# Searching for young runaways across the sky

Marina Kounkel,1 Aidan Mcbride,2 Keivan G. Stassun1 and Nathan Leigh3,4
1Department of Physics and Astronomy, Vanderbilt University, VU Station 1807,
Nashville, TN 37235, USA
2Department of Physics and Astronomy, University of Utah, JFB, 115 1400 E Salt
Lake City, UT 84112, USA
3Departamento de Astronomía, Facultad de Ciencias Físicas y Matemáticas,
Universidad de Concepción, Concepción, Chile
4Department of Astrophysics, American Museum of Natural History, New York, NY
10024, USA E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present a catalog of 3354 candidate young stars within 500 pc that appear
to have been ejected from their parent associations with relative speeds of
$>$5 km s-1. These candidates have been homogeneously selected through
performing a 2d spherical traceback of previously identified pre-main sequence
candidates to various star forming regions, ensuring that the traceback age as
well as the estimated age of a star is consistent with the age of the
population, and excluding contaminants from the nearby moving groups that
follow the dominant velocity currents in the field. Among the identified
candidates we identify a number of pairs that appear to have interacted in the
process of the ejection, these pairs have similar traceback time, and their
trajectory appears to be diametrically opposite from each other, or they have
formed a wide binary in the process. As the selection of these candidates is
performed solely in 2d, spectral follow-up is necessary for their eventual
confirmation. Unfortunately, recently released Gaia DR3 radial velocities
appear to be unsuitable for characterizing the kinematics of low mass stars
with ages $<$100 Myr, as the accretion, activity, and a variety of other
spectral features that make them distinct from the more evolved stars do not
appear to have been accurately accounted for in the data, resulting in
significant artificially inflated scatter in their RV distribution.

###### keywords:

stars: kinematics and dynamics – stars: pre-main-sequence – proper motions

††pubyear: 2022††pagerange: Searching for young runaways across the
sky–Searching for young runaways across the sky

##  1 Introduction

Typically, young stars do not form in isolation – they tend to form in massive
molecular clouds alongside hundreds, if not thousands, of other young stars.
Some of them originate in relatively diffuse environments, some are born in
dense clusters. Regardless of the overall stellar density surrounding them,
many of the youngest stellar objects initially are found in denser “hubs”,
consisting of $>2$ stars resembling proto-multiple systems (Chen et al.,
2013). Some of them will go on to form stable binaries and other higher order
systems, and some will rapidly dissolve (Tobin et al., 2016, 2022).

When three or more stars are found in close proximity to one another and they
do not form a stable orbital configuration (such as a hierarchical triple),
orbital energy exchange will occur (Leigh & Geller, 2013; Stone & Leigh, 2019;
Manwadkar et al., 2020; Manwadkar et al., 2021). Such an exchange will
typically lead to an ejection of one of the stars from the system if they have
finite sizes, and always if they are assumed to be point-particles. Typical
ejection speeds are only a few km s-1 (Reipurth et al., 2010), which would
artificially broaden the overall velocity dispersion of the population in
which these stars have been formed (Kounkel et al., 2021), but otherwise not
be immediately apparent. The most extreme ejection events can accelerate stars
to speeds of several tens of km s-1, such stars are considered walkaway (if
their speed is $<$30 km s-1) and runaway (if their speed is $>$30 km s-1,
e.g., Schoettler et al., 2020), though such boundaries are arbitrary, largely
driven by the precision in velocity in the early studies that were needed for
a robust confirmation that such stars have been ejected, particularly if a
point of origin is uncertain.

One of the most famous pair of runaways are two O stars, AE Aur and $\mu$ Col,
moving in opposite directions from one another, having been ejected $\sim 2$
Myr ago from the Orion Nebula Cluster (ONC) with speeds of $\sim$100 km s-1
(Blaauw, 1961). A number of other OB runaways have also been identified
(Hoogerwerf et al., 2001): as they are bright and fast moving, proper motions
for a number of OB runaways has been readily available for several decades.
And, in many cases, they can be traced back to specific clusters (Bhat et al.,
2022) from which they are believed to originate.

However, identifying low mass ejected stars has presented a greater challenge.
An unequivocal confirmation of youth of a star located outside of notable star
forming regions has been difficult to come by. This is necessary to separate
likely runaway candidates from ordinary field stars that happen to have
similar kinematics – as the latter stars are far more numerous, even a small
degree of contamination can overwhelm the sample of runaways. Furthermore, it
is only recently, with the release of astrometry from Gaia (Gaia Collaboration
et al., 2018, 2021) that the sufficiently precise proper motions have become
available for a large number of stars.

Figure 1: Spatial distribution of the identified structures, arbitrarily
colored by their assigned group. Top: large scale clustering. Bottom: finer
subgroups.

Since then, there have been several studies that searched for young runaway
and walkaway stars, but, such studies tended to primarily focus on those that
originate from the ONC. In particular, McBride & Kounkel (2019) have found 26
stars with tangential velocities $>$10 km s-1 among the known members of the
cluster. Schoettler et al. (2020) have identified 85 candidates younger than 4
Myr that trace back to the ONC, currently located within 100 pc of it.
Similarly, Farias et al. (2020) have identified 17,000 candidates within 45∘
of the cluster that are expected to be young – while most of these sources
suffer from contamination, 25 candidates have a particularly high likelihood
of having been ejected. Other studies that have identified runaway/walkaway
stars in the ONC include Kounkel et al. (2017); Platais et al. (2020); Maíz
Apellániz et al. (2021).

Outside of the ONC, no systematic search of young ejected stars has been
performed, and all the candidates so far are serendipitous, such as, e.g.,
Luhman (2018) identifying a dissolution of three stars in Taurus, or Tajiri et
al. (2020) finding a fast moving dipper star.

In this paper we take advantage of newly available catalogs of likely pre-main
sequence (PMS) stars to search for runaway and walkaway candidates, tracing
them back to a variety of star forming regions within 500 pc. In Section 2 we
describe the data that form the basis of this analysis. In Section 3 we
present the methodology used to identify them. In Section 4 we discuss these
results, and offer conclusions in Section 5.

##  2 Data

###  2.1 Base catalog

The release of data from Gaia has made it increasingly more possible to search
for PMS stars with ages of a few to several tens of Myr. A large number of
catalogs have taken advantage of the fact that young stars form in large
populations that are dynamically cold to identify a large number of young
moving groups (e.g., Kounkel & Covey, 2019; Kounkel et al., 2020; Kerr et al.,
2021; Prisinzano et al., 2022). However, as ejected stars have been
dynamically processed, they may no longer share the kinematics with their
parent association; as such it is necessary to confirm that a specific star is
pre-main sequence without relying on its kinematics.

Other studies, such as Zari et al. (2018) and McBride et al. (2021) have
performed a photometric selection of young stars along their HR diagram. In
particular, McBride et al. (2021) have developed a neural net Sagitta that was
trained on the 2MASS and Gaia photometry, as well as parallaxes, of the stars
that are members of young moving groups from Kounkel et al. (2020).

Sagitta consists of two parts. The first component is a classifier that
assigns a probability of a particular star being pre-main sequence, separating
out the young stars from the stars on the main and binary sequences, as well
as from reddened high mass and red giant branch stars. Sources with the PMS
probability $>0.95$ represent the cleanest sample that have only negligible
contamination from the evolved stars. At lower thresholds, contamination
increases, to as much as $\sim$50% at PMS probability of $\sim 0.85$; however,
it becomes more sensitive towards stars with an age of a few tens of Myr, as
they are closer to the binary sequence.

The second component of Sagitta is estimating the ages of the stars, using a
neural net to interpolate across the empirical isochrones of the full census
of members of young moving groups with known ages.

In total, McBride et al. (2021) have identified $\sim$450,000 PMS candidates
in Gaia EDR3 data down to PMS probability of 0.7 at distances of up to $\sim$3
kpc. In this work we limit the sample to only the sources with PMS$>$0.85 (to
ensure that the majority of sources would be bona fide PMS stars, despite the
contamination), and with parallax $\pi>2$ mas (to ensure high degree of
precision in the distance and tangential velocity of stars, as well as
sensitivity to low mass stars, both of which decrease substantially beyond
that limit), resulting in 70,528 stars. Such cuts ensure high degree of
completeness across regions included in this volume, a relatively limited
degree of contamination, as well as a high precision in the Gaia astrometry
across all of the candidates to ensure a reliable traceback.

###  2.2 Initial clustering

In order to identify runaways, it is necessary to evaluate their position and
velocity relative to a specific population. At the same time, stars that are
members of one star forming region may systematically trace back to a
different star forming region due to the global dynamics – while there may be
some physical process that may relate these populations, such stars are highly
unlikely to be runaways in the true sense of the word, i.e., all of their
individual stars would not have been ejected through the dynamical
interactions. As such, it is necessary to identify all the stars that are the
co-moving members of star forming region, to exclude them from the possible
pool of candidate runaways, as well as to define typical positions and
velocities to which runaways can track. This is possible to do with
hierarchical clustering, such as with HDBSCAN (Campello et al., 2013; McInnes
et al., 2017).

However, star forming regions tend to be extended, often spanning $>$100 pc,
and they may have a significant velocity gradients, which is important to take
into consideration. A single position/velocity combination for an entire
region (e.g., for Orion, or for Sco Cen) does not offer a sufficient degree of
precision in performing such a traceback. But, similarly, if we consider
positions and velocities of individual stars within a given star forming
region, then this introduces too much noise in the traceback. Thus it is
necessary to identify smaller but meaningful subgroups within each region.

It is possible to “tune” HDBSCAN to recover structures of different scales.
But, unfortunately, if it is tuned to recover smaller subgroups, it would
struggle to pick up stars in the more distributed parts of the star forming
regions that could be easily picked up if a full population is considered. As
previously mentioned, clustering is important for rejecting false positives
from global kinematics, thus their exclusion is not ideal.

Because of this, we perform a two step approach to clustering - first to
identify large scale structure, and second to identify subgroups in the
identified populations. This is similar to the approach considered by Zari et
al. (2019) and Kerr et al. (2021).

The clustering was performed in 7 dimensions:

  * •

$\arccos(\cos{(l)})/2$ deg

  * •

$\arcsin(\sin{(l)})/2$ deg

  * •

$b/2$ deg

  * •

$\log_{10}(1000/\pi)$

  * •

$4.74\mu_{l*}/\pi$

  * •

$4.74\mu_{b*}/\pi$

  * •

Age (Myr)$/10$

where $\pi$ is parallax in mas, and $\mu_{l*}$ and $\mu_{b*}$ are proper
motions in $l$ and $b$, in mas yr-1, converted to the local standard of rest
(Schönrich et al., 2010), with galactic rotation subtracted (see Kounkel et
al., 2021). The implemented transformation converts them from proper motion to
the tangential velocity space. In order to prevent splitting of groups at
$l=0^{\circ}=360^{\circ}$ boundary, a circular encoding for $l$ is
implemented. We note that as the sample has $\sigma_{\pi}/\pi<0.1$, the
inversion of the parallax to get the distances is an acceptably precise
approximation for the purposes of the analysis here. Additionally, while some
of the scalings in the above parameters of the mismatched data units are
arbitrary, they were decided on iteratively through the visual examination of
the outputs to ensure a homogeneous selection of known populations.

To identify large scale structure, we used minimum cluster size of 40 stars,
with the minimum sampling of 40 stars, and ‘eom’ as the cluster selection
method. This has selected 27,511 (out of 70,528) stars into 42 different
populations. Afterwards, we applied HDBSCAN again on the already clustered
stars, changing the minimum cluster size to 20 stars, minimum samples of 5
stars, and ‘leaf’ as the cluster selection method. This selected 10451 stars
in 167 different subgroups (Figure 1).

We note that these groups have a considerable degree of similarity with the
structures identified in other all-sky clustering approaches (e.g., Kounkel &
Covey, 2019; Kerr et al., 2021; Prisinzano et al., 2022), but there are minor
differences that sometimes prevent a precise one-to-one match. Fundamentally,
all clusering techniques trace out similar overdensities in the distribution
of stars, but they differ in the ability of recovering particular populations
that tend to be more diffuse, as well as in the ability of separating out the
neighboring structures. The primary purpose of repeating the clustering
compared to these previous works is to ensure the membership in these
identified structures is as complete as possible relative to the base Gaia
EDR3 catalog from McBride et al. (2021) in particular, such that stars that
may have been excluded in the initial selection in other works would be
considered when identifying “core” members of all the groups. Independent
clustering has also enabled us to better control the relative scales of the
recovered structures.

To the best of our ability, we have attempted to cross-match all of the
identified sub-groups to the structures that have been previously recognized
in the literature, but it is not always possible. The membership of all of the
groups is included in Table 1, and their average properties are shown in Table
2.

Table 1: Catalog of clustered PMS candidates Column | Unit | Description
---|---|---
Source |  | Gaia DR3 source id
$l_{*}$ | deg | Galactic longitude
$b_{*}$ | deg | Galactic latitude
$\pi_{*}$ | mas | Gaia DR3 parallax
$\sigma_{\pi,*}$ | mas | Uncertainty in parallax
PMS |  | PMS probability from Sagitta
$\sigma_{PMS}$ |  | Uncertainty in PMS probability
$t_{*}$ | [yr] | (log) Age from Sagitta
$\sigma_{t,*}$ | [yr] | Uncertainty in age
$\mu_{l*}$ | mas yr-1 | Corrected proper motions in l
$\mu_{b*}$ | mas yr-1 | Corrected proper motions in b
Population |  | Name of the large scale population
Subgroup |  | Name of finer subgroup in a larger population
Table 2: Identified groups and their properties Column | Unit | Description
---|---|---
Population |  | Name of the large scale population
Subgroup |  | Name of finer subgroup in a larger population
$l_{g}$ | deg | Average galactic longitude
$\sigma_{l,g}$ | deg | Scatter in l
$b_{g}$ | deg | Average galactic lattitude
$\sigma_{b,g}$ | deg | Scatter in b
$\pi_{g}$ | mas | Average parallax
$\sigma_{\pi,g}$ | mas | Scatter in parallax
$\mu_{l*,g}$ | mas yr-1 | Average corrected proper motions in l
$\sigma_{\mu l,g}$ | mas yr-1 | Scatter in $\mu_{l*}$
$\mu_{b*,g}$ | mas yr-1 | Average corrected proper motions in b
$\sigma_{\mu b,g}$ | mas yr-1 | Scatter in $\mu_{b*}$
$t_{g}$ | [yr] | Average (log) age
$\sigma_{t,g}$ | [yr] | Scatter in age
N |  | Number of stars

##  3 Analysis

###  3.1 Selection of candidate ejected stars

In order to identify all of the candidate runaway and walkway stars, as well
as their origin, we evaluate the position and velocity of all of the stars
that were not identified as members of any of the moving groups relative to
the average position and velocity of all substructures. Stars that have been
ejected from a particular group would appear to be moving on a radial
trajectory away from it in its rest frame.

We perform a traceback solely in the plane of the sky, similarly to Farias et
al. (2020) using an approximation of a spherical surface. In such a geometry,
a path of a star is described by a great circle.

A bearing $\theta$ represents an angle that an object needs to travel along a
great circle that would connect a starting location with a particular point on
a spherical surface. For each star, we calculate the bearing $\theta$
corresponding to its proper motions via111http://www.movable-
type.co.uk/scripts/latlong.html

$\begin{split}\theta=\mathrm{atan2}(\sin(l_{\mu}-l_{*})\cos b_{\mu},\cos
b_{*}\sin b_{\mu}-\\\ \sin b_{*}\cos b_{\mu}\cos(l_{\mu}-l_{*}))\end{split}$
(1)

The galactic coordinates $(l_{*},b_{*})$ correspond to the current position of
the star, and $(l_{\mu},b_{\mu})$ are the position of a star that are
respectively offset by $\mu_{l*}$ and $\mu_{b*}$ in the reference frame of a
particular group defined by that group’s typical proper motions $\mu_{l*,g}$
and $\mu_{b*,g}$. I.e,.

$\displaystyle l_{\mu}$ $\displaystyle=l_{*}-(\mu_{l*}-\mu_{l*,g})\Delta
t/\cos(b_{*})$ (2) $\displaystyle b_{\mu}$
$\displaystyle=b_{*}-(\mu_{b*}-\mu_{b*,g})\Delta t$

with $\Delta t$ set arbitrarily small, to 1 year, and ensuring appropriate
unit conversion.

We also calculate an angular distance $d$ from the star to the median position
of each given group $(l_{g},b_{g})$ as

$\phi=\mathrm{acos}(\sin b_{*}\sin b_{g}+\cos b_{*}\cos
b_{g}\cos(l_{g}-l_{*}))$ (3)

Finally, we determine the initial traceback position $(l_{o},b_{o})$ from
which a star would have originated having moved along the bearing $\theta$
over the angular distance $\phi$.

$\begin{split}b_{o}=\mathrm{asin}(\sin b_{*}\cos\phi+\cos
b_{*}\sin\phi\cos\theta)\\\ l_{o}=l_{*}+\mathrm{atan2}(\sin\theta\sin\phi\cos
b_{*},\cos\phi-\\\ \sin b_{*}\sin b_{o})\end{split}$ (4)

We then establish a set of criteria to evaluate whether a star could have
originated from a given group.

  * •

$|l_{o}-l_{g}|<3\sigma_{l,g}$, $|b_{o}-b_{g}|<3\sigma_{b,g}$, where
$\sigma_{lb,g}$ is the standard deviation in $l,b$ of all the identified
members of a group

  * •

$\phi<50$ deg (to prevent the approximation of the great circle from breaking
down at large angles)

  * •

Relative velocity between the star and a moving group $v_{rel}>$5 km s-1

  * •

Traceback time $t_{rel}\equiv\phi/v_{rel}$ (ensuring appropriate unit
conversion) is less than the age of a star ($t_{*}$)

  * •

Traceback time is less than the typical age of a group ($t_{g})$

  * •

$|t_{*}-t_{g}|<$ 3 Myr, OR $|t_{*}-t_{g}|<(\sigma_{t,g})$, where
$\sigma_{t,g}$ is the standard deviation of ages of all of the identified
members of a group (3 Myr was chosen to ensure that very young stars would be
recovered if they have been ejected from a cluster like the ONC which has
sustained its star formation for a few Myr)

  * •

$|1000/\pi_{*}-1000/\pi_{g}|<$100 pc, where $\pi_{*}$ and $\pi_{g}$ are the
parallaxes of a star and of a group (since RVs are typically not available,
this is an arbitrarily small distance that should be possible to be traversed
by an ejected star in a few Myr, within errors)

  * •

A star is not already a member of any of the moving groups

The catalog of all of the identified candidate runaway and walkaway stars is
presented in Table 3

Table 3: Catalog of candiate runaway and walkaway stars Column | Unit | Description
---|---|---
Source |  | Gaia DR3 source id
$l_{*}$ | deg | Galactic longitude
$b_{*}$ | deg | Galactic latitude
$\pi_{*}$ | mas | Gaia DR3 parallax
$\sigma_{\pi,*}$ | mas | Uncertainty in parallax
PMS |  | PMS probability from Sagitta
$\sigma_{PMS}$ |  | Uncertainty in PMS probability
$t_{*}$ | [yr] | (log) Age from Sagitta
$\sigma_{t,*}$ | [yr] | Uncertainty in age
$\mu_{l*}$ | mas yr-1 | Corrected proper motions in l
$\mu_{b*}$ | mas yr-1 | Corrected proper motions in b
Subgroup |  | Name of finer subgroup in a larger population
$\theta$ | deg | Bearing of a stellar motion relative to the subgroup
$\phi$ | deg | Current separation between the star and the subgroup
$v_{rel}$ | km s-1 | Relative velocity between the star and the subgroup
$t_{rel}$ | Myr | Estimated traceback time to the subgroup
$l_{o}$ | deg | Estimated galactic longitude of the point of origin
$b_{o}$ | deg | Estimated galactic latitude of the point of origin
$\xi_{1}$ |  | Angle-based metric for moving group contaminants
$\xi_{2}$ |  | Distance-based metric for moving group contaminants
Wide |  | Index of wide binary candidates
Opposing |  | Index of opposing pairs candidates

###  3.2 Remaining structure identification

The initial clustering analysis of the catalog has allowed to exclude a number
of false positives runaway candidates through limiting a sample of stars that
trace back to other moving groups due to dominant global kinematics in the
region. However, a significant fraction of runaway candidates is still
dominated by such sources. The primary source of contamination are very sparse
moving groups (e.g., Taurus), comprehensive membership of which is difficult
to recover with clustering in a presence of structures that are much more
massive. Additionally, older groups that are in the process of dissolving have
members that may be missed due to these stars having kinematics just outside
of the typical velocity dispersion, or have positions just outside of where
the bulk of the stars in the moving group still reside. Such stars still
follow the typical dominant velocity currents of a region in which they
reside, but they may not necessarily be identified among the “core” members of
a moving group.

The ability to trace young stars in bulk to other regions may help to reveal
the processes that have shaped the velocity currents in a region. However,
when several stars are found in a similar direction and at a similar distance
away from a particular group, such sources are unlikely to have been ejected
from said group. To evaluate the probability of a given source being a viable
ejected star candidate, we derive two metrics, $\xi_{1}$ and $\xi_{2}$.

$\xi_{1}=n_{\Delta\theta<10}/n_{tot}$ (5)

where $n_{\Delta\theta<10}$ is the number stars with the bearing $\theta$
within 10∘ of $\theta$ of a given star, and $n_{tot}$ is the total number of
stars that trace back to a given sub-group. The more concentrated the
distribution of stars is originating from a particular direction, the larger
$\xi_{1}$ becomes, signifying a possibly unaccounted moving group. If
$n_{tot}<12$, then we set $n_{tot}=12$ – this is to prevent extreme cases of
e.g., if only a single star traces back to a particular sub-group, $\xi_{1}$
becomes 1.

$\xi_{2}=\sigma_{\phi_{\theta}}/|\overline{\phi_{\theta}}-\phi|$ (6)

if $n_{\Delta\theta<10}>5$, $\phi_{\theta}$ are the angular distances of all
stars within 10∘ of $\theta$ of a given star, otherwise, if $n_{tot}>5$,
angular distances of all stars tracing back to a given group are used. Average
$\overline{\phi_{\theta}}$ and the standard deviation $\sigma_{\phi_{\theta}}$
are computed using only 10–90 percentile range of the distribution of
$\phi_{\theta}$. This tests for the commonality of angular distances that
different stars trace back over: the greater the similarity, the larger
$\xi_{2}$ becomes. If $n_{tot}\leq 5$, $\xi_{2}$ is set to 1.

Conservatively, we set the threshold of $\xi_{1}\xi_{2}=0.1$ (Figure 2),
separating members of possibly unaccounted of moving groups and possible
runaway/walkwaway stars, chosen through examining all stars tracing back to
individual groups (e.g., Figure 3), which roughly filters out all of the
overdensities in the distribution of sources that are apparent to the eye.

Figure 2: Distribution of $\xi_{1}$ and $\xi_{2}$ parameters for all of the
6803 candidates (see text for definition). Black line corresponds to
$\xi_{1}\xi_{2}<0.1$.  Figure 3: Candidate ejected stars that trace back to
one group ($\alpha$ Per), color-coded by $\xi_{1}\xi_{2}$. The sources with
large $\xi_{1}\xi_{2}$ reveal the presence of two moving groups in this region
that have not been recovered in the initial clustering.

To ensure that neighboring groups do not “shield” each other, preventing
sources to trace towards them from a particular side, initially, a star could
be matched to multiple subgroups. Following calculation of $\xi_{1}\xi_{2}$,
only the group that is located the closest to a given star is chosen. In
total, this selection has identified 6803 candidate runaway and walkaway
stars, of which 1266 have PMS probability $>$0.95. Of these, 3449 have been
excluded by $\xi_{1}\xi_{2}$ test, with only 3354 stars remaining at PMS
probability $>$0.85, and only 546 at PMS probability $>0.95$ (Figure 5).

##  4 Discussion

Figure 4: A comparison of the Gaia DR3 and APOGEE radial velocities of young
stars that appear to have RVs from Kounkel et al. (2019), from Orion, Taurus,
Perseus, and NGC 2264, separated into the disk-bearing and non disk-bearing
stars. A data for the Pleiades is also shown, to highlight the differences
that the age of a star may create in the sample. Left panel shows the direct
comparison of the two sets of RVs. Right panel shows the magnitude of the
difference between the RVs as a function of the reported uncertainty in RV
from Gaia DR3.

###  4.1 Radial Velocity

Currently, the traceback of motion of the ejected candidates is done solely in
the plane of the sky. A full 3d traceback would be able to more definitively
confirm the point of origin of these fast moving pre-main sequence stars, as
RVs are able to affect the overall trajectory of a star away from following a
great circle.

The RV data for PMS stars remain very sparse. The recent release of Gaia DR3
has provided RVs for over 30 million stars (Katz et al., 2022), representing a
major achievement in providing comprehensive data of the radial velocity
structure of the Galaxy. But, the sources for which RVs are available tend to
be very luminous, disfavoring low mass PMS dwarfs. Only $\sim$10% of the base
catalog of PMS stars has reported Gaia DR3 RVs, and their uncertainties tend
to be 5–10 km s-1. Such data are insufficient to even establish an RV
reference frame for most of the populations, let alone to perform a detailed
traceback of all of the individual stars fully in 3D.

Furthermore, evaluating the available Gaia RVs for PMS stars paints a somewhat
grim picture regarding their quality. We compare these RVs versus those that
have been measured by APOGEE in various star forming regions, such as Orion,
Taurus, Perseus, and NGC 2264 (Kounkel et al., 2019). We limit the sample to
only those sources for which at least 3 high resolution spectroscopic
observations exist to ensure RV stability, excluding any of the SB1s, SB2s (or
higher order multiples) that have been identified in that work.

While the APOGEE RVs of these young stars tend to have low velocity dispersion
of less than a few km s-1, same sources result in the velocity dispersion of
several 10s of km s-1 with Gaia RVs. This scatter is not accounted for in the
uncertainties: even though the the typical reported errors in Gaia RVs are
5–10 km s-1, the difference between Gaia and APOGEE RVs on the order of
10$\sigma$ is common (Figure 4).

Class II young stellar objects, i.e., sources that still have infrared excess
due to the presence of a protoplanetary disk, sources that typically have
strong accretion signature in optical spectrum have RVs that are qualitatively
worse than in the sources that have already depleted their disk. Class III
objects have poor RVs as well, however. These sources still have low surface
gravity, and often may have strong activity features that may have been
unaccounted in the spectral fitting. As such, Gaia RVs of nearly all young
stars are unstable.

This issue persists for $>$100 Myr. For example, in Pleiades, where almost all
of the stars are already on the main sequence, but many still show high
activity. While the solar-type stars ($T_{eff}$$\sim$6000 K) have reasonable
RV precision, with the scatter in RVs reproduced by the uncertainties
(typically $<$2 km s-1), the cooler dwarfs in Pleiades ($\sim$1/4 of the
sample) also have RVs with unrealistically large scatter. More evolved field
dwarfs in the APOGEE data that are not associated with any young cluster or a
moving group do not appear to be affected to a similar degree, even in
comparison to the Pleiades. As such, this issue appears to affect younger and
active stars specifically.

As our sample consists entirely of young cool dwarfs, we chose not to use Gaia
RVs even in the cases where these sparse data are available. Furthermore, we
advise extreme caution to other studies that attempt to over-interpret the
observed features of the RV distribution in the young stars (such as Zucker et
al., 2022) – the improper data processing of Gaia spectra provides yet another
source of RV scatter that needs to be considered carefully in determining such
parameters as e.g., velocity dispersion of young moving groups.

Unfortunately, at the moment, no other RV survey provides a comprehensive
coverage of data across all of the candidates. This will eventually change:
over the next five years, SDSS-V is set to obtain high resolution spectra
$\sim$100,000 young stars (Kounkel M., in prep), however, these data are not
yet available. It is also possible that in the future a more careful
independent processing of the spectroscopic data released by Gaia will enable
a more optimal RV extraction of the young stars – once RVS spectra from Gaia
are released in full, or in the subsequent data releases.

Ultimately, a comprehensive confirmation of the candidate ejected stars, fully
in 3d, may not be possible now, but such an analysis will be possible at a
later date.

Figure 5: Candidate ejected stars with PMS$>$0.95 and relative velocity$>$10
km s-1. The vectors show their apparent path from the estimated $l_{o}$ and
$b_{o}$ to their current position (signified by larger dot), they are color-
coded by the traceback travel time. Greyscale background shows the
distribution of the clustered structures.

###  4.2 Comparison to previous works

Previously, McBride & Kounkel (2019), Schoettler et al. (2020), and Farias et
al. (2020) have searched for stars ejected from the Orion Nebula. We compare
the identified candidates from these works to the catalog we derive here
(Figure 6).

McBride & Kounkel (2019) have identified 26 stars among known members of the
ONC that have high proper motions. Of these, due to limits in color imposed by
Sagitta (typically, the recovered YSO candidates have $G_{BP}-G_{RP}>2$ mag),
and due to the quality of photometry that has been required in the creation of
the base catalog, only 5 of these 26 sources are included in the analysis. Of
these 5 sources, 2 appeared to trace back to the center of the ONC, two
appeared to trace back away from the ONC as visitors to other clusters in the
Orion Complex, and one had its origin unknown. In total, only one of them (a
visitor, Gaia EDR3 3017044689550345856) does not meet the initial set of
criteria due to a slight mismatch in age of a star ($\sim$0.5 Myr) and the
traceback time ($\sim$0.6 Myr) – we note that the uncertainty in age is not
considered for the purpose of this exercise. The remaining 4 stars do meet the
initial selection criteria and indeed can be traced back to various parts of
Orion.

Schoettler et al. (2020) have identified 85 candidates that can be traced to
the ONC from up to 100 pc away from it. Of these only 27 can be matched to the
base catalog from McBride et al. (2021), again, usually due to the color
limits. Of these 27 stars, one has been selected as a core member of the Orion
A molecular cloud. 3 stars do not meet the traceback criteria, due to the
traceback time being longer than the age of a star and/or group. 7 stars have
$\xi_{1}\xi_{2}>0.1$, and, by the strictest set of criteria, 16 sources can be
recovered as runaway/walkaway candidates.

Finally, Farias et al. (2020) have identified $\sim$17,000 candidates. The
sources have been initially selected using a variety of different tracers of
youth, including infrared excess, optical variability, position on the HR
diagram, and others. Some of these criteria have a significant degree of
contamination from the more evolved stars, as such, only 331 can be cross-
matched against our base catalog. Of these, 267 can be recovered as core
members of Orion. 22 stars do not meet traceback criteria, commonly because
the conversion to the local standard of rest reference frame has significantly
affected their trajectory in comparison to the heliocentric reference frame
due to the large separation in distance between a star and the cluster, some
due to the traceback time exceeding the age of a star and/or cluster. Of the
remaining 42 stars, 11 have $\xi_{1}\xi_{2}>0.1$, and 31 meet the strictest
set of criteria as walkaway/runaway candidates.

In total, we identify 249 ejected star candidates with $\xi_{1}\xi_{2}<0.1$
that can be traced back to the Orion Complex, of which 117 appear to originate
from the ONC. Of these, 36 and 11 stars respectively have tangential velocity
$>$30 km s-1, and they can be considered as runaways.

Figure 6: A comparison of the trajectories of the ejected candidates in the
works of McBride & Kounkel (2019), Schoettler et al. (2020), and Farias et al.
(2020) that can be matched to the base catalog of PMS stars. Sources are
color-coded based on whether the stars have met the thresholds to be selected
as runaway/walkaway stars that we use in this work. Pale pink vectors show the
trajectories of all sources that have $\xi_{1}\xi_{2}>0.1$ presented here, to
show the dominant velocity currents in the field when evaluating the
identified candidates in each catalog. Grey circles show the area on the sky
corresponding to 3$\sigma_{lb,g}$ for each of the sub-groups in Orion. The
plots are shown in the Galactic coordinates.

###  4.3 Statistics

We examine the properties of the candidate ejected stars in Figure 7. Across
all ages of the stars, the candidates that the selection is most sensitive to
are the sources that appear to have been ejected recently, within the last 2
Myr. Older stars have a longer tail in the distribution of their traceback
times, as the sources that have been ejected early in their formation would
have more time to travel further outward, but they may be more difficult to
detect with the methodology used here.

The overall projected tangential velocity distribution of the identified
candidates does not appear to vary significantly between the older and the
younger stars. As such, the angular distance between the star and its
projected point of origin correlates strongly with their tangential velocity,
with the bulk of the sources being within 10 degrees of their parent group,
and the older sources having a slightly more pronounced excess of sources at
larger distances than the younger sources.

Restricting the selection to the runaways with the relative tangential speeds
$>$30 km s-1 favors the sample with more recent ejections, as they would
quickly disperse away from their parent population, and they may be more
difficult to recover without a traceback that is fully in 3d, since the
approximation of the spherical geometry may not be sufficiently precise from
identifying them at larger distances.

Figure 7: Top: correlation between the projected tangential velocity,
traceback time, and the angular distance for the candidate ejected stars with
$\xi_{1}\xi_{2}<0.1$, color-coded by their age. The size of the symbol
corresponds to the probability of a star being PMS, with the larger circles
corresponding to the higher confidence sources, and smaller circles having a
greater degree of contamination from the field stars. Bottom: kernel density
estimate showing a distribution of tangential velocity and traceback time for
the sample, separated into three different age bins.

###  4.4 Ejected pairs

####  4.4.1 Selection

Figure 8: Top: spatial distribution of the candidate interacting pairs of
stars that appear to have been ejected concurrently from the same location,
travelling in opposite directions. Bottom: candidate wide binary systems that
may have formed in the process of the ejection (note, due to their proximity,
vectors for both stars may overlap). The current position of each star is
highlighted by a large circle. The vectors show their apparent path from the
estimated $l_{o}$ and $b_{o}$ to their current position (signified by larger
dot), they are color-coded by the traceback travel time.

In the process of disrupting a multiple system, all the stars will undergo
acceleration, the precise magnitude of which depends on both the initial
configuration of their orbits as well as the masses of all stars. As such it
is possible to eject two stars simultaneously, each travelling away from their
parent population in opposite directions from each other, similarly to AE Aur
and $\mu$ Col (Blaauw, 1961; Hoogerwerf et al., 2001).

For most ejected stars, finding their precise pair may be a difficult task, as
most such encounters result in low ejection velocities (Reipurth et al.,
2010), not easily distinguishable from their parent association. In the
process of mixing with their neighbors, the initial trajectory of the sources
with low ejection velocity may be erased. This may be particularly common for
systems with low mass ratio, in which the more massive companion is more
likely to remain in a cluster, ejecting a lower mass neighbor.

As such, the probability of both stars being ejected with sufficiently high
velocity for them to be recovered in the catalog presented here is expected to
be relatively low. Nonetheless, there do appear to be certain pairs of
candidates that may have a common origin.

To identify such pairs, we require

  * •

Using the bearing $\theta_{i}$ that has been inverted, measured from
$(l_{o},b_{o})$ to $(l,b)$ (to prevent the distortion from a spherical
geometry that may occur using $\theta$ using the current position of a star as
a point of origin), the difference in $\theta_{i}$ between the pair of stars
should be in the range between 175–185∘, i.e., they are moving in near-
opposite directions within an arbitrarily small precision that would ensure
recovery of AE Aur and $\mu$ Col.

  * •

The traceback time for both stars should be within 10% of each other, to
ensure a probability of a simultaneous event, within uncertainties.

  * •

The distance between their initial traceback positions $(l_{o},b_{o})$ is
$<2^{\circ}$, or $<$10% of their distance at their current positions $(l,b)$,
whichever one is smaller.

  * •

Although currently RVs of the candidates are in large part not available, it
is possible to evaluate the distances of the candidates to reject obvious
false positives. If two stars are travelling in opposite directions, and they
originate from the same star forming region, the distance to their parent
population has to be between the distances of both stars, within the tolerance
for the uncertainties and the depth of the region.

  * •

Both stars have $\xi_{1}\xi_{2}<$0.1.

In total, 42 pairs satisfy these cuts; they are shown in Figure 8. There may
be chance coincidences in this sample, consisting of pairs that only appear to
move opposite of one another, and that do indeed originate from the cluster,
but wouldn’t necessarily trace back to the same system. However, repeating the
above selection for angles other than 175–180∘ results in a typical number of
19 systems that are selected, i.e., there is a significant excess in the
number of systems moving opposite of one another (Figure 9). We further note
that if we change some of the criteria – e.g., keep all of the requirements
regarding the traceback to a given cluster the same, but force the minimum
separation between stars at the initial position to be $>2^{\circ}$, there is
no preferred direction of motion between these random matches. As such, even
though this sample of the opposing pairs may be contaminated by chance
coincidences, we expect a subset of these systems to truly be simultaneous
ejections.

Figure 9: The number of system that are selected using the criteria for the
opposing pairs, but using different requirement for the angle. Typical number
of systems that appear to move relative to one another at random angles, but
project back to a similar origin within a similar period of time is 19 pairs
(shown as red line), there is a significant excess at the angle of 180
degrees, corresponding to the stars moving in the opposite directions.

Additionally, it may be possible for the two of the ejected stars to form a
binary system in the process of the encounter. Compact binaries can be
detected in the future through spectroscopic followup; however, in wide
binaries, both stars may be fully resolved with Gaia, in which case, both
stars are expected to have very similar kinematics, and have very similar
trajectories. An example of such a system are Brun 259 and V1961 Ori (McBride
& Kounkel, 2019), two stars with separation of $\sim 7000$ au that appear to
have been concurrently ejected from the ONC 0.1 Myr ago. This system is
notable, as binary systems with such wide separations are not expected to be
stable in such a dense cluster. We use criteria similar to the above to
identify new candidates:

  * •

$\theta$ of both stars is within 5 degrees of each other.

  * •

The traceback time for both stars should be within 10% of each other.

  * •

Their projected separation is within 10 pc.

  * •

Their parallaxes are consistent with 10 pc separation, within errors.

  * •

Both stars have $\xi_{1}\xi_{2}<$0.1 and $\xi_{1}<$0.1, this cut is stricter,
to further minimize any contamination from the sources that may belong to
various moving groups, however distantly.

This results in a catalog of 19 candidates, shown in Figure 8. Of these, 6
have projected separation within 1 pc. Recently, El-Badry et al. (2021) have
produced a catalog of wide binary candidates across the entire sky, containing
$>$1 million pairs with 3d $<$1 pc. This catalog is able to recover 5 of these
6 stars. We note that cross-matching against full sample sources that meet our
initial set of criteria results produces 697 stars in common, however, there
is a significant contamination in this sample from various moving groups and
cases of clear mismatch in age of the stars in the pair, such as pre-main
sequence star and a much more evolved counterpart, which are unlikely to be
true binaries. As such the selection of 19 pairs is more conservative, however
there may be other systems in the sample that are missed by our selection.

####  4.4.2 Properties

Figure 10: Top: The comparison of mass ratios between the lower and higher
mass star in the system of the opposing pairs, and among the wide binaries.
Bottom: The comparison of the tangential velocity ratios between lower mass
and higher mass star between the opposing pairs, wide binaries, as well as
pairs chosen as moving at random angles relative to one another. The
uncertainties are derived using sqrt(n) approximation.

Figure 11: Kernel density estimate showing a distribution of tangential
velocity and traceback time for the full sample of candidate ejected stars, as
well as the sources that are a part of opposing pairs and wide binaries.

We examine the properties that these stellar pairs have. To estimate masses of
the individual stars, we cross-match the catalog of the identified systems
against TESS Input Catalog (Stassun et al., 2019), which has an estimate of
mass through photometry. We note that these young stars are moving almost
completely horizontally on the HR diagram along the Hayashi tracks, as such
minor discrepancies in the age of the star does not have a substantial effect
on the mass.

Comparing the mass ratio $q$ within each pair, we find that the opposing pairs
tend to have $q$ consistent with being uniform (with the detection bias
against identifying low $q$ systems, due to the typical masses of the stars
recovered in the base catalog). However, wide binaries show a preference
towards unity, almost half of the systems have $q>0.9$, with a significant
deficit of systems with $q$ between 0.7–0.9 in comparison to the opposing
pairs (Figure 10). There may be a slight trend of wide binaries to also be
increasingly more common at lower $q$, but due to the small number of such
systems, and due to the aforementioned detection bias it is difficult to
definitively state it.

We also examine the tangential velocity ratio between the lower mass and
higher mass star in the system (Figure 10). We find that in wide binaries, by
construction of the catalog, velocities of the pairs are very similar, however
the lower mass star tends to have slightly higher speed than the higher mass
one: typically $v_{2}/v_{1}\sim 1.05$. This is unlikely to be caused by the
orbital motion. For a system with the separation of 1 pc, the orbital period
is $\sim$100 Myr, increasing to 3 Gyr for separations of 10 pc, i.e., much
longer than the age of these stars. Rather, any orbit between these stars is
likely to be unstable, and they are unlikely to necessarily be gravitationally
bound. While they most likely have been ejected in the same event,
conservation of angular momentum has imparted a slightly lower mass star
greater velocity, and they will continue to grow further apart. The only
reason why they can be observed as a pair is due to their youth, since their
velocities have not evolved significantly post ejection. Indeed, on average,
these wide binaries tend to be on average younger than the full sample of
candidate ejected stars, and they tend to be marginally slower, as they are
more likely to lose coherence the further they are from their parent
population (Figure 11).

An opposite trend is observed among the opposing pairs: as the precise
location of the ejection within a region is unknown, at larger distances
minute deviations in the initial angle of ejection become less apparent. As
such to get further out, these opposing pairs tend to be somewhat older and
somewhat faster moving than the full sample (Figure 11). And, surprisingly, a
more massive star tends to have a slightly faster velocity than a lower mass
one (Figure 10). Pairs selected through similar set of criteria, but moving
relative to each other at any angle other than 180∘ tend to have a more
symmetric distribution of velocities between $v_{2}$ and $v_{1}$ (Figure 9).
The excess in the opposing pairs is slight, however, and due to a small number
of pairs, the statistical significance is low.

Nonetheless, we consider possibilities of how such an excess of faster moving
massive stars in the pair may arise. One of them is that initially, both stars
may have belonged to an unstable quadruple system. Initially, the least
massive star in the system may have been ejected, removing energy from the
system in the process, making it more compact but still unstable. The second
least massive star would then also be ejected shortly afterwards, and with a
new configuration the resulting ejection would often (e.g. Leigh et al., 2016;
Ryu et al., 2017a, b) produce slightly higher velocities, provided the virial
ratio is close to zero (Leigh et al., 2016). However, a problem with such a
scenario is that an ejection would not be simultaneous, but rather two
separate ejections in a relatively rapid succession of each other, each
launched in a random direction. Such systems may be present among the systems
that meet the criteria for the opposing pairs but have different angles of
motion; future follow up may help to more definitively identify such systems,
separating them from chance coincidences, enabling a way to trace back their
formation history. However, while this scenario can contribute to a “floor” in
the number of pairs across all orientations (Figure 9), this cannot explain
the preference for the pairs to move 180∘ away from one another.

Another method would be possible through a disintegration of a triple system:
through ejecting one of the stars, the newly formed binary system may have
received enough recoil to also have been ejected – both this system and the
ejected stars would be moving 180∘ relative to one another. And, the binary
system would most likely be more massive than the ejected single, even though
individual stars in that binary may be of lower mass, which could be
responsible for the observed velocity signature.

In large part, a companion with separations $<$0.7” is less likely to be
present as a part of the Gaia catalog. Although Gaia DR3 has produced various
catalogs of stellar multiplicity (Gaia Collaboration et al., 2022), including
astrometric, spectroscopic, and eclipsing binaries, they are highly incomplete
(especially given the aforementioned issues with RVs in the sample of these
young stars). As such, a future search for closer companions among these
opposing pairs may better enable understanding their formation.

##  5 Conclusions

In this work we present a first homogeneous search for candidate young
runaway/walkaway stars across the entire sky within the Solar Neighborhood,
resulting in a catalog of 3354 sources. These candidates were selected from a
photometric catalog of pre-main sequence stars that can be separated from the
field stars with a relatively high degree of confidence. These sources can be
traced back to the nearby star forming regions within the lifetime of a star
and their parent association, most commonly within a few Myr. They are also
distinct from the dominant velocity currents of dissolving young moving groups
that are found in their vicinity.

As part of these analysis, we also identify candidate interacting pairs that
appear to have been ejected concurrently in the same event, of which 42 appear
to be travelling in opposite directions, and 19 appear to have formed a wide
binary in the process of their ejection.

The traceback is performed solely in the plane of the sky, as currently only
sparse data are available pertaining to the radial velocity information for
these stars. Despite the wealth of recently released Gaia DR3 wealth of RVs
across the Solar Neighborhood, these data offer poor constraints on the radial
velocity of low mass stars younger than 100 Myr.

As such these candidates require spectroscopic follow-up observations to
enable the unequivocal confirmation of both their youth that would complement
the initial photometric selection, as well as their status as
walkaway/runaways through a full 3d traceback. Nonetheless, once confirmed,
these candidates can offer insight on the initial dynamical conditions within
their parent associations that have led to these stars being ejected.

## Acknowledgements

This work has made use of data from the European Space Agency (ESA) mission
Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. NWCL gratefully acknowledges
the generous support of a Fondecyt Iniciación grant 11180005, as well as
support from Millenium Nucleus NCN19-058 (TITANs) and funding via the BASAL
Centro de Excelencia en Astrofisica y Tecnologias Afines (CATA) grant
PFB-06/2007. NWCL also thanks support from ANID BASAL project ACE210002 and
ANID BASAL projects ACE210002 and FB210003.

## Data Availability

This work uses the data previously made available in McBride et al. (2021).
The full tables are included in the Supplemental Materials and will be made
available on Vizier.

## References

  * Bhat et al. (2022) Bhat A., Irrgang A., Heber U., 2022, arXiv e-prints, p. arXiv:2204.01594
  * Blaauw (1961) Blaauw A., 1961, Bull. Astron. Inst. Netherlands, 15, 265
  * Campello et al. (2013) Campello R. J. G. B., Moulavi D., Sander J., 2013, in Pei J., Tseng V. S., Cao L., Motoda H., Xu G., eds, Advances in Knowledge Discovery and Data Mining. Springer Berlin Heidelberg, Berlin, Heidelberg, pp 160–172
  * Chen et al. (2013) Chen X., et al., 2013, ApJ, 768, 110
  * El-Badry et al. (2021) El-Badry K., Rix H.-W., Heintz T. M., 2021, MNRAS, 506, 2269
  * Farias et al. (2020) Farias J. P., Tan J. C., Eyer L., 2020, ApJ, 900, 14
  * Gaia Collaboration et al. (2018) Gaia Collaboration et al., 2018, A&A, 616, A1
  * Gaia Collaboration et al. (2021) Gaia Collaboration et al., 2021, A&A, 649, A1
  * Gaia Collaboration et al. (2022) Gaia Collaboration et al., 2022, arXiv e-prints, p. arXiv:2206.05595
  * Hoogerwerf et al. (2001) Hoogerwerf R., de Bruijne J. H. J., de Zeeuw P. T., 2001, A&A, 365, 49
  * Katz et al. (2022) Katz D., et al., 2022, arXiv e-prints, p. arXiv:2206.05902
  * Kerr et al. (2021) Kerr R. M. P., Rizzuto A. C., Kraus A. L., Offner S. S. R., 2021, ApJ, 917, 23
  * Kounkel & Covey (2019) Kounkel M., Covey K., 2019, AJ, 158, 122
  * Kounkel et al. (2017) Kounkel M., et al., 2017, ApJ, 834, 142
  * Kounkel et al. (2019) Kounkel M., et al., 2019, AJ, 157, 196
  * Kounkel et al. (2020) Kounkel M., Covey K., Stassun K. G., 2020, AJ, 160, 279
  * Kounkel et al. (2021) Kounkel M., Stassun K. G., Covey K., Hartmann L., Bird J., 2021, arXiv e-prints, p. arXiv:2111.01159
  * Leigh & Geller (2013) Leigh N. W. C., Geller A. M., 2013, MNRAS, 432, 2474
  * Leigh et al. (2016) Leigh N. W. C., Stone N. C., Geller A. M., Shara M. M., Muddu H., Solano-Oropeza D., Thomas Y., 2016, MNRAS, 463, 3311
  * Luhman (2018) Luhman K. L., 2018, AJ, 156, 271
  * Maíz Apellániz et al. (2021) Maíz Apellániz J., Pantaleoni González M., Barbá R. H., 2021, Research Notes of the American Astronomical Society, 5, 232
  * Manwadkar et al. (2020) Manwadkar V., Trani A. A., Leigh N. W. C., 2020, MNRAS, 497, 3694
  * Manwadkar et al. (2021) Manwadkar V., Kol B., Trani A. A., Leigh N. W. C., 2021, MNRAS, 506, 692
  * McBride & Kounkel (2019) McBride A., Kounkel M., 2019, ApJ, 884, 6
  * McBride et al. (2021) McBride A., Lingg R., Kounkel M., Covey K., Hutchinson B., 2021, AJ, 162, 282
  * McInnes et al. (2017) McInnes L., Healy J., Astels S., 2017, The Journal of Open Source Software, 2
  * Platais et al. (2020) Platais I., et al., 2020, AJ, 159, 272
  * Prisinzano et al. (2022) Prisinzano L., et al., 2022, arXiv e-prints, p. arXiv:2206.00249
  * Reipurth et al. (2010) Reipurth B., Mikkola S., Connelley M., Valtonen M., 2010, ApJ, 725, L56
  * Ryu et al. (2017a) Ryu T., Leigh N. W. C., Perna R., 2017a, MNRAS, 467, 4447
  * Ryu et al. (2017b) Ryu T., Leigh N. W. C., Perna R., 2017b, MNRAS, 470, 2
  * Schoettler et al. (2020) Schoettler C., de Bruijne J., Vaher E., Parker R. J., 2020, MNRAS, 495, 3104
  * Schönrich et al. (2010) Schönrich R., Binney J., Dehnen W., 2010, MNRAS, 403, 1829
  * Stassun et al. (2019) Stassun K. G., et al., 2019, AJ, 158, 138
  * Stone & Leigh (2019) Stone N. C., Leigh N. W. C., 2019, Nature, 576, 406
  * Tajiri et al. (2020) Tajiri T., et al., 2020, arXiv e-prints, p. arXiv:2009.12830
  * Tobin et al. (2016) Tobin J. J., et al., 2016, ApJ, 818, 73
  * Tobin et al. (2022) Tobin J. J., et al., 2022, ApJ, 925, 39
  * Zari et al. (2018) Zari E., Hashemi H., Brown A. G. A., Jardine K., de Zeeuw P. T., 2018, A&A, 620, A172
  * Zari et al. (2019) Zari E., Brown A. G. A., de Zeeuw P. T., 2019, A&A, 628, A123
  * Zucker et al. (2022) Zucker C., Peek J. E. G., Loebman S. R., 2022, arXiv e-prints, p. arXiv:2205.14160


Minimum Atom-Bond Sum-Connectivity
Index of Trees With a Fixed Order and/or
Number of Pendent Vertices

Tariq A. Alraqad1, Igor Ž. Milovanović2,
Hicham Saber1, Akbar Ali1,***Corresponding author, Jaya Percival Mazorodze3

1Department of Mathematics, Faculty of Science,
University of Ha’il, Ha’il, Saudi Arabia
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
2Faculty of Electronic Engineering,
University of Niš, Niš, Serbia
<EMAIL_ADDRESS>
3Department of Mathematics,
University of Zimbabwe, Harare, Zimbabwe
<EMAIL_ADDRESS>
Abstract

Let $d_{u}$ be the degree of a vertex $u$ of a graph. The atom-bond sum-
connectivity (ABS) index of a graph $G$ is the sum of the numbers
$(1-2(d_{v}+d_{w})^{-1})^{1/2}$ over all edges $vw$ of $G$. This paper gives
the characterization of the graphs possessing the minimum ABS index among all
trees of a fixed number of pendent vertices. The problem of determining graphs
possessing the minimum ABS index in the class of all trees of a fixed order
and number of pendent vertices is also addressed, and a partial solution to
this problem is provided.
Keywords: topological index; atom-bond sum-connectivity; tree.
AMS Subject Classification: 05C07, 05C90.

##  1 Introduction

A property of a graph that is preserved by isomorphism is known as a graph
invariant [5]. The order and degree sequence of a graph are examples of graph
invariants. The graph invariants that assume only numerical values are usually
referred to as topological indices in chemical graph theory.

For evaluating the extent of branching of the carbon-atom skeleton of
saturated hydrocarbons, Randić [14] devised a topological index and called it
as the branching index, which nowadays is known as the connectivity index
(also, the Randić index). The connectivity index of a graph $G$ is the
following number:

$\sum_{vw\in E(G)}\frac{1}{\sqrt{d_{v}d_{w}}}\,,$

where $d_{v}$ and $d_{w}$ denote the degrees of the vertices $v$ and $w$ of
$G$, respectively, and $E(G)$ is the set of edges of $G$. It is believed that
the connectivity index is the most-studied topological index (in both
theoretical and applied aspects) [9]. Detail about the study of the
connectivity index can be found in the survey papers [12, 13], books [10, 11],
and related papers cited therein.

Because of the success of the connectivity index, many modified versions of
this index have been introduced in the literature. The atom-bond connectivity
(ABC) index [7, 8] and the sum-connectivity (SC) index [17] are among the
well-studied modified versions of the connectivity index. The ABC and SC
indices of a graph are defined as

$ABC(G)=\sum_{vw\in E(G)}\sqrt{\frac{d_{v}+d_{w}-2}{d_{v}d_{w}}}\,,$

and

$SC(G)=\sum_{vw\in E(G)}\frac{1}{\sqrt{d_{v}+d_{w}}}.$

The readers interested in detail about the ABC and SC indices are referred to
the survey papers [1] and [4], respectively.

Using the main idea of the SC index, a modified version of the ABC index was
proposed in [2] recently and it was referred to as the atom-bond sum-
connectivity (ABS) index. The ABS index of a graph $G$ is defined as

$ABS(G)=\sum_{uv\in E(G)}\sqrt{\frac{d_{u}+d_{v}-2}{d_{u}+d_{v}}}=\sum_{uv\in
E(G)}\sqrt{1-\frac{2}{d_{u}+d_{v}}}\,.$

The graphs possessing the maximum/minimum ABS index in class of all (i)
(molecular) trees (ii) general graphs, with a given order, were characterized
in [2]. Analogous results for unicyclic graphs were reported in [3], where the
chemical applicability of the ABS index was also investigated.

A vertex of degree in a graph is called a pendent vertex. Ali et. al. [3]
posed a problem asking to determine trees possessing the minimum value of the
ABS index among all trees with a fixed number of pendent vertices.

The main goal of the present paper is to determine trees possessing the
minimum value of the $ABS$ index in two classes of trees. For positive
integers $n$ and $p$, $\Gamma_{p}$ denotes the class of all trees with $p$
vertices and $\Gamma_{n,p}$ denotes the class of all trees of order $n$ and
$p$. In section 2 we give a complete solution to the problem posed by Ali et.
al. [3], where we show that the star graph $S_{p+1}$ uniquely attain the
minimum value of the $ABS$ index in the class $\Gamma_{p}$. In section 3 we
provide results on trees that minimize the value of the $ABS$ index in
$\Gamma_{n,p}$.

##  2 Trees with a Fixed Number of Pendent Vertices

We will need the next already known result.

###### Lemma 1

[2, Corollary 8] Let $u$ and $v$ be non-adjacent vertices in a connected graph
$G$. Then $ABS(G+uv)>ABS(G)$, where $G+uv$ is the graph obtained from $G$ by
adding the edge $uv$.

###### Lemma 2

For $p\geq 2$ be an greater. If $T^{\ast}$ is a tree attaining the minimum
value of the $ABS$ index in the class $\Gamma_{P}$, then $T^{\ast}$ has no
vertex of degree $2$.

Figure 1: The graph transformation used in the proof of Lemma 2. The subtree
$H_{1}$ may or may not consist of only one vertex $w$.

Proof:  Suppose to the contrary that $T^{\ast}$ has at least one vertex of
degree $2$. Take $v\in V(T^{\ast})$ such that $N(v)=\\{u,w\\}$ and $d_{u}\geq
d_{w}\geq 1$. Let $T^{\prime}$ be the tree formed by removing the vertex $v$
(and its incident edges) and adding the edge $uw$ (see Figure 1). In what
follows, by $d_{x}$ we denote the degree of a vertex $x\in
V(T^{\prime})=V(T^{\ast})$ in $T^{\ast}$, not in $T^{\prime}$. Using the
definition of the ABS index, we have

$ABS(T^{\ast})-ABS(T^{\prime})=\sqrt{1-\frac{2}{d_{u}+2}}+\sqrt{1-\frac{2}{d_{w}+2}}-\sqrt{1-\frac{2}{d_{u}+d_{w}}}~{}>0,$

a contradiction to the assumption that $T^{\ast}$ attains the minimum value of
the atom-bond sum-connectivity index among all trees with $p$ pendent
vertices.

$\square$

###### Lemma 3

The function $f$ defined by

$\displaystyle f(x,y)$
$\displaystyle=(x-1)\left(\sqrt{1-\frac{2}{x+1}}-\sqrt{1-\frac{2}{x+y-1}\,}\,\right)$
$\displaystyle\quad\
+(y-1)\left(\sqrt{1-\frac{2}{y+1}}-\sqrt{1-\frac{2}{x+y-1}\,}\,\right)$
$\displaystyle\quad\ +\sqrt{1-\frac{2}{x+y}}$

with $x\geq y\geq 3$, is strictly decreasing in $y$.

Proof:  We have

$\displaystyle\frac{\partial f}{\partial y}$
$\displaystyle=\frac{1}{(x+y)^{3/2}\sqrt{x+y-2}}-\frac{x+y-2}{(x+y-1)^{3/2}\sqrt{x+y-3}}$
$\displaystyle\quad\ -\sqrt{\frac{x+y-3}{x+y-1}}+g(y),$ (1)

where

$g(y)=\frac{y+2}{y+1}\sqrt{\frac{y-1}{y+1}}\,.$

Certainly, the function $g$ is strictly increasing because $y\geq 3$. Hence,
$g(y)<g(x+y-2)$ as $x\geq 3$. Consequently, Equation (2) gives

$\displaystyle\frac{\partial f}{\partial y}$
$\displaystyle<\frac{1}{(x+y)^{3/2}\sqrt{x+y-2}}-\frac{x+y-2}{(x+y-1)^{3/2}\sqrt{x+y-3}}$
$\displaystyle\quad\
-\sqrt{\frac{x+y-3}{x+y-1}}+\frac{x+y}{x+y-1}\sqrt{\frac{x+y-3}{x+y-1}}$
$\displaystyle=\frac{1}{(x+y)^{3/2}\sqrt{x+y-2}}-\frac{1}{(x+y-1)^{3/2}\sqrt{x+y-3}}$
(2)

Since the function $\psi$ defined by

$\psi(t)=\frac{1}{t^{3/2}\sqrt{t-2}}$

is strictly decreasing for $t>2$, from (2) it follows that $\frac{\partial
f}{\partial y}<0$.

$\square$

###### Theorem 1

Let $p\geq 2$ be an integer. Then for every $T\in\Gamma_{p}$,

$ABS(T)\geq p\sqrt{\frac{p-1}{p+1}},$

with equality holds if and only if $T\cong S_{p+1}$.

Figure 2: The graph transformation used in the proof of Theorem 1. For
$i\in\\{1,\ldots,d_{u}-1\\}$ and $j\in\\{1,\ldots,d_{v}-1\\}$, the subtree
$J_{i}$ may or may not consist of only one vertex $u_{i}$ and the subtree
$H_{j}$ may or may not consist of only one vertex $v_{j}$, respectively.

Proof:  Let $T$ be a graph attaining the minimum $ABS$ value among all trees
with $p$ pendent vertices. By Lemma 2, $T$ has no vertex of degree $2$. We
claim that $T$ has only one vertex of degree greater than $2$. Contrarily, we
assume that $T$ contains at least two vertices of degree greater than $2$.
Among the vertices of degrees at least $3$, we pick $u,v\in V(T)$ such that
$uv\in E(G)$ (Lemma 2 guaranties the existence of the vertices $u$ and $v$
when $T$ has at least one pair of vertices of degrees greater than $2$).
Without loss of generality, we suppose that $d_{u}\geq d_{v}$. Let
$v_{1},\cdots,v_{d_{v}-1}$ be the neighbors of $v$ different from $u$.
Construct a new tree $T^{\prime}$ by dropping the vertex $v$ (and its incident
edges) and inserting the edges $v_{1}u,\cdots,v_{d_{v}-1}u$. Certainly, both
the trees $T$ and $T^{\prime}$ have the same number of pendent vertices.
However, in the following, we show that $ABS(T)>ABS(T^{\prime})$, which gives
a contradiction to minimality of $ABS(T)$ and hence $T$ must contain exactly
one vertex of degree greater than $2$, as desired.

In what follows, by $d_{w}$ we denote the degree of a vertex $w\in
V(T^{\prime})=V(T)$ in $T$, not in $T^{\prime}$. If $u_{1},\cdots,u_{d_{u}-1}$
are the neighbors of $u$ different from $v$, then

$\displaystyle ABS(T)-ABS(T^{\prime})$
$\displaystyle=\sum_{i=1}^{d_{u}-1}\left(\sqrt{1-\frac{2}{d_{u}+d_{u_{i}}}}-\sqrt{1-\frac{2}{d_{u}+d_{v}+d_{u_{i}}-2}\,}\,\right)$
$\displaystyle\quad\
+\sum_{j=1}^{d_{v}-1}\left(\sqrt{1-\frac{2}{d_{v}+d_{v_{j}}}}-\sqrt{1-\frac{2}{d_{u}+d_{v}+d_{v_{j}}-2}\,}\,\right)$
$\displaystyle\quad\ +\sqrt{1-\frac{2}{d_{u}+d_{v}}}\,.$ (3)

Note that the function $\psi$ defined by

$\psi(x,y,z)=\sqrt{1-\frac{2}{x+y}}-\sqrt{1-\frac{2}{x+y+z-2}\,}$

with $\min\\{y,z\\}\geq 3$ and $x\geq 1$, is increasing in $x$ because the
function $\phi$ defined by

$\phi(t)=\sqrt{1-\frac{2}{t}}$

is concave for $t>2$ and

$\psi(x,y,z)=\phi(x+y)-\phi(x+y+z-2).$

Thus, Equation (2) yields

$\displaystyle ABS(T)-ABS(T^{\prime})$
$\displaystyle\geq(d_{u}-1)\left(\sqrt{1-\frac{2}{d_{u}+1}}-\sqrt{1-\frac{2}{d_{u}+d_{v}-1}\,}\,\right)$
$\displaystyle\quad\
+(d_{v}-1)\left(\sqrt{1-\frac{2}{d_{v}+1}}-\sqrt{1-\frac{2}{d_{u}+d_{v}-1}\,}\,\right)$
$\displaystyle\quad\ +\sqrt{1-\frac{2}{d_{u}+d_{v}}}\,.$ (4)

Since $d_{u}\geq d_{v}$, by Lemma 3, Inequality (2) gives

$\displaystyle ABS(T)-ABS(T^{\prime})$ $\displaystyle\geq
2(d_{u}-1)\left(\sqrt{1-\frac{2}{d_{u}+1}}-\sqrt{1-\frac{2}{2d_{u}-1}\,}\,\right)$
$\displaystyle\quad\ +\sqrt{1-\frac{1}{d_{u}}}\,.$ (5)

Note that the function $f$ defined by

$f(s)=2(s-1)\left(\sqrt{1-\frac{2}{s+1}}-\sqrt{1-\frac{2}{2s-1}\,}\,\right)+\sqrt{1-\frac{1}{s}}\,,$

with $s\geq 3$, is strictly decreasing and its series expansion at $s=\infty$
is

$\frac{7}{4s}-\frac{21}{8s^{2}}+\frac{77}{64s^{3}}-\frac{271}{128s^{4}}+\frac{283}{512s^{5}}-\frac{2525}{1024s^{6}}+O\left(\left(\frac{1}{s}\right)^{7}\right).$

Therefore,

$\lim_{s\rightarrow\infty}f(s)=0$

and hence the right-hand side of (2) is positive for $d_{u}\geq 3$. This
completes the proof.

$\square$

##  3 Trees With a Fixed Order and Pendent Vertices

In this section, we characterize trees attaining the minimum value of the ABS
index in $\Gamma_{n,p}$, where $p\geq 3$ and $n\geq 3p-2$.

###### Lemma 4

Let

$f(x,y)=\sqrt{1-\dfrac{2}{x+y}}\,,$

and define the function $g_{s}$ as $g_{s}(x,y)=f(x+s,y)-f(x,y)$, where
$\min\\{x,y\\}\geq 1$. Then $g_{s}(x,y)$ is strictly decreasing in $x$ and in
$y$.

Proof:  The first partial derivatives function of $f$ are calculated as

$\frac{\partial f}{\partial x}(x,y)=\frac{\partial f}{\partial
y}(x,y)=(x+y-2)^{-\frac{1}{2}}(x+y)^{-\frac{3}{2}},$

which are both strictly decreasing in $x$. This implies that

$\frac{\partial g_{s}}{\partial x}(x,y)=\frac{\partial f}{\partial
x}(x+s,y)-\frac{\partial f}{\partial x}(x,y)<0$

and

$\frac{\partial g_{s}}{\partial y}(x,y)=\frac{\partial f}{\partial
y}(x+s,y)-\frac{\partial f}{\partial y}(x,y)<0.$

Thus $g_{s}(x,y)$ is strictly decreasing in $x$ and in $y$. $\square$

###### Lemma 5

If $y$ is a fixed real number greater than or equal to $3$ then the function
$f$, defined in Lemma 4, is strictly increasing in $x$.

For a tree $T$, denote by $W_{1}(T)$ the set of pendent vertices of $T$,
$W_{2}(T)=\cup_{v\in W_{1}(T)}N(v)$, and $W_{3}(T)=V(T)\setminus(W_{1}(T)\cup
W_{2}(T))$. A vertex in $T$ of degree at least three is called a branching
vertex. A path $v_{1}-v_{2}-...-v_{r}$ is called an internal path, if $v_{1}$
and $v_{r}$ are branching vertices and every other vertex has degree two.

###### Lemma 6

Let $T^{\ast}\in\Gamma_{n,p}$ attaining minimum $ABS$ value. Then every
internal path of $T^{\ast}$ has length one.

Proof:  For a contradiction, assume $T^{\ast}$ contains an internal path
$v=v_{0}-v_{1}-v_{2}-..-v_{r}=u$ of length $r\geq 2$. Let $w\in W_{2}(T)$ and
let $y$ be a pendent vertex adjacent to $w$. Let
$T^{\prime}=T^{\ast}-\\{vv_{1},uv_{r-1}\\}+\\{vu,yv_{1}\\}$. Then
$T^{\prime}\in\Gamma_{n,p}$. If $r>2$, then

$\displaystyle ABS(T^{\prime})-ABS(T^{\ast})$
$\displaystyle=f(d_{v},d_{u})+f(2,2)+f(d_{w},2)-f(d_{v},2)$
$\displaystyle-f(d_{u},2)-f(2,2)+f(1,2)-f(d_{w},1)$
$\displaystyle=(f(d_{v},d_{u})-f(d_{v},2))-(f(2,d_{u})-f(2,2))$
$\displaystyle+(f(d_{w},2)-f(d_{w},1))-(f(2,2)-f(2,1))$
$\displaystyle=g_{d_{u}-2}(2,d_{v})-g_{d_{u}-2}(2,2)+g_{1}(1,d_{w})-g_{1}(1,2)<0,$

which yields a contradiction. If $r=2$ then

$\displaystyle ABS(T^{\prime})-ABS(T^{\ast})$
$\displaystyle=f(d_{v},d_{u})+f(2,1)+f(d_{w},2)-f(d_{v},2)-f(d_{u},2)-f(d_{w},1)$
$\displaystyle=(f(d_{v},d_{u})-f(d_{v},2))-(f(2,d_{u})-f(2,2))$
$\displaystyle+(f(d_{w},2)-f(d_{w},1))-(f(2,2)-f(2,1))$
$\displaystyle=g_{d_{u}-2}(2,d_{v})-g_{d_{u}-2}(2,2)+g_{1}(1,d_{w})-g_{1}(1,2)<0.$

Again this contradicts the minimality of $ABS(T^{\ast})$. Thus $T^{\ast}$ does
not have an internal path of length grater than one. $\square$

###### Lemma 7

Let $T^{\ast}\in\Gamma_{n,p}$ attaining minimum $ABS$ value. Then $d_{u}\leq
d_{v}$ for every $u\in W_{2}(T^{\ast})$ and $v\in W_{3}(T^{\ast})$.

Proof:  Let $y$ be a pendent vertex adjacent to $u$ and let $x$ be a non
pendent vertex adjacent to $v$ and not on the $u-v$ path. Let
$T^{\prime}=T^{\ast}-\\{xz\mid z\in N(x)\setminus\\{v\\}\\}+\\{zy\mid z\in
N(x)\setminus\\{v\\}\\}$. Clearly, $T^{\prime}\in\Gamma_{n,p}$, and so
$ABS(T^{\prime})\geq ABS(T^{\ast})$. Thus we have,

$0\leq
ABS(T^{\prime})-ABS(T^{\ast})=f(d_{v},1)-f(d_{v},d_{x})+f(d_{u},d_{x})-d(d_{u},1)=g_{d_{x}-1}(1,d_{u})-g_{d_{x}-1}(1,d_{v}).$

Hence $d_{u}\leq d_{v}$. $\square$

###### Lemma 8

Let $T\in\Gamma_{n,p}$. If $n=3p-2+t$ for some $t\geq 0$, then the following
assertions holds.

  1. 1.

$d_{v}=2$ for all $v\in W_{2}(T)$

  2. 2.

$\sum_{u\in W_{3}(T)}d_{u}=3p-6+2t.$

Proof:  (1) We have $p+\sum_{v\in W_{2}(T)}d_{v}+\sum_{u\in
W_{3}(T)}d_{u}=6p-6+2t$. Seeking a contradiction, assume $d_{v_{0}}\geq 3$ for
some ${v_{0}}\in W_{2}(T)$. Let $\beta=min\\{d_{u}\mid u\in W_{3}(T)\\}$. Then
by Lemma 7, $\beta\geq d_{v_{0}}\geq 3$. On the other hand, since $d_{v}\geq
v$ for all $v\in W_{2}(T)$, we have

$p+2(|W_{2}(T)|-1)+3+\beta(2p-2+t-|W_{2}(T)|)\leq 5p-6+2t.$

So
$\beta\leq\frac{5p-7+2t-2|W_{2}(T)|}{2p-2+t-|W_{2}(T)|}=\frac{p-3}{2p-2+t-|W_{2}(T)|}+2<3$,
a contradiction. Thus $d_{v}=2$ for all $v\in W_{2}(T)$.

(2) We conclude from part (1) that $|W_{2}(T)|=p$. So $p+2p+\sum_{u\in
W_{3}(T)}d_{u}=6p-6+2t$, which yields $\sum_{u\in W_{3}(T)}d_{u}=3p-6+2t$
$\square$

###### Theorem 2

Let $p\geq 3$ and $n\geq 3p-2$. If $T^{\ast}\in\Gamma_{n,p}$ attaining the
minimum $ABS$ value, then the maximum degree of $T^{\ast}$ is $3$.

Proof:  For a contradiction, assume $T^{\ast}$ has a vertex of degree at least
$4$. Let $\beta=min\\{d_{v}\mid v\in W_{3}(T)\\}$. Then, by Lemma 8 part (2),
$4+\beta(|W_{3}(T^{\ast})|-1)\leq 3p-6+2t$. So
$\beta\leq\frac{3p-10+2t}{p-3+t}<3$. So there is a vertex $u\in
W_{3}(T^{\ast})$ such that $d_{u}=2$. We select $u$ so that one of its
neighbors has degree at least three. Now let $v$ be the farthest vertex from
$u$ that has degree at least $4$. Clearly, for each vertex $x\in N(v)$ that
does not lie on the $u-v$ path, we have $d_{x}\leq 3$.
Case 1. $u\in N(v)$. In this case choose $y\in N(v)\setminus\\{u\\}$ and let
$T^{\prime}=T^{\ast}-\\{yv\\}+\\{yu\\}$. Then

$\displaystyle ABS(T^{\prime})-ABS(T^{\ast})=$ $\displaystyle
f(d_{y},3)-f(d_{y},d_{v})+f(2,3)-f(2,2)+f(3,d_{v}-1)-f(2,d_{v})$
$\displaystyle+\sum_{x\in
N(v)\setminus\\{y\\}}(f(d_{x},d_{v}-1)-f(d_{x},d_{v}))$ $\displaystyle\leq$
$\displaystyle
f(3,3)+f(2,3)-f(2,2)-f(3,d_{v})-(d_{v}-2)(f(3,d_{v})-f(3,d_{v}-1))$

Consider the function
$h_{1}(t)=f(3,3)+f(2,3)-f(3,t)-f(2,2)-(t-2)(f(3,t)-f(3,t-1))$.

$h_{1}^{\prime}(t)=\frac{A_{1}-B_{1}}{\sqrt{t(t+1)}(t+2)^{2}(t+3)^{2}},$

where $A_{1}=(t^{2}+3t-2)(t+3)^{2}\sqrt{(t+1)(t+2)}$ and
$B_{1}=(t^{2}+5t+2)(t+2)^{2}\sqrt{t(t+3)}$. Since
$A^{2}_{1}-B_{1}^{2}=-14t^{5}-137t^{4}-456t^{3}-577t^{2}-140t+108\leq 0$, for
$t\geq 1$, we get $h_{1}^{\prime}(t)\leq 0$ for $t\geq 1$, and thus $h_{1}(t)$
is strictly decreasing for $t\geq 1$. This implies that
$ABS(T^{\prime})-ABS(T^{\ast})<h_{1}(d_{v})\leq h_{1}(4)<0$, a contradiction.

Case 2. $u\not\in N(v)$. Let $w\in N(v)$ and $z\in N(u)$ such that $w$ and $z$
lie on the $u-v$ path. Now we divide this case to four subcases.

Subcase 2.1. $d_{v}\geq 5$. Let $y\in N(v)\setminus\\{w\\}$ and take
$T^{\prime}=T^{\ast}-\\{yv\\}+\\{yu\\}$. Then $T^{\prime}\in\Gamma_{n,p}$.
Moreover,

$\displaystyle ABS(T^{\prime})-ABS(T^{\ast})=$ $\displaystyle
f(d_{y},3)-f(d_{y},d_{v})+f(2,3)-f(2,2)+f(3,3)-f(3,2)$
$\displaystyle+\sum_{x\in
N(v)\setminus\\{y\\}}(f(d_{x},d_{v}-1)-f(d_{x},d_{v}))$ $\displaystyle\leq$
$\displaystyle 2f(3,3)-f(3,d_{v})-f(2,2)-(d_{v}-2)(f(3,d_{v})-f(3,d_{v}-1))$
$\displaystyle-(f(d_{w},d_{v})-f(d_{w},d_{v}-1)$

The function $h_{2}(t)=2f(3,3)-f(3,t)-f(2,2)-(t-2)(f(3,t)-f(3,t-1))$ is
strictly decreasing for $t\geq 1$, and so
$ABS(T^{\prime})-ABS(T^{\ast})<h_{2}(5)<0,$ a contradiction.

Subcase 2.2. $d_{v}=4$ and $d_{w}\leq 3$. Let $y\in N(v)\setminus\\{w\\}$ and
take $T^{\prime}=T^{\ast}-\\{yv\\}+\\{yu\\}$. Then
$T^{\prime}\in\Gamma_{n,p}$. Moreover,

$\displaystyle ABS(T^{\prime})-ABS(T^{\ast})=$ $\displaystyle
f(d_{y},3)-f(d_{y},d_{v})+f(2,3)-f(2,2)+f(3,3)-f(3,2)$
$\displaystyle+\sum_{x\in
N(v)\setminus\\{y\\}}(f(d_{x},d_{v}-1)-f(d_{x},d_{v}))$ $\displaystyle\leq$
$\displaystyle 2f(3,3)-f(3,d_{v})-f(2,2)-(d_{v}-1)(f(3,d_{v})-f(3,d_{v}-1))$
$\displaystyle-(f(d_{w},d_{v})-f(d_{w},d_{v}-1)$

Consider the function

$h_{3}(t)=2f(3,3)-f(3,t)-f(2,2)-(t-1)(f(3,t)-f(3,t-1))$

.

We have

$h_{3}^{\prime}(t)=\frac{A_{3}-B_{3}}{\sqrt{t(t+1)}(t+2)^{2}(t+3)^{2}},$

where $A_{3}=(t^{2}+3t-1)(t+3)^{2}\sqrt{(t+1)(t+2)}$ and
$B_{3}=(t^{2}+5t+3)(t+2)^{2}\sqrt{t(t+3)}$. Since
$A_{3}^{2}-B_{3}^{2}=-(t+2)(t+3)(10t^{5}+97t^{4}+328t^{3}+447t^{2}+180t-27)\leq
0$, for $t\geq 1$, we get $h_{3}^{\prime}(t)\leq 0$ for $t\geq 1$, and thus
$h_{3}(t)$ is strictly decreasing for $t\geq 1$. Hence
$ABS(T^{\prime})-ABS(T^{\ast})<h_{3}(d_{v})\leq h_{3}(4)<0$, a contradiction.

Subcase 2.3: $d_{v}=4$ and $d_{w}\geq 7$. Note that $d_{x}\leq 4$ for each
$x\in N(w)\setminus\\{w_{1}\\}$, because otherwise we reach a contradiction as
in Subcase 2.1. Let $T^{\prime}=T^{\ast}-\\{vw\\}+\\{vu\\}$. Then

$\displaystyle ABS(T^{\prime})-ABS(T^{\ast})=$ $\displaystyle
f(4,3)-f(4,d_{4})+f(3,3)-f(2,2)$ $\displaystyle+\sum_{x\in
N(v)\setminus\\{v\\}}(f(d_{x},d_{v}-1)-f(d_{x},d_{v}))$ $\displaystyle<$
$\displaystyle
f(4,3)-f(4,d_{w})+f(3,3)-f(2,2)-(d_{w}-2)(f(4,d_{v})-f(4,d_{v}-1))$

The function

$h_{4}(t)=f(4,3)-f(4,t)-+f(3,3)-f(2,2)-(t-2)(f(4,t)-f(4,t-1))$

is strictly decreasing for $t\geq 1$. Thus
$ABS(T^{\prime})-ABS(T^{\ast})<h_{4}(7)<0$, a contradiction.

Subcase 2.4: $d_{v}=4$ and $4\leq d_{w}\geq 6$. Let $w_{1}\in N(w)$ that lies
on the $u-w$ path, and let $b_{1},...,b_{k}$ be all vertices in
$N(w)\setminus\\{w_{1}\\}$ of degree $4$. Then by Lemma 8 Part (2), there are
$k+1$ distinct vertices $x_{i}\in W_{3}(T^{\ast})\\{u\\}$ such that
$d_{x_{i}}=2$. The vertex $x_{k+1}$ exists because $d_{w}\geq 4$. We may
assume these vertices form a path $u=x_{0}-x_{1}-...-x_{k}-x_{k+1}$. If
$x_{k+1}\in N(v)$, then we get a case similar to case (1). For each
$i=1,...,k$ select $c_{i}\in N(b_{i})\setminus\\{w\\}$ and let
$T^{\prime}=T^{\ast}-(\\{c_{i}b_{i},i=1,...,k\\}\cup\\{vw\\})+(\\{c_{i}x_{i},i=1,...,k\\}\cup\\{vu\\})$.
Note that since $d(c_{i},u)>d(u,v)$, we have $d_{c_{i}}\leq 3$. Thus

$\displaystyle ABS(T^{\prime})-ABS(T^{\ast})\leq$
$\displaystyle(4k+1)f(3,3)-3kf(3,4))-kf(2,2)-f(4,d_{w})$
$\displaystyle-(k-2)(f(4,d_{w})-f(3,d_{w}-1))$
$\displaystyle-(d_{w}-k-1)(f(3,d_{w})-f(3,d_{w}-1))$

Calculations shows that the right hand side of this inequality is negative
when $d_{w}\in\\{4,5,6\\}$ and $k\in\\{0,...,d_{w}\\}$. This also yields a
contradiction. Therefore the maximum degree of $T^{\ast}$ is $3$ as desired.
$\square$

Now we are ready to characterize threes that minimize $ABS$ index in
$\Gamma_{n,p}$ where $p\geq 3$ and $n\geq 3p-2$. For $p\geq 3$ and $n=3p-2+t$
with $t\geq 0$, let $\Gamma_{n,p}^{*}\subset\Gamma_{n,p}$ such that an
arbitrary tree $T^{\ast}$ belonging to the class $\Gamma_{n,p}^{*}$ is defined
as follows.
(1) Let $T_{0}$ be a tree of order $p-2$ and maximum degree three.
(2) Construct a tree $T_{1}$ from $T_{0}$ by attaching $3-i$ pendent path(s)
of length two at each vertex of degree $i$ for $i=1,2,3$. (Note that the order
of $T_{1}$ is $5n_{1}(T_{0})+3n_{2}(T_{0})+n_{3}(T_{0})$, which is equal to
$3p-2$ because $n_{1}(T_{0})+n_{2}(T_{0})+n_{3}(T_{0})=p-2$ and
$n_{1}(T_{0})+2n_{2}(T_{0})+3n_{3}(T_{0})=2(p-3)$, where $n_{k}(T_{0})$
denotes the number of vertices of degree $k$ in $T_{0}$.)
(3) Finally $T^{\ast}$ is obtained from $T_{1}$ by inserting $t$ vertices of
degree two into one or more pendent paths.

Figure 3 gives a tree of the class $\Gamma_{n,p}^{*}$.

Figure 3: A tree attaining the minimum value of the ABS index in the class
$\Gamma_{n,p}$ where $p\geq 3$ and $n\geq 3p-2$.

###### Theorem 3

Let $p\geq 3$ and $n=3p-2+t$ where $t\geq 0$. If $T\in\Gamma_{n,p}$ then

$ABS(T)\geq\left(\frac{2}{\sqrt{6}}+\sqrt{\frac{3}{5}}+\frac{1}{\sqrt{3}}\right)p+\frac{\sqrt{2}}{2}t+\frac{4}{\sqrt{6}},$

with equality if and only if $T\in\Gamma_{n,p}^{*}$.

Proof:  Let $T^{\ast}\in\Gamma_{n,p}$ having the minimum $ABS$ value. Then by
Theorem 2, the maximum degree in $T$ is three. For $i,j=1,2,3$ let $n_{i,j}$
be the number of edges in $T^{\ast}$ that joints vertices of degrees $i$ and
$j$. Clearly, $n_{1,1}=0$. From Lemma 8 (Part 1), we get $n_{1,2}=p$ and
$n_{1,3}=0$. Let $d$ be the number of vertices of degree $3$. Then by Lemma 8
(Part 2), we get $2(p-2+t-d)+3d=3p-6+2t$. Thus $d=p-2$. Since $T$ has no
internal paths of length more than one, then the induced subgraph of $T$ over
the set of vertices of degree three is a tree on $p-2$ vertices. Thus
$n_{3,3}=p-3$. Also we get $n_{2,3}+2n_{3,3}=3(p-2)$, and hence $n_{2,2}=p$.
By subtracting the values $n_{1,2}=p$, $n_{2,3}=p$, $n_{3,3}=p-3$ from
$n-1=3p-3+t$, we get $n_{2,2}=t$. Thus

$\displaystyle ABS(T^{\ast})$
$\displaystyle=pf(1,2)+tf(2,2)+pf(2,3)+(p-2)f(3,3)$
$\displaystyle=\left(\frac{2}{\sqrt{6}}+\sqrt{\frac{3}{5}}+\frac{1}{\sqrt{3}}\right)p+\frac{\sqrt{2}}{2}t+\frac{4}{\sqrt{6}}.$

$\square$

##  4 Funding Statement

This work is supported by Scientific Research Deanship, University of Ha’il,
Saudi Arabia, through project number RG-22 005.

##  5 Data Availability Statement

Data about this study may be requested from the authors.

##  6 Conflict-of-Interest Statement

The authors declare that they do not have any conflict of interest.

## References

  * [1] A. Ali, K. C. Das, D. Dimitrov, B. Furtula, Atom-bond connectivity index of graphs: a review over extremal results and bounds, Discrete Math. Lett. 5 (2021) 68–93.
  * [2] A. Ali, B. Furtula, I. Redžepović, I. Gutman, Atom-bond sum-connectivity index, J. Math. Chem. 60 (2022) 2081–2093.
  * [3] A. Ali, I. Gutman, I. Redžepović, Atom-bond sum-connectivity index of unicyclic graphs and some applications, Electron. J. Math. 5 (2023) 1–7.
  * [4] A. Ali, L. Zhong, I. Gutman, Harmonic index and its generalization: extremal results and bounds, MATCH Commun. Math. Comput. Chem. 81 (2019) 249–311.
  * [5] J. L. Gross, J. Yellen, Graph Theory and Its Applications, Second Edition, CRC Press, 2005.
  * [6] J. A. Bondy, U. S. R. Murty, Graph Theory, Springer, 2008.
  * [7] E. Estrada, L. Torres, L. Rodríguez, I. Gutman, An atom-bond connectivity index: modelling the enthalpy of formation of alkanes, Indian J. Chem. Sec. A 37 (1998) 849–855.
  * [8] E. Estrada, Atom-bond connectivity and the energetic of branched alkanes, Chem. Phys. Lett. 463 (2008) 422–425.
  * [9] I. Gutman, Degree-based topological indices, Croat. Chem. Acta 86 (2013) 351–361.
  * [10] I. Gutman, B. Furtula (Eds.), Recent Results in the Theory of Randić Index, Univ. Kragujevac, Kragujevac, 2008.
  * [11] X. Li, I. Gutman, Mathematical Aspects of Randić-Type Molecular Structure Descriptors, Univ. Kragujevac, Kragujevac, 2006.
  * [12] X. Li, Y. Shi, A survey on the Randić index, MATCH Commun. Math. Comput. Chem. 59 (2008) 127–156.
  * [13] M. Randić, The connectivity index 25 years after, J. Mol. Graph. Model. 20 (2001) 19–35.
  * [14] M. Randić, M. On characterization of molecular branching, J. Am. Chem. Soc. 97 (1975) 6609–6615.
  * [15] Y. Tang, D. B. West, B. Zhou, Extremal problems for degree-based topological indices, Discrete Appl. Math. 203 (2016) 134–143.
  * [16] S. Wagner, H. Wang, Introduction to Chemical Graph Theory, CRC Press, Boca Raton, 2018.
  * [17] B. Zhou, N. Trinajstić, On a novel connectivity index, J. Math. Chem. 46 (2009) 1252–1270.