following
cases hold: (i) $x,y\in V(G)$ and $x$ is adjacent with $y$ in $G$. (ii)
$x,y\in E(G)$ and $x,y$ are adjacent in $G$. (iii) $x\in V(G)$, $y\in E(G)$,
and $x,y$ are incident in $G$. Clearly, total graph is generalized line graph.
T. Nonaka et al., in [32] determined the connectivity and edge connectivity of
the total graph $T(G)$. Motivated by this, in this paper, we investigate the
generalized 3-(edge)-connectivity of total graph for some graphs, such as
tree, unicycle graph, complete graph and complete bipartite graph in section
3. And further we discuss the bounds on 3-generalized-(edge-)connectivity for
the total graph in section 4.

##  2 Preliminary and known results

###### Observation 2.1

If $G$ is a connected graph, then
$\kappa_{k}(G)\leq\lambda_{k}(G)\leq\delta(G)$.

###### Observation 2.2

If $H$ is a spanning subgraph of $G$, then $\kappa_{k}(H)\leq\kappa_{k}(G)$
and $\lambda_{k}(H)\leq\lambda_{k}(G)$.

###### Proposition 2.3

[7] For every two integers $n$ and $k$ with $2\leq k\leq n$,
$\kappa_{k}(K_{n})=n-\lceil k/2\rceil.$

###### Proposition 2.4

[25] For every two integers $n$ and $k$ with $2\leq k\leq n$,
$\lambda_{k}(K_{n})=n-\lceil k/2\rceil.$

###### Proposition 2.5

[23] Let $G$ be a connected graph of order $n\geq 6$. Then for $3\leq k\leq
6$, $\kappa_{k}(G)\leq\kappa(G)$. Moreover, the upper bound is always sharp
for $3\leq k\leq 6$.

###### Proposition 2.6

[25] For any graph $G$ of order $n$. $\lambda_{k}(G)\leq\lambda(G)$. Moreover,
the upper bound is tight.

###### Proposition 2.7

[23] Let $G$ be a connected graph of order $n$ with minimum degree $\delta$.
If there are two adjacent vertices of degree $\delta$, then
$\kappa_{3}(G)\leq\delta-1$. Moreover, the upper bound is sharp.

###### Proposition 2.8

[26] Let $G$ be a connected graph of order $n$ with minimum degree $\delta$.
If there are two adjacent vertices of degree $\delta$, then
$\lambda_{k}(G)\leq\delta-1$ for $3\leq k\leq n$. Moreover, the upper bound is
sharp.

###### Proposition 2.9

[23] Let $G$ be a connected graph with $n$ vertices. For every two integers
$k$ and $r$ with $k\geq 0$ and $r\in\\{0,1,2,3\\}$, if $\kappa(G)=4k+r$, then
$\kappa_{3}(G)\geq 3k+\lceil\frac{r}{2}\rceil$. Moreover, the lower bound is
sharp.

###### Theorem 2.10

[21] Given any three positive integers $a,b,k$ such that $a\leq b$ and $2\leq
k\leq a+b$, let $K_{a,b}$ denote a complete bipartite graph with a bipartition
of sizes $a$ and $b$. Then $\kappa_{k}(K_{a,b})$ is $a$ for $k\leq b-a+2$ and

$(i)$
$\frac{a+b-k+1}{2}+\Big{\lfloor}\frac{(a-b+k-1)(b-a+k-1)}{4(k-1)}\Big{\rfloor}$
for $k>b-a+2$ and $a-b+k$ is odd.

$(ii)$
$\frac{a+b-k}{2}+\Big{\lfloor}\frac{(a-b+k)(b-a+k)}{4(k-1)}\Big{\rfloor}$ for
$k>b-a+2$ and $a-b+k$ is even.

###### Corollary 2.11

Let $a,b$ be two integers with $2\leq a\leq b$, and $K_{a,b}$ denote a
complete bipartite graph with a bipartition of sizes $a$ and $b$,
respectively. Then

$\kappa_{3}(K_{a,b})=\left\\{\begin{array}[]{ll}a-1,&if~{}b=a;\\\
a,&if~{}b>a.\end{array}\right.$

###### Proposition 2.12

[27] Let $a,b$ be two integers with $2\leq a\leq b$, and $K_{a,b}$ denote a
complete bipartite graph with a bipartition of sizes $a$ and $b$,
respectively. Then

$\lambda_{3}(K_{a,b})=\left\\{\begin{array}[]{ll}a-1,&if~{}b=a;\\\
a,&if~{}b>a.\end{array}\right.$

##  3 Generalized 3-(edge)-connectivity for total graph

In this section, we determine the generalized 3-connectivity and generalized
3-edge-connectivity of some total graph $T(G)$ such as total graphs of tree,
unicycle graph, complete graph and complete bipartite graph.

For $S\subset V(G)$, we by $G[S]$ denote the subgraph of $G$ which induced by
$S$. Let $T$ be a subtree of $G$, then $E(T)\subset V(L(G))$. A spanning tree
of induced subgraph $L(G)[E(T)]$ be called _a corresponding tree_ of $T$ in
$L(G)$, denoted by $CT$. Now we start our investigation with tree $T_{n}$ with
order $n$.

###### Theorem 3.1

Let $T(T_{n})$ be a total graph of tree $T_{n}$ with order $n(\geq 2)$. Then
the generalized 3-connectivity of $T(T_{n})$ is

$\kappa_{3}(T(T_{n}))=\begin{cases}1,&\mbox{ if $n=2$,}\\\ 2,&\mbox{ if $n\geq
3$.}\\\ \end{cases}$

Proof. Since $T(T_{2})=T(K_{2})=C_{3}$, so $\kappa_{3}(T(T_{n}))=1$ while
$n=2$. Here we consider $n\geq 3$. Since the minimum degree of $T(T_{n})$ is
2, by Observation 2.1, we have $\kappa_{3}(T(T_{n}))\leq 2$. Next we prove
$\kappa_{3}(T(T_{n}))\geq 2$.

Suppose $V(T_{n})=\\{u_{1},u_{2},\cdots,u_{n}\\}$,
$V(L(T_{n}))=\\{e_{ij}|e_{ij}=u_{i}u_{j}\in E(T_{n})\\}$, then
$V(T(T_{n}))=V(L(T_{n}))\cup V(T_{n})$. Let $S$ be a 3-subset of
$V(T(T_{n}))$, we only need to show that there exist at least two internally
disjoint $S$-trees in $T(T_{n})$.

If $|S\cap V(T_{n})|=3$, assume $S=\\{u_{i},u_{j},u_{k}\\}\subseteq V(T_{n})$.
By $P_{ij}$ denote the path connecting vertices $u_{i}$ and $u_{j}$ in
$T_{n}$, then we obtain one $S$-tree $T=P_{ij}\cup P_{ik}$ in $T_{n}$. Let
$CT$ be the corresponding tree of $T$ in $L(T_{n})$ and suppose $u_{a},u_{b}$
and $u_{c}$ are neighbor vertices of $u_{i},u_{j}$ and $u_{k}$ in $T$,
respectively. Follow this we get another $S$-tree $T^{\prime}=CT\cup
u_{i}e_{ia}\cup u_{j}e_{jb}\cup u_{k}e_{kc}$. Clearly, these two $S$-trees are
internally disjoint, as desire. See Fig. 1 (a).

If $|S\cap V(T_{n})|=2$, assume $S=\\{u_{i},u_{j},e_{ks}\\}$. Then let
$S^{\prime}=\\{u_{i},u_{j},u_{k}\\}\subseteq V(T_{n})$ and
$S^{\prime\prime}=\\{e_{ip},e_{jq},e_{ks}\\}\subseteq V(L(T_{n}))$. Since
$T_{n}$ and $L(T_{n})$ are both connected subgraphs of in $T(T_{n})$, there
exist a $S^{\prime}$-tree in $T_{n}$, written as $T_{S^{\prime}}$, and a
$S^{\prime\prime}$-tree in $L(T_{n})$, written as $T_{S^{\prime\prime}}$.
Follow this we can obtain two internally disjoint $S$-trees: $u_{k}e_{ks}\cup
T_{S^{\prime}}$ and $u_{i}e_{ip}\cup u_{j}e_{jq}\cup T_{S^{\prime\prime}}$, as
desire. See Fig. 1 (b).

If $|S\cap V(T_{n})|=1$, assume $S=\\{u_{i},e_{jq},e_{ks}\\}$. Then let
$S^{\prime}=\\{u_{i},u_{j},u_{k}\\}\subseteq V(T_{n})$ and
$S^{\prime\prime}=\\{e_{ip},e_{jq},e_{ks}\\}\subseteq V(L(T_{n}))$. Similarly
we obtain a $S^{\prime}$-tree $T_{S^{\prime}}$ in $T_{n}$ and a
$S^{\prime\prime}$-tree $T_{S^{\prime\prime}}$ in $L(T_{n})$. Thus we
construct two internally disjoint $S$-trees in $T(T_{n})$ as: $u_{j}e_{jq}\cup
u_{k}e_{ks}\cup T_{S^{\prime}}$ and $u_{i}e_{ip}\cup T_{S^{\prime\prime}}$, as
desire. See Fig. 1 (c).

If $|S\cap V(T_{n})|=0$, assume $S=\\{e_{ip},e_{jq},e_{ks}\\}$ and let
$S^{\prime}=\\{u_{i},u_{j},u_{k}\\}\subseteq V(T_{n})$, then we can obtain a
$S^{\prime}$-tree in $T_{n}$, written as $T_{S^{\prime}}$. Follow this we can
obtain one $S$-tree in $T_{n}$ is $u_{i}e_{ip}\cup u_{j}e_{jq}\cup
u_{k}e_{ks}\cup T_{S^{\prime}}$. Since $S\subset V(L(T_{n}))$ and $L(T_{n})$
be connected, we get another $S$-tree $T$ in $L(T_{n})$. Clearly, these two
$S$-trees are internally disjoint, as desire. See Fig. 1 (d).

Therefore, we get $\kappa_{3}(T(T_{n}))=2$ for $n\geq 3$.

Figure 1: Two internally disjoint $S$-trees for 3-subset $S=\\{x,y,z\\}\subset
V(L(T_{n}))\cup V(T_{n})$: One is dotted, another is dashed.

###### Theorem 3.2

Let $T(T_{n})$ be a total graph of tree $T_{n}$ with order $n(\geq 2)$. Then
the generalized 3-edge-connectivity of $T(T_{n})$ is

$\lambda_{3}(T(T_{n}))=\begin{cases}1,&\mbox{ if $n=2$,}\\\ 2,&\mbox{ if
$n\geq 3$.}\\\ \end{cases}$

Proof. Since $T(T_{2})=C_{3}$, so $\lambda_{3}(T(T_{2}))=1$. While $n\geq 3$,
note that the minimum degree of $T(T_{n})$ is 2, by Observation 2.1, we get
$\lambda_{3}(T(T_{n}))\leq 2$. On the other hand, by the Observation 2.1 and
Theorem 3.1, $\lambda_{3}(T(T_{n}))\geq\kappa_{3}(T(T_{n}))=2$. Thus
$\lambda_{3}(T(T_{n}))=2$ for $n\geq 3$.

The following we determine the generalized 3-connectivity and the generalized
3-edge-connectivity of unicycle graph.

###### Theorem 3.3

Let $T(G_{n})$ be a total graph of unicycle graph $G_{n}$ with order $n(\geq
3)$. Then the generalized 3-connectivity of $T(G_{n})$ is

$\kappa_{3}(T(G_{n}))=\begin{cases}3,&\mbox{ if $G_{n}=C_{n}$,}\\\ 2,&\mbox{
otherwisw.}\\\ \end{cases}$

Proof. First consider $G_{n}=C_{n}$. Since $T(C_{n})$ is 4-regular graph, by
Proposition 2.7 we get $\kappa_{3}(T(C_{n}))\leq 3$. On the other hand, since
$T(C_{n})$ is 4-connected, by Proposition 2.9 we get $\kappa_{3}(T(C_{n}))\geq
3$. Thus $\kappa_{3}(T(C_{n}))=3$.

For the general case, if $G_{n}\neq C_{n}$, since the minimum degree of
$T(G_{n})$ is 2 and thus by Observation 2.1 get $\kappa_{3}(T(G_{n}))\leq 2$.
On the other hand, suppose $C$ is unique cycle of $G_{n}$ and $e_{ij}$ is an
edge of $C$. Now let $H_{n}=G_{n}-e_{ij}$, clearly, $H_{n}$ is a spanning tree
of $G_{n}$ and by Theorem 3.1 we have $\kappa_{3}(T(H_{n}))=2$. Combine this
with Observation 2.2 we get $\kappa_{3}(T(T_{n}))\geq 2$. Hence
$\kappa_{3}(T(T_{n}))=2$ for $G_{n}\neq C_{n}$.

###### Theorem 3.4

Let $T(G_{n})$ be a total graph of unicycle graph $G_{n}$ with order $n(\geq
3)$. Then the generalized 3-edge-connectivity of $T(G_{n})$ is

$\lambda_{3}(T(G_{n}))=\begin{cases}3,&\mbox{ if $G_{n}=C_{n}$,}\\\ 2,&\mbox{
otherwisw.}\\\ \end{cases}$

Proof. While $G_{n}=C_{n}$. Since $T(C_{n})$ is 4-regular graph, by
Proposition 2.8 we get $\lambda_{3}(T(C_{n}))\leq 3$. At the same time, by
Observation 2.1 and Theorem 3.3 we have
$\lambda_{3}(T(C_{n}))\geq\kappa_{3}(T(C_{n}))=3$. Thus
$\lambda_{3}(T(C_{n}))=3$.

While $G_{n}\neq C_{n}$, since the minimum degree of $T(G_{n})$ is 2, by
Observation 2.1 we have $\lambda_{3}(T(G_{n}))\leq 2$. On the other hand, by
Observation 2.1 and Theorem 3.3 we have
$\lambda_{3}(T(G_{n}))\geq\kappa_{3}(T(G_{n}))=2$, so
$\lambda_{3}(T(G_{n}))=2$ for $G_{n}\neq C_{n}$.

Next we determine the generalized 3-connectivity and the generalized 3-edge-
connectivity of complete graph. Before investigation, we first list a useful
Lemma.

###### Lemma 3.5

[27] Let $L(K_{n})$ be a line graph of complete graph $K_{n}$ with
$V(L(K_{n}))=\\{e_{ij}|e_{ij}=u_{i}u_{j}\\}$ for $V(K_{n})=\\{u_{i}|1\leq
i\leq n\\}$. Suppose $S_{0}=\\{e_{pq},e_{rs},e_{tk}\\}\subseteq V(L(K_{n}))$
and $V_{S_{0}}=\\{u_{p},u_{q},u_{r},u_{s},u_{t},u_{k}\\}\subseteq V(K_{n})$.
If the induced subgraph $K_{n}[V_{S_{0}}]=K_{3}$, then generalized local
connectivity $\kappa(S_{0})=\lfloor\frac{3(n-2)}{2}\rfloor$.

###### Lemma 3.6

[27] Let $K_{n}$ be complete graph with order $n(\geq 3)$. Then the
generalized 3-connectivity of line graph $L(K_{n})$ is
$\kappa_{3}(L(K_{n}))=\lfloor\frac{3(n-2)}{2}\rfloor$.

###### Lemma 3.7

[27] Let $K_{n}$ be complete graph with order $n(\geq 3)$. Then the
generalized 3-edge-connectivity of line graph $L(K_{n})$ is
$\lambda_{3}(L(K_{n}))=2n-5$.

###### Theorem 3.8

Let $T(K_{n})$ be a total graph of complete graph $K_{n}$ with order $n(\geq
2)$. Then the generalized 3-connectivity of $T(K_{n})$ is

$\kappa_{3}(T(K_{n}))=\begin{cases}3,&\mbox{ if $n=3$,}\\\
\lfloor\frac{3(n-2)}{2}\rfloor+1,&\mbox{ otherwise.}\\\ \end{cases}$

Proof. By Theorem 3.1 and 3.3, the result holds for cases when $n=2,3$. Now we
consider $n\geq 4$. Suppose $V(K_{n})=\\{u_{1},u_{2},\cdots,u_{n}\\}$ and
$V(L(K_{n}))=\\{e_{ij}|e_{ij}=u_{i}u_{j}\in E(K_{n})\\}$, then
$V(T(K_{n}))=V(L(K_{n}))\cup V(K_{n})$. First let
$S_{0}=\\{e_{ij},e_{jk},e_{ik}\\}\subseteq V(L(K_{n}))$. Clearly,
$V_{S_{0}}=\\{u_{i},u_{j},u_{k}\\}$ and $K_{n}[V_{S_{0}}]=K_{3}$, by Lemma 3.5
there exist at most $\lfloor\frac{3(n-2)}{2}\rfloor$ internally disjoint
$S_{0}$-trees in $L(K_{n})$. Besides these $S_{0}$-trees, add tree
$e_{ij}u_{i}u_{j}e_{jk}\cup e_{ik}u_{i}$ together, we obtain at most
$\lfloor\frac{3(n-2)}{2}\rfloor+1$ internally disjoint $S_{0}$-trees in
$T(K_{n})$. Thus we get
$\kappa_{3}(T(K_{n}))\leq\lfloor\frac{3(n-2)}{2}\rfloor+1$. The following we
distinguish four cases to show
$\kappa_{3}(T(K_{n}))\geq\lfloor\frac{3(n-2)}{2}\rfloor+1$. Let
$S=\\{x,y,z\\}$ be a 3-subset of $V(T(K_{n}))$, we only need to show that
there exist at least $\lfloor\frac{3(n-2)}{2}\rfloor+1$ internally disjoint
$S$-trees in $T(K_{n})$.

Case 1. $|S\cap V(K_{n})|=3$

This means $x,y,z\in V(K_{n})$, assume $x=u_{a},y=u_{b},z=u_{c}$ with $1\leq
a,b,c\leq n$. Firstly, path $zxy$ together with trees $T_{i}=u_{i}z\cup
u_{i}x\cup u_{i}y$ for $i\in\\{1,2,\cdots,n\\}\setminus\\{a,b,c\\}$ are $n-2$
internally disjoint $S$-trees. Secondly, paths $xe_{ab}yz$, $xe_{ac}ze_{bc}y$
and trees $T_{j}=xe_{ja}e_{jb}y\cup e_{jb}e_{jc}z$ for
$j\in\\{1,2,\cdots,n\\}\setminus\\{a,b,c\\}$ are $n-1$ internally disjoint
$S$-trees. Total up all we get $2n-3>\lfloor\frac{3(n-2)}{2}\rfloor+1$
internally disjoint $S$-trees in $T(K_{n})$, as desire.

Case 2. $|S\cap V(K_{n})|=2$

Without loss of generality, assume $x,y\in V(K_{n})$, $z\in V(L(K_{n}))$ and
then let $x=u_{a},y=u_{b},z=e_{cd}$ with $1\leq a,b,c,d\leq n$. Here first
consider the case for $u_{a}u_{b}\neq u_{c}u_{d}$. If edges $u_{a}u_{b}$ and
$u_{c}u_{d}$ are nonadjacent in $K_{n}$, then we form internally disjoint
$S$-trees as: For every $i\in\\{1,2,\cdots,n\\}\setminus\\{a,b\\}$ to form
$T_{i}=xu_{i}y\cup u_{i}e_{ic}z$ and thus get $n-2$ internally disjoint
$S$-trees; For every $i\in\\{1,2,\cdots,n\\}\setminus\\{a,b,d\\}$ to form
$T^{\prime}_{i}=xe_{ai}e_{bi}y\cup e_{bi}e_{id}z$ and thus get $n-3$
internally disjoint $S$-trees. Put all $T_{i}$, $T^{\prime}_{i}$ with trees
$yxe_{ad}z$ and $xe_{ab}ye_{bd}z$ together, we get $2n-3$ internally disjoint
$S$-trees in $T(K_{n})$. If edges $u_{a}u_{b}$ and $u_{c}u_{d}$ are adjacent
in $K_{n}$, by similar procedure as the above we also get $2n-3$ internally
disjoint $S$-trees in $T(K_{n})$. Note that
$2n-3>\lfloor\frac{3(n-2)}{2}\rfloor+1$, so the result holds.

Now we consider the case $u_{a}u_{b}=u_{c}u_{d}$, it is clear that for every
two integers $i,j\in\\{1,2,\cdots,n\\}\setminus\\{a,b\\}$, we can get three
internally disjoint $S$-trees such as $xu_{i}e_{ai}z\cup u_{i}y$,
$xu_{j}e_{bj}z\cup u_{j}y$ and $xe_{aj}ze_{bi}y$ and thus get at least
$\lfloor\frac{3(n-2)}{2}\rfloor$ internally disjoint $S$-trees. Put these
trees with $xyz$ together, we obtain at least
$\lfloor\frac{3(n-2)}{2}\rfloor+1$ internally disjoint $S$-trees in
$T(K_{n})$, as we desire.

Case 3. $|S\cap V(K_{n})|=1$

Assume $x\in V(K_{n})$, $y,z\in V(L(K_{n}))$ and then let
$x=u_{a},y=e_{bc},z=e_{df}$ with $1\leq a,b,c,d,f\leq n$. If $e_{bc}$,
$e_{df}$ and $u_{a}$ are nonadjacent each other in $T(K_{n})$, we can form
internally disjoint $S$-trees as: For every
$i\in\\{1,2,\cdots,n\\}\setminus\\{a,d\\}$ to form $T_{i}=xu_{i}e_{ic}y\cup
u_{i}e_{id}z$ and thus get $n-2$ internally disjoint $S$-trees; For every
$i\in\\{1,2,\cdots,n\\}\setminus\\{a,b\\}$ to form $T^{\prime}_{i}=xe_{ai}\cup
ze_{if}e_{ai}e_{ib}y$ and thus get $n-2$ internally disjoint $S$-trees. Put
all these trees with tree $ye_{ab}xu_{d}z$ together we get
$2n-3>\lfloor\frac{3(n-2)}{2}\rfloor+1$ internally disjoint $S$-trees in
$T(K_{n})$, as desire.

If $e_{bc}$ and $e_{df}$ are adjacent but nonadjacent to $u_{a}$, then assume
$d=c$. Thus we form internally disjoint $S$-trees as: For every
$i\in\\{1,2,\cdots,n\\}\setminus\\{a,f\\}$ to form $T_{i}=xu_{i}e_{ic}y\cup
e_{ic}z$ and thus get $n-2$ internally disjoint $S$-trees; For every
$i\in\\{1,2,\cdots,n\\}\setminus\\{a,b,f\\}$ to form
$T^{\prime}_{i}=xe_{ai}\cup ze_{if}e_{ai}e_{ib}y$ and thus get $n-3$
internally disjoint $S$-trees. Put all these trees with trees
$xu_{f}e_{bf}y\cup e_{bf}z$ and $xe_{ab}e_{af}z\cup e_{ab}y$ together, we get
$2n-3>\lfloor\frac{3(n-2)}{2}\rfloor+1$ internally disjoint $S$-trees in
$T(K_{n})$, as desire.

If $e_{bc}$ and $e_{df}$ are adjacent and one of them adjacent to $u_{a}$,
then assume $d=c$ and $a=b$. Thus trees $T_{i}=xu_{i}e_{ic}y\cup e_{ic}z$ for
every $i\in\\{1,2,\cdots,n\\}\setminus\\{f\\}$ are $n-1$ internally disjoint
$S$-trees. Besides these, trees $T^{\prime}_{i}=xe_{ai}\cup
ze_{if}e_{ai}e_{ib}y$ for every $i\in\\{1,2,\cdots,n\\}\setminus\\{a,c\\}$ are
also $n-2$ internally disjoint $S$-trees. Altogether, we get
$2n-3>\lfloor\frac{3(n-2)}{2}\rfloor+1$ internally disjoint $S$-trees in
$T(K_{n})$, as desire.

If $e_{bc}$, $u_{a}$ and $e_{df}$ are adjacent each other, then assume
$a=d=c$. Trees $T_{i}=xu_{i}e_{ib}y\cup u_{i}e_{if}z$ for every
$i\in\\{1,2,\cdots,n\\}\setminus\\{a,f\\}$ and $T^{\prime}_{i}=xe_{ai}y\cup
e_{ai}z$ for every $i\in\\{1,2,\cdots,n\\}\setminus\\{c,f\\}$ are $2n-4$
internally disjoint $S$-trees. In addition to these, add tree $yu_{b}xz$
together, we get $2n-3>\lfloor\frac{3(n-2)}{2}\rfloor+1$ internally disjoint
$S$-trees in $T(K_{n})$, as desire.

Case 4. $|S\cap V(K_{n})|=0$

It is clear $S\subseteq V(L(K_{n}))$ in this case, so by Lemma 3.6 there exist
at least $\lfloor\frac{3(n-2)}{2}\rfloor$ internally disjoint $S$-trees in
$V(L(K_{n}))$. Put these $S$-trees with tree $e_{ab}u_{a}u_{c}e_{cd}\cup
u_{c}u_{g}e_{gf}$ together we get $\lfloor\frac{3(n-2)}{2}\rfloor$+1
internally disjoint $S$-trees in $T(K_{n})$, as desire.

This complete the proof.

###### Theorem 3.9

Let $T(K_{n})$ be a total graph of complete graph $K_{n}$ with order $n(\geq
2)$. Then the generalized 3-edge-connectivity of $T(K_{n})$ is
$\lambda_{3}(T(K_{n}))=2n-3$.

Proof. Since $T(K_{n})$ is $2n-2$ regular graph, by proposition 2.8
$\lambda_{3}(T(K_{n}))\leq 2n-3$. Next we by constructing $2n-3$ edge disjoint
$S$-trees in $T(K_{n})$ for any 3-subset $S$ of $V(T(K_{n}))$ to prove
$\lambda_{3}(T(K_{n}))\geq 2n-3$.

Recall of the proof of Theorem 3.8, except case $|S\cap V(K_{n})|=0$ and case
$|S\cap V(K_{n})|=2$ for $S=\\{u_{a},u_{b},e_{cd}\\}$ with
$u_{a}u_{b}=u_{c}u_{d}$, there always exist at least $2n-3$ internally
disjoint $S$-trees in $T(K_{n})$, which are also edge disjoint $S$-trees in
$T(K_{n})$. Thus here we only need to consider the above two exception cases
and show there still exist at least $2n-3$ edge disjoint $S$-trees in
$T(K_{n})$.

If $|S\cap V(K_{n})|=0$, assume $S=\\{e_{ab},e_{cd},e_{gf}\\}\subset
V(L(K_{n}))$, by Lemma 3.7 there exist at least $2n-5$ edge disjoint $S$-trees
in $L(K_{n})$, in addition these trees, add two edge disjoint $S$-trees
$e_{ab}u_{a}u_{c}e_{cd}\cup u_{c}u_{g}e_{gf}$ and $e_{ab}u_{b}u_{d}e_{cd}\cup
u_{d}u_{f}e_{gf}$ together, we obtain at least $2n-3$ edge disjoint $S$-trees
in $T(K_{n})$, as desire.

If $|S\cap V(K_{n})|=2$ for $S=\\{u_{a},u_{b},e_{cd}\\}$ with
$u_{a}u_{b}=u_{c}u_{d}$, this means $S=\\{u_{a},u_{b},e_{ab}\\}$, then we form
$S$-trees $T_{i}^{1}=xu_{i}e_{bi}z\cup e_{bi}y$ and
$T_{i}^{2}=xe_{ai}u_{i}y\cup e_{ai}z$ for every $i\in[n]\setminus\\{a,b\\}$
and thus get $2n-4$ edge disjoint $S$-trees. Add tree $xzy$ together with all
$T_{i}^{1}$ and $T_{i}^{2}$ we get $2n-3$ edge disjoint $S$-trees in
$T(K_{n})$, as desire.

Thus we get $\lambda_{3}(T(K_{n}))\geq 2n-3$. This complete the proof.

At end of this section, we determine the generalized 3-connectivity and the
generalized 3-edge-connectivity of the total graph of the complete bipartite
graph. We start with definition and Lemmas.

_The Cartesian product_ $G_{1}\times G_{2}$ of $G_{1}$ and $G_{2}$ is a graph
which has vertex set $V(G_{1})\times V(G_{2})$ with two vertices
$x=(u,u^{\prime})$ and $y=(v,v^{\prime})$ adjacent iff for $u=v$, $u^{\prime}$
is adjacent with $v^{\prime}$ in $G_{2}$ or $u^{\prime}=v^{\prime}$, $u$ is
adjacent with $v$ in $G_{1}$.

It is clear that line graph $L(K_{m,n})$ of complete bipartite graph $K_{m,n}$
is the Cartesian product of $K_{m}$ and $K_{n}$ and the generalized
3-connectivity of $L(K_{m,n})$ has been determined in our another paper, which
listed as follow.

###### Lemma 3.10

[27] Let $L(K_{m,n})$ be the line graph of complete bipartite graph
$K_{m,n}(m\leq n)$, then the generalized 3-connectivity of $L(K_{m,n})$ is
$\kappa_{3}(L(K_{m,n}))=\kappa_{3}(K_{m}\times K_{n})=m+n-3$.

###### Theorem 3.11

Let $T(K_{m,n})$ be a total graph of complete bipartite graph $K_{m,n}$ with
$1\leq m\leq n$. Then the generalized 3-connectivity of $T(K_{m,n})$ is

$\kappa_{3}(T(K_{m,n}))=\begin{cases}2m-1,&\mbox{ if $m=n$,}\\\ 2m,&\mbox{ if
$m<n$.}\\\ \end{cases}$

Proof. Suppose $U=\\{u_{1},u_{2},\cdots,u_{m}\\}$ and
$V=\\{v_{1},v_{2},\cdots,v_{n}\\}$ be the two parts of $K_{m,n}$, ie.,
$V(K_{m,n})=U\cup V$. Then $V(L(K_{m,n}))=\\{e_{ij}|e_{ij}=u_{i}v_{j}\\}$ and
$V(T(K_{m,n}))=U\cup V\cup V(L(K_{m,n}))$. Clearly, by Theorem 3.1, the
results hold for case $m=1$. The following we consider cases for $n\geq m\geq
2$.

Case 1. $m=n$

Since $T(K_{m,m})$ is $2m$ regular, by proposition 2.7 we get
$\kappa_{3}(T(K_{m,m}))\leq 2m-1$. The following we prove
$\kappa_{3}(T(K_{m,m}))\geq 2m-1$. In fact, for any 3-subset $S=\\{x,y,z\\}$
of $V(T(K_{m,m}))$, here we only need to show that there exist at least $2m-1$
internally disjoint $S$-trees in $T(K_{m,m})$.

Subcase 1.1. $|S\cap V(K_{m,m})|=3$

$|S\cap V(K_{m,m})|=3$ means $S\subseteq V(K_{m,m})$. Then either $|S\cap
V|=3$ or $|S\cap V|=2$ and $|S\cap U|=1$. If $|S\cap V|=3$, assume
$x=v_{1},y=v_{2},z=v_{3}$ (see Fig 2 (a)), we can form $2m(>2m-1)$ internally
disjoint $S$-trees such as: $T_{i}=xu_{i}z\cup u_{i}y$ and
$T^{\prime}_{i}=xe_{i1}e_{i2}y\cup e_{i2}e_{i3}z$ for $1\leq i\leq m$. If
$|S\cap V|=2$ and $|S\cap U|=1$, assume $x=v_{1},y=v_{2},z=u_{m}$ (see Fig 2
(b)), we form $2m-1$ internally disjoint $S$-trees as follow: $xzy$,
$xe_{m1}e_{11}e_{12}y\cup e_{m1}z$, $xu_{1}ye_{m2}z$ and $xu_{i}y\cup
u_{i}v_{i+1}z$, $xe_{i1}e_{i2}e_{i(i+1)}e_{m(i+1)}z\cup e_{i2}y$ for $2\leq
i\leq m-1$. All as we desire.

Figure 2: The subcase for $|S\cap V(K_{m,n})|=3$.

Subcase 1.2. $|S\cap V(K_{m,m})|=2$

Here we need discuss five possible cases. If $x,y,z$ are nonadjacent each
other in $T(K_{m,m})$, assume $x=v_{1},y=v_{2},z=e_{mm}(m\neq 1,2)$ (see Fig 3
(a)). Then $2m-1$ internally disjoint $S$-trees be formed as follows:
$xu_{m}z\cup u_{m}y$, $xu_{1}v_{m}z\cup u_{1}y$, $xe_{m1}e_{m2}z\cup e_{m2}y$,
$xe_{11}e_{12}e_{1m}z\cup e_{12}y$ and $xu_{i}e_{im}z\cup u_{i}y$ for $2\leq
i\leq m-1$, $xe_{i1}e_{i2}e_{i(i+1)}e_{m(i+1)}z\cup e_{i2}y$ for $2\leq i\leq
m-2$.

If $x$ is nonadjacent to $y,z$ but $y$ and $z$ are adjacent in $T(K_{m,m})$,
assume $x=v_{1},y=v_{2},z=e_{22}$ (see Fig 3 (b)). Then $2m-1$ internally
disjoint $S$-trees are formed as: $xu_{2}z\cup u_{2}y$, $xu_{1}yz$,
$xe_{11}e_{12}z\cup e_{12}y$ and $xu_{i}v_{i}e_{2i}z\cup u_{i}y$,
$xe_{i1}e_{i2}z\cup e_{i2}y$ for $3\leq i\leq m$.

If $z$ is nonadjacent to $x,y$ but $x$ and $y$ are adjacent in $T(K_{m,m})$,
assume $x=v_{1},y=u_{1},z=e_{mm}$ (see Fig 3 (c)). Then $2m-1$ internally
disjoint $S$-trees be formed as: $xyv_{m}z$, $ye_{11}xu_{m}z$,
$xe_{m1}ze_{1m}y$ and $yv_{i}u_{i}x\cup u_{i}e_{im}z$,
$xe_{i1}e_{ii}e_{1i}y\cup e_{1i}e_{mi}z$ for $2\leq i\leq m-1$.

If $x$ and $z$ are both adjacent to $y$ but $x$ and $z$ are nonadjacent in
$T(K_{m,m})$, assume $x=v_{1},y=u_{1},z=e_{1m}$ (see Fig 3 (d)). Then $2m-1$
internally disjoint $S$-trees be formed as: $xyz$, $xe_{11}y\cup e_{11}z$,
$xu_{m}v_{m}z\cup u_{m}y$ and $xu_{i}v_{i}y\cup u_{i}e_{im}z$,
$xe_{i1}e_{ii}e_{1i}y\cup e_{ii}z$ for $2\leq i\leq m-1$.

If $x,y,z$ are adjacent each other in $T(K_{m,m})$, assume
$x=v_{1},y=u_{1},z=e_{11}$ (see Fig 3 (e)). Then $xzy$, $xe_{1m}z\cup
e_{1m}e_{mm}u_{m}y$, $ye_{m1}z\cup e_{m1}e_{m2}v_{2}x$ and $xe_{1i}z\cup
e_{1i}e_{ii}u_{i}y$, $ye_{i1}e_{i(i+1)}v_{i+1}x\cup e_{i1}z$ for $2\leq i\leq
m-1$ are $2m-1$ internally disjoint $S$-trees in $T(K_{m,m})$.

Figure 3: The subcases for $|S\cap V(K_{m,n})|=2$.

Subcase 1.3. $|S\cap V(K_{m,m})|=1$

Similarly, we also need discuss five possible cases while $|S\cap
V(K_{m,m})|=1$. If $x,y,z$ are nonadjacent each other in $T(K_{m,m})$, assume
$x=v_{1},y=e_{22},z=e_{mm}$ (see Fig 4 (a)). Then $2m-1$ internally disjoint
$S$-trees be formed as: $xu_{2}v_{m}z\cup u_{2}y$, $xu_{m}v_{2}y\cup u_{m}z$,
$xu_{1}e_{12}y\cup e_{12}e_{1m}z$, $xe_{21}ye_{2m}z$, $xe_{m1}ze_{m2}y$ and
$xu_{i}v_{i}e_{mi}z\cup v_{i}e_{2i}y$, $xe_{i1}e_{ii}e_{im}z\cup
e_{ii}e_{i2}y$ for $3\leq i\leq m-1$.

If $y,z$ are adjacent but they are nonadjacent to $x$ in $T(K_{m,m})$, assume
$x=v_{1},y=e_{22},z=e_{m2}$ (see Fig 4 (b)). Then $2m-1$ internally disjoint
$S$-trees be constructed as: $xu_{1}v_{2}z\cup v_{2}y$, $xe_{11}e_{12}z\cup
e_{12}y$, $xu_{2}v_{m}u_{m}z\cup u_{2}y$, $xe_{21}y\cup e_{21}e_{2m}e_{mm}z$,
$xe_{m1}zy$ and $xu_{i}e_{i2}y\cup e_{i2}z$, $xe_{i1}e_{ii}e_{2i}y\cup
e_{ii}e_{mi}z$ for $3\leq i\leq m-1$.

If $x,y$ are adjacent but they are nonadjacent to $z$ in $T(K_{m,m})$, assume
$x=v_{1},y=e_{21},z=e_{mm}$ (see Fig 4 (c)). Then $2m-1$ internally disjoint
$S$-trees be constructed as: $yxu_{m}z$, $xu_{2}v_{m}z\cup u_{2}y$,
$xe_{m1}ze_{2m}y$, $xu_{1}v_{2}e_{22}y\cup e_{22}e_{m2}z$, $xe_{11}y\cup
e_{11}e_{1m}z$ and $xu_{i}v_{i}e_{2i}y\cup v_{i}e_{mi}z$, $xe_{i1}y\cup
e_{i1}e_{im}z$ for $3\leq i\leq m-1$.

If $x$ and $z$ are both adjacent to $y$ but $x$ and $z$ are nonadjacent in
$T(K_{m,m})$, assume $x=v_{1},y=e_{21},z=e_{22}$ (see Fig 4 (d)). Then $2m-1$
internally disjoint $S$-trees be constructed as: $xyz$, $xu_{2}y\cup u_{2}z$,
$xe_{11}u_{1}v_{2}z\cup e_{11}y$ and $xu_{i}v_{i}e_{2i}y\cup e_{2i}z$,
$xe_{i1}y\cup e_{i1}e_{i2}z$ for $3\leq i\leq m$.

If $x,y,z$ are adjacent each other in $T(K_{m,m})$, assume
$x=v_{1},y=e_{11},z=e_{21}$ (see Fig 4 (e)). Then $2m-1$ internally disjoint
$S$-trees be constructed as: $yxz$, $xu_{1}y\cup u_{1}v_{2}u_{2}z$,
$zye_{12}e_{22}u_{2}x$ and $xu_{i}v_{i}e_{2i}z\cup v_{i}e_{1i}y$,
$xe_{i1}y\cup e_{i1}z$ for $3\leq i\leq m$.

Figure 4: The subcases for $|S\cap V(K_{m,n})|=1$.

Subcase 1.4. $|S\cap V(K_{m,m})|=0$

$|S\cap V(K_{m,m})|=0$ means $S\subseteq V(L(K_{m,m}))$, without loss
generality, suppose $S=\\{e_{pq},e_{rs},e_{tk}\\}$, then by Lemma 3.10 there
always exist $2m-3$ internally disjoint $S$-trees in $L(K_{m,m})$, named as
$T_{i}(1\leq i\leq 2m-3)$. Now we add two $S$-trees
$e_{pq}u_{p}v_{j}u_{t}e_{tk}\cup v_{j}u_{r}e_{rs}$ and
$e_{pq}v_{q}u_{j}v_{k}e_{tk}\cup u_{j}v_{s}e_{rs}$ to all $T_{i}$, here $j\neq
p,q,r,s,t,k$. In total, we get $2m-1$ internally disjoint $S$-trees in
$T(K_{m,m})$, as desire.

Case 2. $m<n$

Since the minimum degree of $T(K_{m,n})$ is $2m$, by Observation 2.1 we get
$\kappa_{3}(T(K_{m,n}))\leq 2m$. Let $S=\\{x,y,z\\}$ be a 3-subset of
$V(T(K_{m,n}))$, now we by forming $2m$ internally disjoint $S$-trees in
$T(K_{m,n})$ to prove $\kappa_{3}(T(K_{m,n}))\geq 2m$.

Subcase 2.1. $|S\cap V(K_{m,n})|=3$

Clearly, either $|S\cap V|=3$ or $|S\cap V|=2$ and $|S\cap U|=1$. If $|S\cap
V|=3$, by the proof of Subcase 1.1, there exist $2m$ internally disjoint
$S$-trees in $T(K_{m,n})$. Here we consider the case for $|S\cap V|=2$ and
$|S\cap U|=1$, assume $x=v_{1},y=v_{2},z=u_{m}$ (see Fig 2 (b)),we can form
$2m$ internally disjoint $S$-trees as: $xzy$, $xe_{m1}ze_{m2}y$ and
$xu_{i}y\cup u_{i}v_{i+2}z$, $xe_{i1}e_{i2}e_{i(i+2)}e_{m(i+2)}z\cup e_{i2}y$
for $1\leq i\leq m-1$. As desire.

Subcase 2.2. $|S\cap V(K_{m,n})|=2$

Here we need discuss five possible cases. If $x,y,z$ are nonadjacent each
other in $T(K_{m,n})$, assume $x=v_{1},y=v_{2},z=e_{mm}$ (see Fig 3 (a)). Then
we form $2m$ internally disjoint $S$-trees as: $xu_{m}z\cup u_{m}y$,
$xe_{m1}e_{m2}y\cup e_{m2}z$ and $xu_{i}v_{i+2}e_{m(i+2)}z\cup u_{i}y$,
$ye_{i2}e_{i1}x\cup e_{i1}e_{im}z$ for $1\leq i\leq m-1$.

If $y$ and $z$ are nonadjacent to $x$ but $y$ and $z$ are adjacent in
$T(K_{m,n})$, assume $x=v_{1},y=v_{2},z=e_{11}$ (see Fig 3 (b)). Then $2m$
internally disjoint $S$-trees be constructed as: $xu_{1}z\cup u_{1}y$,
$ye_{12}zx$ and $xu_{i}v_{i+1}e_{1(i+1)}z\cup u_{i}y$, $xe_{i1}e_{i2}y\cup
e_{i1}z$ for $2\leq i\leq m$.

If $x$ and $y$ are nonadjacent to $z$ but $x$ and $y$ are adjacent in
$T(K_{m,n})$, assume $x=v_{1},y=u_{1},z=e_{mm}$ (see Fig 3 (c)). Then $2m$
internally disjoint $S$-trees be constructed as: $xyv_{m}z$,
$yv_{m+1}u_{m}z\cup u_{m}x$, $xe_{11}y\cup e_{11}e_{m1}z$,
$xe_{m1}ze_{m(m+1)}e_{1(m+1)}y$ and $yv_{i}u_{i}x\cup u_{i}e_{im}z$,
$xe_{i1}e_{ii}e_{1i}y\cup e_{1i}e_{mi}z$ for $2\leq i\leq m-1$.

If $x$ and $z$ are both adjacent to $y$ but $x$ and $z$ are nonadjacent in
$T(K_{m,n})$, assume $x=v_{1},y=u_{1},z=e_{1m}$ (see Fig 3 (d)). Then $2m$
internally disjoint $S$-trees be constructed as follows: $xyz$, $xe_{11}y\cup
e_{11}z$, $xu_{m}v_{m}z\cup u_{m}y$, $xe_{m1}e_{m(m+1)}e_{1(m+1)}y\cup
e_{1(m+1)}z$ and $xu_{i}v_{i}y\cup u_{i}e_{im}z$, $xe_{i1}e_{ii}e_{1i}y\cup
e_{ii}z$ for $2\leq i\leq m-1$.

If $x,y,z$ are adjacent each other in $T(K_{m,n})$, assume
$x=v_{1},y=u_{1},z=e_{11}$ (see Fig 3 (e)). Then $xzy$, $xy\cup xe_{1(m+1)}z$,
$xe_{1m}z\cup e_{1m}e_{mm}u_{m}y$, $ye_{m1}z\cup e_{m1}e_{m2}v_{2}x$ and
$xe_{1i}z\cup e_{1i}e_{ii}u_{i}y$, $ye_{i1}e_{i(i+1)}v_{i+1}x\cup e_{i1}z$ for
$2\leq i\leq m-1$ are $2m$ internally disjoint $S$-trees in $T(K_{m,n})$, as
desire.

Subcase 2.3. $|S\cap V(K_{m,n})|=1$

Similarly, we also need discuss five possible cases. If $x,y,z$ are
nonadjacent each other in $T(K_{m,n})$, assume $x=v_{1},y=e_{22},z=e_{mm}$
(see Fig 4 (a)). Then $2m$ internally disjoint $S$-trees be constructed as:
$xu_{2}v_{m}z\cup u_{2}y$, $xu_{m}v_{2}y\cup u_{m}z$, $xu_{1}e_{12}y\cup
e_{12}e_{1m}z$, $xe_{21}ye_{2m}z$, $xe_{m1}ze_{m2}y$,
$xe_{11}e_{1(m+1)}e_{2(m+1)}y\cup e_{2(m+1)}e_{m(m+1)}z$ and
$xu_{i}v_{i}e_{mi}z\cup e_{mi}e_{2i}y$, $xe_{i1}e_{im}z\cup e_{im}e_{i2}y$ for
$3\leq i\leq m-1$.

If $y$ and $z$ are adjacent but they are nonadjacent to $x$ in $T(K_{m,n})$,
assume $x=v_{1},y=e_{22},z=e_{m2}$ (see Fig 4 (b)). Then $2m$ internally
disjoint $S$-trees be constructed as: $xu_{1}v_{2}z\cup v_{2}y$,
$xe_{11}e_{12}z\cup e_{12}y$, $xu_{2}v_{m+1}e_{m(m+1)}z\cup u_{2}y$,
$xu_{m}z\cap u_{m}e_{m2}y$, $xe_{21}yz$, $xe_{m1}z\cup e_{m1}e_{mm}e_{2m}y$
and $xu_{i}e_{i2}y\cup e_{i2}z$, $xe_{i1}e_{ii}e_{2i}y\cup e_{ii}e_{mi}z$ for
$3\leq i\leq m-1$.

If $x$ and $y$ are adjacent but they are nonadjacent to $z$ in $T(K_{m,n})$,
assume $x=v_{1},y=e_{11},z=e_{mm}$ (see Fig 4 (c)). Then $2m$ internally
disjoint $S$-trees be constructed as: $xye_{1m}z$, $xu_{1}v_{m}z\cup u_{1}y$,
$xe_{m1}z\cup e_{1m}y$, $xu_{m}e_{m(m+1)}e_{1(m+1)}y\cup e_{m(m+1)}z$ and
$xu_{i}v_{i}e_{1i}y\cup v_{i}e_{mi}z$, $xe_{i1}y\cup e_{i1}e_{im}z$ for $2\leq
i\leq m-1$.

If $x$ and $z$ are both adjacent to $y$ but $x$ and $z$ are nonadjacent in
$T(K_{m,n})$, assume $x=v_{1},y=e_{21},z=e_{22}$ (see Fig 4 (d)). Then $2m$
internally disjoint $S$-trees be constructed as: $xyz$, $xu_{2}y\cup u_{2}z$,
$xe_{11}e_{12}z\cup e_{11}y$, $xu_{1}v_{m+1}e_{2(m+1)}y\cup e_{2(m+1)}z$ and
$xu_{i}v_{i}e_{2i}y\cup e_{2i}z$, $xe_{i1}y\cup e_{i1}e_{i2}z$ for $3\leq
i\leq m$.

If $x,y,z$ are adjacent each other in $T(K_{m,n})$, assume
$x=v_{1},y=e_{11},z=e_{21}$ (see Fig 4 (e)). Then $2m$ internally disjoint
$S$-trees be constructed as: $yxz$, $xu_{1}y\cup u_{1}e_{12}e_{22}z$,
$xe_{21}z\cup e_{21}y$, $xu_{2}e_{2(m+1)}z\cup e_{2(m+1)}e_{1(m+1)}y$ and
$xu_{i}v_{i}e_{2i}z\cup v_{i}e_{1i}y$, $xe_{i1}y\cup e_{i1}z$ for $3\leq i\leq
m$.

Subcase 2.4. $|S\cap V(K_{m,n})|=0$

$|S\cap V(K_{m,n})|=0$ means $S\subseteq V(L(K_{m,n}))$, assume
$S=\\{e_{pq},e_{rs},e_{tk}\\}$, then by Lemma 3.10 there exist $m+n-3$
internally disjoint $S$-trees in $L(K_{m,n})$, named as $T_{i}(1\leq i\leq
m+n-3)$. Now we add two $S$-trees $e_{pq}u_{p}v_{j}u_{t}e_{tk}\cup
v_{j}u_{r}e_{rs}$ and $e_{pq}v_{q}u_{j}v_{k}e_{tk}\cup u_{j}v_{s}e_{rs}$ to
all $T_{i}$, here $j\neq p,q,r,s,t,k$. Thus we get $m+n-1(\geq 2m)$ internally
disjoint $S$-trees in $T(K_{m,n})$, as desire.

By now we complete the proof.

Note that the fact $T(K_{m,n})$ is $2m$ -regular graph while $m=n$ and its
minimum degree is $2m$ while $m<n$. Combine this with Observation 2.1 and
Theorem 3.11, we immediately get Theorem 3.12.

###### Theorem 3.12

Let $T(K_{m,n})$ be a total graph of complete bipartite graph $K_{m,n}(m\leq
n)$. Then the generalized 3-edge-connectivity of $T(K_{m,n})$ is

$\lambda_{3}(T(K_{m,n}))=\begin{cases}2m-1,&\mbox{ if $m=n$,}\\\ 2m,&\mbox{ if
$m<n$.}\\\ \end{cases}$

##  4 Bound for generalized 3-connectivity of total graph

In this section we give some bounds for the generalized 3-connectivity of
total graph $T(G)$.

###### Lemma 4.1

[6] Let $G$ be graph for which $\kappa(G)\geq m$. Then $\kappa(T(G))\geq 2m$
and $\lambda(T(G))\geq 2m$.

###### Lemma 4.2

[27] Let $G$ be connected graph, then $\kappa_{3}(L(G))\geq\lambda_{3}(G)$.

###### Theorem 4.3

Let $G$ be connected graph with connectivity $\kappa(G)$ and minimum degree
$\delta(G)$. Then

$\lfloor\frac{3\kappa(G)-1}{2}\rfloor\leq\kappa_{3}(T(G))\leq 2\delta(G)$

Proof. Since the minimum degree of $T(G)$ is $2\delta(G)$, so by Proposition
2.6 we get $\kappa_{3}(T(G))\leq 2\delta(G)$. On the other hand, by Lemma 4.1,
we know $\kappa(T(G))\geq 2\kappa(G)$ and let $\kappa(T(G))=4a+b$ with
$b\in\\{0,1,2,3\\}$. Then we get $a\geq\frac{2\kappa(G)-b}{4}$. Combine this
with Proposition 2.9 we have

$\kappa_{3}(T(G))\geq
3a+\lceil\frac{b}{2}\rceil\geq\frac{3(2\kappa(G)-b)}{4}+\lceil\frac{b}{2}\rceil$

Since $b\in\\{0,1,2,3\\}$, so the value of
$\frac{3(2\kappa(G)-b)}{4}+\lceil\frac{b}{2}\rceil$ can meet
$\frac{3\kappa(G)}{2},\frac{3\kappa(G)}{2}+\frac{1}{4},\frac{3\kappa(G)}{2}-\frac{1}{4}$
and $\frac{3\kappa(G)}{2}-\frac{1}{2}$, respectively. Consider this with
$\kappa_{3}(T(G))$ is an integer, we get

$\kappa_{3}(T(G))\geq\lfloor\frac{3\kappa(G)-1}{2}\rfloor$

###### Remark 4.4

The upper bound is sharp for complete bipartite graph $K_{m,n}$ with $n>m$.
The lower bound is also sharp for complete graph $K_{n}$ with $n\neq 3$.

Similar we also get the following result for the generalized 3-edge-
connectivity of total graph $T(G)$.

###### Theorem 4.5

Let $G$ be connected graph with edge connectivity $\lambda(G)$ and minimum
degree $\delta(G)$. Then

$\min\\{2\lambda(G)-1,2\lambda_{3}(G),\lambda_{3}(G)+2\\}\leq\lambda_{3}(T(G))\leq
2\delta(G)$

Proof. Since the minimum degree of $T(G)$ is $2\delta(G)$, by Observation 2.1
we get $\lambda_{3}(T(G))\leq 2\delta(G)$. Next we prove
$\lambda_{3}(T(G))\geq\min\\{2\lambda(G)-1,2\lambda_{3}(G),\lambda_{3}(G)+2\\}$.

Suppose $V(G)=\\{v_{1},v_{2},\cdots,v_{n}\\}$ and
$V(L(G))=\\{e_{ij}|v_{i}v_{j}\in E(G)\\}$, let $S=\\{x,y,z\\}$ be a 3-subset
of $V(T(G))$. The following we by constructing edge disjoint $S$-trees in
$T(G)$ to prove
$\lambda_{3}(T(G))\geq\min\\{2\lambda(G)-1,2\lambda_{3}(G),\lambda_{3}(G)+2\\}$.
Now distinguish four cases to complete the proof.

Case 1. $|S\cap V(G)|=3$

$|S\cap V(G)|=3$ means $S\subseteq V(G)$, since there are $\lambda_{3}(G)$
edge disjoint $S$-trees in $G$, name them as $T_{i}$. In addition to these, it
is clear that every $T_{i}$ has a corresponding tree $CT_{i}$ in $L(G)$ and
every $CT_{i}$ can be formed as a $S$-tree by connecting it with $S$. Thus
altogether we can get at least $2\lambda_{3}(G)$ edge disjoint $S$-trees in
$T(G)$.

Case 2. $|S\cap V(G)|=2$

Suppose $x,y\in V(G)$, $z\in V(L(G))$ and let $x=v_{i},y=v_{j},z=e_{pq}$. If
$T(G)[S]$ is a triangle, since there are $\lambda(G)$ edge disjoint $xy$-paths
in $G$ and thus assume $xv_{1}v_{2}\cdots v_{k}y$ is a $xy$-path. Based on
this $xy$-path we can form 2 edge disjoint $S$-trees in $T(G)$ as:
$xe_{i1}v_{1}v_{2}\cdots v_{k}y\cup e_{i1}z$ and $xv_{1}e_{12}e_{23}\cdots
e_{kj}y\cup e_{kj}z$. Thus we total get $2\lambda(G)$ edge disjoint $S$-trees
in $T(G)$.

If $p=i$ but $q\neq j$, assume $S^{\prime}=\\{v_{i},v_{j},v_{q}\\}\subseteq
V(G)$. Since there exist $\lambda_{3}(G)$ edge disjoint $S^{\prime}$-trees in
$G$, name them as $T_{i}$. For every pair trees $T_{i}$ and its corresponding
tree $CT_{i}$, we by symmetric difference operation on $G$ and $L(G)$ can
obtain $2\lambda_{3}(G)$ edge disjoint $S$-trees in $T(G)$.

If $p\neq i$ and $q\neq j$, assume $S^{\prime}=\\{v_{i},v_{j},v_{q}\\}$. Since
there exist $\lambda_{3}(G)$ edge disjoint $S^{\prime}$-trees in $G$, named as
$T_{i}$. For every pair trees $T_{i}$ and its corresponding tree $CT_{i}$, we
also can obtain at least$\lambda_{3}(G)+2$ edge disjoint $S$-trees in $T(G)$.

Case 3. $|S\cap V(G)|=1$

Assume $x\in V(G)$, $y,z\in V(L(G))$ and let $x=v_{i},y=e_{jk},z=e_{pq}$. No
matter what vertices $x,y,z$ they are, there exist 3 elements among
$v_{i},v_{j},v_{k},v_{p},v_{q}$ which can form a 3-subset $S^{\prime}\subseteq
V(G)$. Since there exist $\lambda_{3}(G)$ edge disjoint $S^{\prime}$-trees in
$G$, named as $T_{i}$. Similarly, based on every pair $T_{i}$ and its
corresponding tree $CT_{i}$ we get at least $\lambda_{3}(G)+2$ edge disjoint
$S$-trees in $T(G)$.

Case 4. $|S\cap V(G)|=0$

This case means $S\subseteq V(L(G))$, by Lemma 4.2 there exist
$\lambda_{3}(G)$ internally disjoint $S$-trees in $L(G)$, which are also edge
disjoint $S$-trees. Based on this, using vertices in graph $G$ can obtain at
least $\lambda_{3}(G)+2$ edge disjoint $S$-trees in $T(G)$.

By argument of the above , we can claim that there exist at least
$\min\\{2\lambda(G)-1,2\lambda_{3}(G),\lambda_{3}(G)+2\\}$ edge disjoint
$S$-trees in $T(G)$. Thus
$\lambda_{3}(T(G))\geq\min\\{2\lambda(G)-1,2\lambda_{3}(G),\lambda_{3}(G)+2\\}$.

###### Remark 4.6

The upper bound is sharp for complete bipartite graph $K_{m,n}$ with $n>m$.
The lower bound is also sharp for
$\lambda_{3}(T(C_{n}))=2\lambda(C_{n})-1=\lambda_{3}(C_{n})+2$,
$\lambda_{3}(T(K_{n}))=2\lambda(K_{n})-1$ and
$\lambda_{3}(T(K_{m,n}))=2\lambda_{3}(K_{m,n})$ with $n>m$.

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