# Strong log-convexity of genus sequences

Bojan Mohar
Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada
Supported in part by an NSERC Discovery Grant R832714 (Canada), and in part by
the ERC Synergy grant KARST (European Union, ERC, KARST, project number
101071836).On leave from: FMF, Department of Mathematics, University of
Ljubljana, Ljubljana, Slovenia.

###### Abstract

For a graph $G$, and a nonnegative integer $g$, let $a_{g}(G)$ be the number
of $2$-cell embeddings of $G$ in an orientable surface of genus $g$ (counted
up to the combinatorial homeomorphism equivalence). In 1989, Gross, Robbins,
and Tucker [21] proposed a conjecture that the sequence
$a_{0}(G),a_{1}(G),a_{2}(G),\dots$ is log-concave for every graph $G$. This
conjecture is reminiscent to the Heron–Rota–Welsh Log Concavity Conjecture
that was recently resolved in the affirmative by June Huh et al. [24, 1],
except that it is closer to the notion of $\Delta$-matroids than to the usual
matroids. In this short paper, we disprove the Log Concavity Conjecture of
Gross, Robbins, and Tucker by providing examples that show strong deviation
from log-concavity at multiple terms of their genus sequences.

##  1 Introduction

The _genus sequence_ (also called the _genus distribution_) of a graph $G$ is
the sequence $(a_{g}(G))_{g\geq 0}$, whose term $a_{g}(G)$ gives the number of
2-cell embeddings of $G$ in the orientable surface of genus $g$ (up to ambient
homeomorphisms). Due to a variety of applications within mathematics,
theoretical physics and theoretical computer science (see, e.g., [29]), there
is abundant literature on the genus distribution of graphs. The earlier papers
were mainly concerned with the _average genus_ computations [40, 2]. One of
the initial motivations for considering the variety of all 2-cell embeddings
of a graph was the possibility of performing local optimization for the
minimum genus problem, with the hope to obtain good approximations (see, e.g.,
[17] or [48, 10]). Later it became evident that genus distributions are
intimately related to the enumeration of rooted maps on surfaces [32]. We
refer to [29] for extensive treatment of map enumeration and its applications
in different areas of mathematics, such as theoretical physics (quantum field
theory, string theory, Feynman diagrams, the KdV equation), matrix integrals,
algebraic number theory (algebraic curves, Galois theory via Grothendiek’s
_dessins d’enfants_ , geometry of moduli spaces of complex algebraic curves),
singularity theory. Even the very special case of one-vertex graphs (bouquet
of loops) bears interesting applications, in particular the relationship to
the Hopf algebra of chord diagrams and algebraic invariants of knots and links
[29]. The one-vertex maps are isomorphic, via surface duality, to unicellular
maps that have received a lot of attention, see, e.g. [3, 7, 8].

Embedding distributions for specific families have been the subject of
extensive research. Families containing vertices of arbitrarily large degrees
include bouquets of loops [21, 9, 28], dipoles [27, 45], bouquet of dipoles
[26], and others [39]. More often, families consist of graphs whose structure
resembles a long path (see, e.g., [16, 41, 42, 43, 30, 32, 22, 46]) or a long
cycle ([38, 49]); in such structures, the vertices of a path or a cycle are
replaced by copies of a small graph.

While considering embedding distributions of various infinite families, Gross,
Robbins, and Tucker [21] proposed what became known as the Log-Concavity Genus
Distribution Conjecture (shortly LCGD Conjecture):

######  Conjecture 1 (Gross, Robbins, and Tucker, 1989).

The genus distribution of every graph is log-concave.

The conjecture has been confirmed for many graph families, including bouquets
of loops [21], dipoles [27, 45], graphs constructed by vertex and edge-
amalgamations [19], linear sequences of graphs [20, 13, 12], ring-like
families [18, 11], Halin graphs [18], and others.

The LCGD Conjecture is reminiscent on another famous problem, the
Heron–Rota–Welsh log-concavity conjecture [23, 37, 47], which asserts that the
coefficients of the characteristic polynomial of a matroid form a log-concave
sequence. A weaker conjecture that the coefficients are unimodal has been
proposed by Read. In 2012, Huh used algebraic geometry to prove Read’s
unimodality conjecture [24] for graphs, and the more general Heron–Rota–Welsh
conjecture for matroids represented over a field of characteristic 0. The case
of matroids representable over a field of a non-zero characteristic was
resolved a bit later by Huh and Katz [25]. Finally, in 2018 the
Heron–Rota–Welsh conjecture was proved in full generality by Adiprasito, Huh,
and Katz [1]. All these exciting developments gave hope to resolve the LCGD
Conjecture since the variety of 2-cell embeddings is related to the notion of
_$\Delta$ -matroids_ [5, 6, 33, 14] and to the Bollobás-Riordan polynomial
[4], both of which deal with graphs on surfaces and involve contraction-
deletion recurrences.

The main results of this paper are Theorems 3 and 8 that refute the LCGD
Conjecture in a rather spectacular (and unexpected) way. Let us state it first
in a weaker form.

######  Theorem 2.

For every $k\geq 1$ there is a $4$-connected graph whose genus sequence has at
least $k$ terms at which the log-concavity is violated.

We use standard graph theory language [15] and follow the usual terminology of
topological graph theory [35]. Throughout the paper, $G$ will always be a
connected simple graph, and by a surface we will refer to a closed orientable
surface. Let us observe that the analogue of our main result holds for
nonorientable surfaces as well.

##  2 Multiple log-convexity of genus sequences is viable

Let $(a_{n})_{n\in\mathbb{N}}$ be a sequence of non-negative numbers. The
$k$th term $a_{k}$ ($k\geq 1$) of this sequence is said to be _log-concave_ if
$a_{k}^{2}\geq a_{k-1}a_{k+1}$, and it is _log-convex_ if $a_{k}^{2}\leq
a_{k-1}a_{k+1}$. If the corresponding inequality is strict, we say that the
term is _strictly log-concave_ or _strictly log-convex_ , respectively. The
sequence is _log-concave_ if all of its terms are log-concave. These
definitions extend to any finite sequence, which can be viewed as an infinite
sequence with zeros appended at the end to make it infinite.

The afore-mentioned LCGD Conjecture 1 of Gross, Robbins, and Tucker [21]
states that the genus sequence of every graph is log-concave. In this section
we refute this long-standing open problem by proving that there are graphs
whose genus sequence has many strictly log-convex terms.

######  Theorem 3.

For every integer $g\geq 0$ and every $k\geq 2$ there is a $4$-connected graph
$G_{g,k}$ of genus $g$ such that the terms
$a_{g+1},a_{g+3},a_{g+5},\dots,a_{g+2k-1}$ of its genus sequence are all
strictly log-convex.

In the proof of this result we will need several properties about
combinatorics of 2-cell embeddings.111An embedding of a graph in a surface $S$
is a _$2$ -cell embedding_ if each face of the embedding is homeomorphic to an
open disk. Suppose that a graph $G$ is 2-cell embedded in a closed
(orientable) surface $S$ of genus $g$. A closed walk or a cycle in $G$ that
bounds a face in this embedding is said to be a _facial walk_ or _facial
cycle_. A cycle $C$ of $G$ is _contractible_ if it is contractible viewed as a
curve on the surface. This is the same as saying that $C$ splits the surface
into two components, one of which is a disk. More generally, $C$ is _surface
separating_ if it splits the surface into two components.

Combinatorially, $C$ is a _non-separating cycle_ of the graph $G$ if the
subgraph $G-V(C)$ is connected; the cycle $C$ is an _induced cycle_ of $G$ if
it is induced as the subgraph of $G$, i.e. it has no chords in $G$. Realizing
that facial cycles of 3-connected plane graphs (i.e. graphs of 3-dimensional
convex polyhedra) are induced and non-separating, Tutte [44] called such
cycles _peripheral_. Due to this relationship, we say that a 2-cell embedding
of a graph in a surface is a _polyhedral embedding_ if every face is bounded
by a peripheral cycle.

If a peripheral cycle in a 2-cell embedded graph is surface separating, then
it must be facial, see [35] for more details. For further use we state this
simple fact as a lemma.

######  Lemma 4.

Suppose that $C$ is a peripheral cycle of a graph $G$ that is $2$-cell
embedded in a surface $S$. If $C$ is not facial, then it is surface non-
separating.

More generally, we say that a family $\mathcal{C}$ of peripheral cycles in $G$
is _peripheral_ if any two cycles in the family are disjoint and nonadjacent
and $G-\cup\\{V(C)\mid C\in\mathcal{C}\\}$ is connected.

The following result is a simple generalization of Lemma 4.

######  Lemma 5.

Suppose that $\mathcal{C}$ is a peripheral family of cycles of a graph $G$
that is $2$-cell embedded in a surface $S$ of genus $g$. If none of the cycles
in $\mathcal{C}$ is facial, then $|\mathcal{C}|\leq g$.

###### Proof.

Each cycle in $\mathcal{C}$ is surface nonseparating and has a neighbor on
each side (otherwise it would be facial). Those neighbors are not on any of
the other cycles in the family. This implies that the union of all cycles in
$\mathcal{C}$ is surface non-separating. Consequently, repeatedly cutting the
surface along any cycle in $\mathcal{C}$ decreases the genus by $1$ (easily
seen by using Euler characteristic computation), and hence the number of
cycles is at most $g$. ∎

Two facial walks $W_{1},W_{2}$ are said to be _cofacial_ if there is a facial
walk $W$ that contains a vertex in $W_{1}$ and a vertex in $W_{2}$. In
particular, two facial walks that intersect are cofacial. The facial walk $W$
is said to be a _cofacial connector_ for $W_{1}$ (and at the same time also
for $W_{2}$). Given a set $\mathcal{C}$ of facial cycles, and
$C\in\mathcal{C}$, we denote by $\Delta(C)$ the number of different cofacial
connectors for $C$. The set $\mathcal{C}$ is _sparse_ if $\Delta(C)\leq 1$ for
each $C\in\mathcal{C}$.

######  Lemma 6.

Suppose that $G$ is a 3-connected graph with a polyhedral embedding in some
surface and $\mathcal{C}$ is a sparse collection of its facial cycles. Suppose
that $G$ has another embedding $\Pi$ in which none of the cycles in
$\mathcal{C}$ is facial. Then the genus of the embedding $\Pi$ is at least
$|\mathcal{C}|$.

###### Proof.

Let us first observe that the cycles in $\mathcal{C}$ are peripheral since
they are faces of a polyhedral embedding. Moreover, sparsity implies that
cycles in $\mathcal{C}$ are pairwise disjoint and pairwise nonadjacent. The
basic property used in the proof will be the following. For $C\in\mathcal{C}$,
let $N(C)$ be the set of vertices that are not on $C$ but have a neighbor on
$C$, and let $W(C)$ be the set of vertices that are cofacial with a vertex on
$C$ but do not belong to $\cup\mathcal{C}$. The vertices in $N(C)$ are ordered
clockwise as $u_{1},u_{2},\dots,u_{p}$, and each $u_{i}$ is connected with
$u_{i+1}$ via part $W_{i}$ of a facial walk that does not intersect $C$. Since
$\mathcal{C}$ is sparse, we may assume that the facial walks $W_{i}$ (except
possibly for $W_{p}$) are disjoint from $\cup\mathcal{C}$. This shows that for
every $v_{i},v_{j}\in N(C)$ there is a path in $G-V(\cup\mathcal{C})$
connecting them.

Each cycle in $\mathcal{C}$ is surface nonseparating in $\Pi$. Moreover, the
family $\mathcal{C}$ is peripheral. To see this, it is clear that
$\mathcal{C}$ is induced. To show that it is nonseparating, take vertices
$x,y\in V(G)\setminus V(\cup\mathcal{C})$ and consider a path from $x$ to $y$
in $G$. If the path contains a segment $Q$ whose first and last vertex are out
of $\mathcal{C}$ but other vertices on $Q$ are in one of the cycles
$C\in\mathcal{C}$, then we can replace $Q$ by a path in $G-V(\cup\mathcal{C})$
connecting them (as proved above). By repeating this for all such subpaths $Q$
we obtain an $(x,y)$-walk in $G-V(\cup\mathcal{C})$, showing that $x$ and $y$
are connected in $G-V(\cup\mathcal{C})$. This shows that $\mathcal{C}$ is a
peripheral family.

The proof is now easily concluded by applying Lemma 5. ∎

Our last ingredient is a bound on the number of disjoint nonhomotopic cycles
on a surface. By a _homotopy_ between two cycles of the graph embedded in a
surface we mean the homotopy between some of their orientations viewed as
loops on the surface.

######  Lemma 7.

Suppose that $\mathcal{C}$ is a family of pairwise disjoint and pairwise
nonhomotopic and noncontractible cycles on a surface of genus $g\geq 1$. Then
$|\mathcal{C}|\leq 3g-2$.

The proof of the lemma can be found in [31] (see also [35, Proposition
4.2.6]). The bound of $3g-2$ can actually be improved to $3g-3$ if $g\geq 2$
(which is best possible).

We are ready to proceed with our main construction.

###### Proof of Theorem 3..

For $m\geq 3$, let $\widehat{C}_{m}$ be the 4-regular _antiprism_ graph on
$2m$ vertices that is obtained from the prism $C_{m}\Box K_{2}$ by adding
diagonals into quadrangular faces. More precisely, $\widehat{C}_{m}$ is
obtained from two copies of the cycle of length $m$ with vertices
$v_{1}^{0},\dots,v_{m}^{0}$ and $v_{1}^{1},\dots,v_{m}^{1}$ (respectively) and
adding all edges $v_{i}^{0}v_{i}^{1}$ and $v_{i}^{0}v_{i+1}^{1}$ (index $i+1$
considered modulo $m$), $i=1,\dots,m$.

Figure 1: (a) The antiprismatic path of length 9 gives the antiprism
$\widehat{C}_{9}$ after identifying the left edge $v_{0}^{0}v_{0}^{1}$ with
the one on the right. (b) The stacked antiprism $\widehat{C}_{9}^{4}$ of
height 4. (c) The genus 0 cylinder $H_{m}^{k,b,d}$ with parameters $m=9$,
$k=2$, $b=2$, and $d=3$. The vertices and edges on the left are identified
with the vertices and edges on the right side to give a planar graph on the
cylinder.

Next, we define the _stacked antiprism_ $\widehat{C}_{m}^{b}$ of _height_ $b$
by stacking $b$ copies of $\widehat{C}_{m}$ on the top of each other
(identifying the cycle $v_{1}^{0}v_{2}^{0}\dots v_{m}^{0}$ of the bottom one
with the cycle $v_{1}^{1}v_{2}^{1}\dots v_{m}^{1}$ of the copy at the top of
it). See Figure 1(b) for an example.

Now, we take the disjoint union of $k+1$ stacked antiprisms
$\widehat{C}_{m}^{b}$, where $b$ is a large positive integer (to be specified
later), and where $m\geq 6$ is a multiple of 3. For $j=1,\dots,k$, we join the
$j$th stacked antiprism with three _antiprismatic paths_ of length $d$ (the
value to be specified later) to the $(j-1)$st stacked antiprism. These
antiprismatic paths will be referred to as the _connectors_. The connectors
start and end on the stacked antiprisms equally spaced (at distance
$\tfrac{1}{3}m-1$ from each other). See Figure 1(c) for an example. The
resulting graph $H_{m}^{k,b,d}$ has a natural embedding into the cylinder, and
is therefore planar. It is easy to see that $H_{m}^{k,b,d}$ is 4-connected, so
its embedding in the plane is polyhedral. There are two faces $F_{1},F_{2}$ of
length $m$, one at the top and one at the bottom, and for each
$j\in\\{1,\dots,k\\}$, there are $3$ faces of length $\tfrac{2}{3}m+2d-2$. All
other faces are of length $3$.

In order to obtain $G_{g,k}$ for $g\geq 0$, we start with $H_{m}^{k,b,d}$ and
fill in each of its faces $F_{1}$ and $F_{2}$ of length $m$ with a dense
triangulation such that the following properties hold:

  * (a)

Apart from the $3k$ long faces within $H_{m}^{k,b,d}$, all other faces in
$G_{g,k}$ are triangles;

  * (b)

the embedding of $G_{g,k}$ is of genus $g$;

  * (c)

the graph $G_{g,k}$ is 4-connected;

  * (d)

the maximum vertex degree in $G_{g,k}$ is at most 7 and any two vertices of
degree 7 are at distance at least 6 from each other;

  * (e)

every noncontractible cycle in $G_{g,k}$ has length at least
$f(g,k):=6(g+2k+1)$;

  * (f)

$G_{g,k}$ contains $g$ isometric non-surface-separating cycles whose pairwise
distance and their distance from the subgraph $H_{k}$ is at least $f(g,k)$.

Such triangulations are easy to construct. One can start with any
triangulation whose maximum degree is at most 7 and contains a triangle
$T_{0}$ whose degrees are all 4. then subdivide each edge and connect the
three subdivision vertices in each triangle except $T_{0}$ to subdivide that
triangle into four smaller triangles. This can be repeated so long that
conditions needed for (a)–(f) are obtained. We omit the details.

Note that for every $g\geq 0$ and $k\geq 1$, $G_{g,k}$ is 4-connected and
polyhedrally embedded in the surface of genus $g$. Herewith we will also
assume that $m\geq f(g,k)$ and that $b=f(g,k)$. Let $\nu(g,k)$ be the number
of vertices of $G_{g,k}$ not counting the vertices on the $3k$ connectors in
$H_{m}^{k,b,d}$, i.e.

$\nu(g,k)=|V(G_{g,k})|-6k(d+1).$

Now, we take the last parameter $d$ such that the following inequality holds:

$d>36k^{2}(\log_{2}d+7)+(g+2k)(217+\log_{2}\nu(g,k)).$ (1)

This completes the description of the graphs $G_{g,k}$. The constructed
polyhedral embedding of $G_{g,k}$ has genus $g$ and it will be denoted as
$\Pi_{0}$.

Next, we will obtain upper and lower bounds for the embedding coefficients
$a_{h}(G_{g,k})$ for $g\leq h\leq g+2k+1$. Let us first observe that the
constructed embedding of $G_{g,k}$ of genus $g$ is polyhedral and its face-
width (see [35] for the definition, which we omit here since it is not really
essential) is at least $f(g,k)>2g$. By a result of Robertson and Vitray [36]
(see also [34]), the embedding of $G_{g,k}$ is a minimum genus embedding, and
the graph has no other embeddings of genus (at most) $g$ save the one with the
same facial cycles as $\Pi_{0}$ but with reversed orientation. Thus,

$a_{g}(G_{g,k})=2.$

Let us now consider 2-cell embeddings of $G_{g,k}$ of genus $h=g+r$, where
$1\leq r\leq 2k$. Let us first give a lower bound on $a_{h}(G_{g,k})$.

######  Claim 1.

Suppose that $1\leq r\leq 2k$. Let $q:=\lfloor r/2\rfloor$. Then we have:

$a_{g+r}(G_{g,k})\geq 2^{2d(r+q)}\binom{k}{q}.$

Figure 2: Merging three big faces into a single face $F$.

###### Proof.

Let us consider the embedding of $G_{g,k}$ of genus $g+r$ that is obtained
from the embedding $\Pi_{0}$ as follows. First, choose $q$ indices $j$ between
$1$ and $k$ and for each such $j$ consider the two edges of the connectors on
the $j$th stacked antiprism $\widehat{C}_{m}^{b}$. Now change the local
rotation at the endpoints of these two edges so that these edges would emanate
into the incident big faces. This change merges four triangular faces of
$\Pi_{0}$ containing these two edges into two faces of length 4 and merges the
three big faces into a single face. Hence the effect of this change is that
the genus increases by two. Let $W_{i},W_{i}^{\prime}$ be the two paths on the
$i$th connector ($i=1,2,3$), oriented from “bottom” to “top” (see Fig. 2).
Then the big new face $F$ contains these paths in the following order:
$F=W_{1}^{-}e_{1}W_{1}^{\prime}\cdots W_{2}^{-}e_{2}W_{2}^{\prime}\cdots
W_{3}^{-}\cdots W_{3}^{\prime}\cdots$, where $W_{i}^{-}$ denotes the traversal
of the path in the opposite direction and $e_{i}$ is the flipped edge (see
Fig. 2). Now, each of the $2d$ edges connecting $W_{i}$ and $W_{i}^{\prime}$
can stay where it is or it can be flipped and reembedded into $F$, giving
$2^{2d}$ different choices. By doing this for $i=1,2,3$ and for each of the
$q$ chosen connectors, we get $2^{6dq}$ different embeddings, all of genus
$g+2q$. This proves the claim when $r$ is even.

If $r$ is odd, we select one of the remaining connecting triples and in that
case flip the first edge of just one of the connectors. This will further
increase the genus by one. In the same way as above we see that we can
independently reembed each of the $2d$ edges between $W_{1}$ and
$W_{1}^{\prime}$ into the new big face $F$ which has $W_{1}$ and
$W_{1}^{\prime}$ on its boundary traversed in the opposite directions. This
gives additional $2^{2d}$ possible choices to each of the previously described
embeddings of genus $g+r-1$. ∎

The next claim gives an upper bound on $a_{h}(G_{g,k})$, where we will not try
to optimize the bound.

######  Claim 2.

Suppose that $1\leq r\leq 2k$. Let $q:=\lfloor r/2\rfloor$ and $\nu=\nu(g,k)$.
Then we have:

$a_{g+r}(G_{g,k})\leq\left((5!)^{31}\nu\right)^{g+2k}\cdot\left(108d\right)^{36k^{2}}\cdot
2^{2d(q+r)}.$

###### Proof.

We split the vertices of $G=G_{g,k}$ into four sets,
$V_{A},V_{B},V_{C},V_{D}$, which are defined as follows. First, $V_{D}$
contains for each of the $3k$ connectors the six vertices in the two triangles
containing the first and the last edge of the connector (respectively) and the
vertex adjacent to that edge but outside of the connector; the set $V_{C}$
contains the remaining $2d-2$ vertices on each of the connectors; the set
$V_{B}$ contains all vertices that are incident to any of the nontriangular
faces and are not in $V_{C}\cup V_{D}$; lastly, $V_{A}$ contains all remaining
vertices. Note that $|V_{D}|=18k$, $|V_{C}|=6k(d-1)$, $|V_{B}|=2k(m-6)$, and
$|V_{A}|=\nu(g,k)-2k(m+3)$.

Suppose that $\Pi$ is a 2-cell embedding of $G$ whose genus is (at most)
$g+r$. We will compare which facial triangles under $\Pi_{0}$ are no longer
facial under $\Pi$. We say that a $\Pi_{0}$-facial triangle is _unstable_ if
it is not $\Pi$-facial; otherwise it is _stable_. We say that a vertex $v\in
V(G)$ is _unstable_ if it is contained in an unstable triangle. In the first
part of the proof, we will show that there are not too many unstable vertices
in $V_{A}\cup V_{B}\cup V_{D}$. By “guessing” which vertices in $V_{A}\cup
V_{B}\cup V_{D}$ are unstable, we get a bound on the number of possible
embeddings $\Pi$ having the guessed unstable set. On the other hand, the
number of unstable vertices in $V_{C}$ can be much larger. For the number of
rotation systems on the connectors we will need more specific homotopy
arguments.

Let us first consider the set $U$ of all unstable vertices in $V_{A}\cup
V_{B}$. Let $U^{\prime}$ be the largest set of unstable triangles, each of
which containing at least one vertex in $V_{A}$ and such that any two
triangles in $U^{\prime}$ are at distance at least 2 in $G$. It is easy to see
that the only possible cofacial connector of any two triangles in $U^{\prime}$
is one of the long faces and such a long face is unique for each triangle in
$U^{\prime}$. (Note that the possible exceptions to this property would be the
triangles with their vertices in $V_{C}\cup V_{D}$. This was actually the
reason to define $V_{D}$.) Consequently, the set $U^{\prime}$ is sparse, and
Lemma 6 shows that

$|U^{\prime}|\leq g+r.$

By maximality of $U^{\prime}$, each vertex in $U$ is at distance at most 2
from $U^{\prime}$. Since $G$ has maximum degree 7 and vertices of degree 7 are
at distance at least 4 from each other, it is easy to see that

$|U|\leq 30|U^{\prime}|.$

Among the 30 or fewer vertices at distance at most 2 from a triangle in
$U^{\prime}$, there is at most one of degree 7 (by property (d)), and thus
they give fewer than $6!(5!)^{29}$ different choices for rotations around
these vertices. This “overcount” also counts possibilities that stable
triangles can have two orientations as the faces in $\Pi$. This yields that
there are less than

$\binom{\nu(g,k)}{|U^{\prime}|}\cdot\left(6\cdot
6!\cdot(5!)^{29}\right)^{|U^{\prime}|}$

ways to select rotations around vertices in $V_{A}\cup V_{B}$. Similarly,
vertices in $V_{D}$ have at most $(5!)^{18k}$ ways to choose local rotations
around them.

Figure 3: Triangles on a connector (drawn horizontally). The three black
vertices belong to $V_{D}$.

To count the number of ways to change the local rotations around vertices in
$V_{C}$ and still have genus at most $g+r$, we first note that the above
bounds and properties (e) and (f) show that the genus of the subgraph at
distance $f(g,k)$ from the long faces of $G$ is at least $g$. Thus we can
afford to increase the genus at most by $r$ when changing the rotation around
the vertices on the connectors.

Let us now consider one of the connectors. We will use the notation shown on
Fig. 3, where the connector is drawn horizontally. Starting on the left with
triangle $T_{1}$, we find the first unstable triangle $T_{i_{1}}$. Suppose
that there is another unstable triangle $T_{j}$ that is homotopic to
$T_{i_{1}}$ under the embedding $\Pi$. (Here the homotopy is free homotopy
among some orientation of two triangles.) We let $T_{i_{2}}$ be the homotopic
triangle with largest possible $i_{2}$, possibly $i_{2}=i_{1}$. Since the two
homotopic noncontractible triangles bound a cylinder, any triangle $T_{j}$
with $i_{1}\leq j\leq i_{2}$ is either contractible or homotopic to
$T_{i_{1}}$. Then we look further and find the smallest $i_{3}\geq i_{2}+3$
such that $T_{i_{3}}$ is unstable. That unstable triangle is disjoint from
$T_{i_{1}}$ and $T_{i_{2}}$. If $i_{3}$ exists, then we define $i_{4}$ and
$T_{i_{4}}$ in the same way as we did to obtain $i_{2}$ after having fixed
$T_{i_{1}}$. Proceeding, we obtain pairs
$(i_{1},i_{2}),(i_{3},i_{4}),\dots,(i_{2t-1},i_{2t})$. The triangles
$T_{i_{1}},T_{i_{3}},\dots,T_{i_{2t-1}}$ are pairwise disjoint and
nonhomotopic, and they increase the genus at most by $r$. Thus, by Lemma 7,
$t\leq 3r-2$.

Figure 4: Degrees of freedom when the homotopy of unstable triangles is
changed. The shaded triangles are facial (with the shown or with the opposite
orientation on the surface).

The above paragraph shows that we can determine the number of possibilities of
different local rotations on one of the connectors by first finding
$i_{1},i_{2},\dots,i_{2t-1},i_{2t}$ (less than $(2d)^{6r-4}$ possibilities),
then estimating that each edge between $T_{i_{2j-1}}$ and $T_{i_{2j}}$ has two
choices for joining the two paths on the connector, which amounts all together
to at most $2^{2d}$ possibilities, and finally, when changing homotopy,
considering the possible ways of local rotations at vertices between
$T_{i_{2j}}$ and $T_{i_{2j+1}}$. The latter possibility gives at most $6^{3}$
ways, for local rotations, including for the three vertices on $T_{i_{2j+1}}$.
Figure 4 shows different possibilities for the latter count, where $6^{3}$ is
the worst. The $6^{3}$ ways can be replaced by $3^{3}$ to compensate with the
overcount of $2^{2d}$ stated above.

Without trying to optimize and use the same bounds on other connectors, we
take it from the proof of Claim 1 that at most $q+r\leq 3k$ connectors have
unstable triangles. There are at most $2^{3k}$ possible subsets of connectors
with unstable triangles. This yields that the number of embeddings is at most

$2^{3k}\Bigl{(}2^{2d}\bigl{(}2d\cdot 3^{6}\bigr{)}^{6r-4}\Bigr{)}^{q+r}.$

All over, we obtain the upper bound of

$\displaystyle\binom{\nu}{|U^{\prime}|}\cdot\left(6\cdot
6!\cdot(5!)^{29}\right)^{|U^{\prime}|}$ $\displaystyle\cdot(5!)^{18k}\cdot
2^{3k}\Bigl{(}2^{2d}\bigl{(}2d\cdot 3^{3}\bigr{)}^{6r-4}\Bigl{)}^{q+r}$
$\displaystyle\leq$
$\displaystyle\left(36(5!)^{30}\nu\right)^{g+2k}\cdot\left(2(5!)^{6}(54d)^{12k-4}\right)^{3k}\cdot
2^{2d(q+r)}$ $\displaystyle<$
$\displaystyle\left((5!)^{31}\nu\right)^{g+2k}\cdot\left(108d\right)^{36k^{2}}\cdot
2^{2d(q+r)},$

where $\nu=\nu(g,k)$. ∎

Finally, suppose that $1\leq r<2k$, where $r$ is odd. Then, using Claim 1 for
$r-1$ and $r+1$ (and noting that $r-1$ and $r+1$ are even), we see that

$a_{g+r-1}(G_{g,k})\cdot a_{g+r+1}(G_{g,k})>2^{6dr}.$ (2)

By using Claim 2 for $r$, we have

$a_{g+r}(G_{g,k})\leq\left((5!)^{31}\nu\right)^{g+2k}\cdot(108d)^{36k^{2}}2^{(3r-1)d}.$
(3)

Comparing (2), the square of (3) and recalling (1), the conclusion of the
theorem follows easily. ∎

##  3 Many consecutive log-convex terms

One obvious question is whether having only every second term being strongly
log-convex is necessary. Our examples can be generalized to obtain the
following result.

######  Theorem 8.

For every $g\geq 0$ and $k\geq 1$, there exists a $4$-connected graph of genus
$g$, whose genus distribution sequence has $k$ consecutive terms that are
strongly log-convex.

###### Proof.

The proof is essentially the same as that of Theorem 3 and we do not give all
details here. However, we provide the basic idea.

The graphs for this theorem are essentially the same as those in the proof of
Theorem 3. The graphs $G_{g,k}$ have $k$ connecting parts, in each of which
three connectors join one stacked antiprism with another one. We replace the
$j$th triple of connectors with $k+j$ connectors ($1\leq j\leq k$). We also
make their lengths slightly different, the lengths increasing with $j$ (this
will make the proof work). For some very large $d$, we let the length of all
these $q=k+j$ connectors be $c_{q}d$, where

$c_{q}=\frac{q-1}{q}+\frac{q}{4k^{2}},\qquad k\leq q\leq 2k.$

Note that $c_{k}=1-\tfrac{3}{4k}<c_{k+1}<\cdots<c_{2k}=1$ and that the
function $r(q)=q.c_{q}$ is strictly convex:

$2qc_{q}<(q-1)c_{q-1}+(q+1)c_{q+1}.$ (4)

Details are left to the reader.

The genus coefficients $a_{g},a_{g+1},\dots,a_{g+k-1}$ grow “roughly” with the
factor of $2^{(1+o_{d}(1))d}$, i.e.

$a_{g+t}=2^{(1+o_{d}(1))dt}$

for $1\leq t<k$. This growth is realized by joining $t+1$ large faces of the
last connectors where we have $2k+1$ connectors of height $c_{2k}d=d$.
However, $a_{g+k},a_{g+k+1},\dots,a_{g+2k}$ grow exponentially faster. We
claim that

$a_{g+k+t}=2^{c_{k+t+1}d(k+t+1)(1+o_{d}(1))}$ (5)

for $0\leq t\leq k$. To see that $a_{g+k+t}\geq 2^{c_{k+t+1}d(k+t+1)}$, we use
a similar proof as for Claim 1. With $k+t+1$ connectors of height $c_{k+t+1}d$
we increase genus $g$ to $g+k+t$ to obtain one face where the sides of these
$k+t+1$ connectors appear as in the mentioned proof, yielding
$2^{c_{k+t+1}d(k+t+1)}$ different embeddings of the same genus.

To prove that the right hand side of (5) is an upper bound, we have to prove
that using the increase of the genus by $k+t$ as in the lower bound
construction above is best possible. To show this we have to argue that
merging $k+t+1$ big faces with the highest connectors of height $c_{k+t+1}d$
gives much smaller numbers, and that using the gain where we merge $k+s$ faces
of height $c_{k+s}d$ and adding the rest of the genus by merging $t-s+1$ faces
with the connectors of height $c_{2k}d$ also gives lower numbers. This
comparison reduces to the following two inequalities:

$\displaystyle c_{k+t+1}(k+t+1)$ $\displaystyle>$ $\displaystyle
c_{2k}(k+t)\quad\mathrm{and}$ $\displaystyle c_{k+t+1}(k+t+1)$
$\displaystyle>$ $\displaystyle c_{k+s+1}(k+s)+c_{2k}(t-s),\quad 0\leq s<t.$

Using the fact that $c_{2k}=1$, they are both easy to verify and details are
left to the reader.

Finally, (5) implies that for every $0\leq t<k$ the following holds

$\displaystyle 2\log_{2}a_{g+k+t}$ $\displaystyle\approx$ $\displaystyle
2c_{k+t+1}d(k+t+1)$ $\displaystyle\ll$ $\displaystyle
c_{k+t}d(k+t)+c_{k+t+2}d(k+t+2)$ $\displaystyle\approx$
$\displaystyle\log_{2}a_{g+k+t-1}(1+o_{d}(1))+\log_{2}a_{g+k+t+1}(1+o_{d}(1)),$

where $\approx$ means equality up to a factor $1+o(1)$, and $a\ll b$ means
that $a/b<1-\varepsilon$ for some $\varepsilon>0$. This harsh inequality
follows from (4). This completes the proof. ∎

##  4 In conclusion

Let us end with a more positive viewpoint. Our counterexamples to the LGCD
Conjecture are still unimodal. As far as the author can tell, the unimodal
weakening of the LGCD Conjecture could (and actually should) still be true.

######  Conjecture 9.

The genus distribution of every graph $G$ is unimodal.

The following also looks plausible.

######  Conjecture 10.

Every graph that triangulates some surface has log-concave genus distribution.

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# AstroSat Observation of Recent Outburst in the Be/X-ray Binary LS V +44
17/RX J0440.9+4431

Arshad Hussain<EMAIL_ADDRESS>Umananda Dev Goswami  0000-0003-0012-7549
<EMAIL_ADDRESS>Department of Physics, Dibrugarh University, Dibrugarh
786004, Assam, India

###### Abstract

A Be/X-ray binary system known as RX J0440.9+4431 (or LS V +44 17) is a
potential member of the uncommon gamma-ray binary class. With an orbital
period of 150 days, this system consists of a neutron star and a Be star
companion. The MAXI observatory discovered an X-ray outburst from the source
in December of 2022. Early in January, the outburst reached its peak, which
was then followed by a decrease and a subsequent rebrightening. The X-ray flux
exceeded 1 Crab in the 15-50 keV range at this second peak. AstroSat
observations were conducted close to the peak of the second outburst, from
January 11 to January 12, 2023. We report here the results of our search for
3-80 keV X-ray emission in the data of the AstroSat’s LAXPC detector. It is
found that the pulse period of the source is around 208 seconds. The source is
found to be emitting more in the softer part of the X-ray energy range. The
spectral characteristics can be described by employing a power-law model with
an exponential cutoff, along with a blackbody component, interstellar
absorption and an additional 6.4 keV iron fluorescence line.

Be/X-ray binary; AsroSat’s LAXPC; Temporal and spectral behaviours

##  I Introduction

X-ray binaries (XRBs) are the peculiar X-ray systems among the most puzzling
and brightest objects in the Universe. XRBs can be simply defined as systems
that consist of a compact object (black hole or neutron star) and an optical
companion (typically a normal main-sequence star) which are gravitationally
bound to each other [1, 2, 3, 4]. They are “close” systems due to mass
transfer from the companion to the compact object [5]. X-ray binaries can be
divided into low-mass and high-mass XRBs depending on the donor star’s mass.
In the case of low-mass XRBs (LMXBs) ($M\leq$ 1 $M_{\odot}$), the accretion of
materials happens through Roche-lobe overflow of the companion star [5, 6].
For high-mass XRBs (HMXBs) ($M\geq$ 5 $M_{\odot}$), the accretion takes place
through stellar wind or Be-disk of the donor star [1]. They contain early-type
stars as companions, such as O or B stars and are strong X-ray emitters. Based
on the luminosity class, the HMXBs can be further subdivided into Be/X-ray
binaries (BeXBs), where the optical companion can be a dwarf, giant, or sub-
giant OBe star in the luminosity classes III-IV, and supergiant XRBs (SGXBs),
in which the companion is a luminosity class I-II supergiant star [1, 5].
Moreover, in BeXBs, the optical partners are fast-rotating stars of the type
mentioned above with distinctive emission lines in their spectrum at some
point of their existence (accordingly the lowercase “e” used to denote
emission) [5, 7, 8, 9]. Usually, the neutron star as the compact object in the
binary system is considered as a defining characteristic of BeXBs. These non-
supergiant BeXBs possess larger orbital periods [5, 1]. Roughly two-thirds of
HMXBs populations are Be/X-ray binaries [1]. In the binary systems known as
the accreting X-ray pulsars (XRPs), a neutron star accretes matter from the
donor companion. The vast majority of XRPs fall under the subclass of Be/X-ray
binaries. Here, the companion’s circumstellar disk emits H$\alpha$ Balmer
lines at least once in its life [10, 11, 7]. In BeXBs, the optical emissions
predominantly come from the Be star, whereas the X-ray emissions provide
insights into the conditions in the vicinity of the neutron star [11, 12].
BeXBs systems exhibit two different X-ray outburst types: Type-I and Type-II
X-ray outbursts. Type-I commonly occurs during the periastron passage of the
neutron star, while Type-II can occur at various orbital phases which may be
linked to the warping of the circumstellar disk’s outermost region [12].

RX J0440.9+4431 is a BeXB which was discovered by Motch et al. in 1997 [13]
during the ROSAT’s survey of the galactic plane [14, 1]. Its distance was
estimated to be $\sim$ 3.3 $\pm$ 0.5 kpc [1]. Strong X-ray emissions ($\sim$
1034-35 erg s-1) [14] with pulsations (period $\sim$ 202.5 $\pm$ 0.5 s [15])
were discovered from this source which confirmed it as an X-ray pulsar [15,
16]. Reig et al. (2005) [17, 16] had done a thorough investigation of the
characteristics of its optical counterpart. This optical companion, BSD
24-491/LS V +44 17, was classified as a Be star which belongs to the B0.2V
spectral class [17, 1]. Due to the occurrence of the limited number of
relatively dim Type-I outbursts (which is typical for BeXBs) that have been
observed, the characteristics of this binary system have not been extensively
investigated in the X-ray domain [16, 18, 19, 20]. RX J0440.9+4431 stands out
as one of the rare X-ray binary systems where accretion continues even during
quiescence [16, 15, 21]. It is crucial to note that the accretion in quiescent
is powered by a cold and non-ionized disk [16, 22] along with the wind
accretion.

The MAXI detector observed the first outburst from RX J0440.9+4431 during
March 26, 2010 and April 15, 2010 [14, 11, 18] with a peak luminosity of 3.9
$\times$ 1036 erg s-1 (3-30 keV) [14, 23]. _Swift_ /BAT also detected the next
two minor X-ray flares following this first outburst [11, 24]. It is
remarkable to note that considering the RX J0440.9+4431 is located at a
distance of 2.44${}^{+0.06}_{-0.08}$ kpc [14, 25], the observed luminosity of
the source was found to be remained considerably below the characteristic 1037
erg s-1, which is typically associated with Type-I outbursts in the case of
BeXBs [14, 26, 27, 24, 28, 5].

The transient activity exhibited by the source was observed mainly in the
years 2010 and 2011 [16] with luminosity peaks varying from (1-5) $\times$
1036 erg s-1. During this span of time, the presence of a probable cyclotron
line was reported as an absorption feature at $\sim$ 30 keV which denotes a
magnetic field intensity of 3 $\times$ 1012 G [14, 19]. However, later studies
have raised questions about the presence of this specific spectral feature
[16, 20]. The pulse profile that was recorded displayed a generally simple and
nearly sinusoidal pattern in 0.3-60 keV range. Nevertheless, some luminosity-
dependent structures were evident in the 3-15 keV range [16, 19]. Usui et al.
(2012) [23] noted an abrupt fall of luminosity after the initial peak. This
was described as a consequence of the accretion stream blocking the emission
zone. The source was most likely in a sub-critical accretion condition,
meaning that the emission originated from a hotspot, as evidenced by the
relatively low measured luminosity and very simple shape of the pulse profile
[16, 29]. The soft X-ray spectra were also observed by various observatories.
The spectra can be explained as a composite model of power-law and black-body
components with the addition of the 6.4 keV iron line [14, 23, 19, 21].

After more than ten years of inactivity, RX J0440.9+4431 went through a
massive Type-I outburst on December 29, 2022, which was detected by the MAXI
all-sky monitor [30]. This was followed by a multi-wavelength campaign using
NuSTAR, NICER, Chandra, _Swift_ -XRT, and AstroSat which lasted for almost
four months [30, 31, 14, 32, 33]. _Swift_ /BAT data indicates that on January
4, 2023, the outburst peaked $\sim$ 0.6 Crab in the 15-50 keV band with
luminosity $\sim$ 5 $\times$ 1036 erg s-1. The source then had a rapid drop in
luminosity until the trend changed around January 13, 2023, indicating the
start of a significant Type-II outburst. This outburst achieved its peak of
$\sim 2.2$ Crab on January 20, 2023, having a luminosity of $\sim$ 2 $\times$
1037 erg s-1 in the 15-50 keV band [14, 32]. The source was observed by
AstroSat LAXPC on January 11, 2023 [34]. In this paper, we report the results
of the analysis of this observational data of RX J0440.9+4431.

The remaining part of this paper is organized as follows. In Section II, we
provide a summary of the instrument, the observation data used and the data
reduction procedure. The temporal and spectral analysis of the LAXPC data are
presented in the Section III. The Section IV is devoted to discuss the results
of both timing and spectral analysis in more detail. In Section V, we
summarise our findings.

##  II LAXPC detectors, Observations and Data Reduction

###  II.1 LAXPC

India’s first multi-wavelength astronomical observatory, the AstroSat, has
been in active operation for about the last eight years. Launched by the
Indian Space Research Organisation (ISRO) on September 28, 2015, it was placed
in a circular orbit at an altitude of 650 km above the Earth’s surface with a
6∘ inclination and an orbital period of 98 minutes. AstroSat uses a collection
of four equipment, which includes a UV imaging telescope and three X-ray
sensors all of which cover a range of energies from around 1 eV to 100 keV
[35]. The hard X-ray instrument, the Large Area X-ray Proportional Counter
(LAXPC) is one of the three X-ray sensors [36, 37, 38]. It consists of three
identical large-area proportional counters that are co-aligned, each having
multilayer geometry [36, 37, 39] and with a collection area of 100 $\times$ 36
cm2 [40]. The detectors are identified as LX10 (or LAXPC10), LX20 (or LAXPC20)
and LX30 (or LAXPC30). Each detector is equipped with five anode layers which
are divided into seven anodes, A1-A7, the upper two layers each having two
anodes. On each of the three sides of the detector, there are three veto
anodes, A8-A10, as well [36]. A gas mixture of 90% xenon and 10% methane kept
at a pressure of around 2 atmospheres, fills the entire volume of LAXPC10 and
LAXPC20 detectors. In contrast, a gas combination composed of 84.4% xenon,
9.4% methane and 6.2% argon is present in LAXPC30, which is also kept at a
pressure around $2$ atmospheres [40].

The LAXPC operates within the energy range of 3-80 keV. The lower energy
threshold of this set of devices is influenced by the negligible transmission
of X-ray (below 3 keV) through 50 $\mu$m thick aluminized mylar. On the other
hand, the reduced detection efficiency at higher energy leads to the higher
energy threshold [41]. The LAXPC possesses the capability to study
fluctuations in kHz scale at a high counting rate. A large photon collection
area with robust detection efficiency over the complete energy range is
necessary for this. The effective area of all three units is $\sim$ 4500 cm2
at $\sim$ 5 keV, $\sim$ 6000 cm2 at $\sim$ 10 keV and $\sim$ 5000 cm2 at
$\sim$ 40 keV [41, 42].

This instrument’s main objectives are broadband spectrum observations and
high-time-resolution research to examine rapid changes in intensity. This is
accomplished by the instrument’s ability to precisely identify each observed
X-ray photon’s arrival time, with an accuracy of 10 $\mu$s. Additionally, it
uses a $1024$-channel pulse-height analyser to calculate each photon’s energy
[42]. Agrawal (2006) [37] and Singh et al. (2014) [38] both provide detailed
descriptions of the AstroSat observatory and its equipment. Yadav et al.
(2016) [39], Agrawal et al. (2017) [41], and Roy et al. (2016) [43] give a
more thorough explanation of the characteristics of the LAXPC instrument.
Antia et al. (2017) [40] provides comprehensive information on calibration
specifications.

###  II.2 AstroSat Observation

AstroSat monitored the source RX J0440.9+4431 from 11:52:07.64 UT on January
11, 2023 to 00:43:20.53 UT on January 12, 2023, covering its eight orbits. In
terms of Modified Julian Day (MJD) this observation of AstroSat lies within
the MJD period 59955.49 to 59956.03. Background data used for this source was
collected between May 2, 11:55:52.52 UT to May 3, 09:50:52.35 UT in the year
2022. This background data was obtained by pointing the instrument at a region
devoid of any sources. The background data have been used for obtaining the
source’s background-subtracted light curve. We obtain all these data from the
archival of the AstroSat in the Indian Space Science Data Centre (ISSDC) [34].
It should be mentioned that between 2022 and 2023, MAXI/GSC detected a strong
X-ray burst from RX J0440.9+4431 in the 2 to 20 keV energy range [18]. Fig. 1
shows the source’s one-day binned MAXI/GSC light curve in 2-20 keV from MJD
59500.0 to MJD 60200.0 [18, 44]. It is coupled with the AstroSat observation
on MJD 59955.49 as mentioned above, denoted by a thick red vertical line. The
source was about to have a Type-II outburst when the AstroSat observation was
made, as seen from the MAXI/GSC light curve.

Figure 1: The light curve of the RX J0440.9+4431 observed by MAXI/GSC during
the period of 2022 to 2023 [18, 44]. A significant and substantial outburst
event was identified emanating from RX J0440.9+4431 within the energy range of
2 to 20 keV. The vertical strip indicates the specific time period of the
AstroSat observations.

###  II.3 Data Reduction

The data reduction process wass carried out using the LAXPC data reduction
pipeline software “LAXPCsoftware22Aug15”. The “lxplevel2datapipeline” utility
was used to convert Level-1 raw data files into Level-2 data. Data was
analysed for each of the three LAXPC units individually. The Level-2 data
consists of light curve, information about each detected X-ray photon, pulse
height and the originating anode layer in (i) broadband counting mode
(modeBB), (ii) event mode (modeEA), and (iii) mkf files (which stores
housekeeping data and parameter files). Using the LaxpcSoft standard tasks,
the light curves and energy spectrum were obtained from the Level-2 event
file. The software routine generates the Good Time Interval, which contains
timing details for Earth occultation and the South Atlantic Anomaly. We
utilise average background counts to determine the source light curve with the
background subtraction process. In addition to background correction, to
account for the variations in the arrival times of the photons due to the
motion of the Solar system, these photon arrival times have undergone
barycentric correction to the Solar system barycenter using the AstroSat
barycentric correction utility “as1bary”. It requires HEASOFT software [45]
package version 6.17 or higher. We use the latest version (6.30.1) available
at the time of our analysis. HEASOFT incorporates various tools and programs.
XRONOS tool [46] was used for the timing analysis. FTOOLS [47] were primarily
used for general data extraction and analysis. The XSPEC package [48] played a
key role in spectral analysis. We use data only from the LAXPC20 detector as
LAXPC30 was impacted by gas leakage and hence turned off on March 8, 2018 due
to anomalous gain fluctuations. We did not include the data from LAXPC10 as it
was operating at low gain [49, 50]. We pick the 3-30 keV energy band for the
spectrum since the background is dominant above the 30 keV.

##  III Analysis of Data

###  III.1 Timing Analysis

The light curve of the source was produced from the above-mentioned AstroSat
data, i.e. the data obtained from all the observations of LAXPC20 by using the
10 seconds averaged count rates in the 3-80 keV band, which is presented in
the top panel of Fig. 2. To have the best signal-to-noise ratio, we extract
data only from the top layer of the LAXPPC20 instrument. As mentioned earlier,
there are eight orbits of observation and for those combined orbits, we also
generate the corresponding backgrounds shown in the middle panel of Fig. 2.
The background varies between $\sim$ 48 - 61 counts s-1 during the
observations. To get the final light curve, this background was subtracted
from the light curve using the “lcmath” tool. The background-subtracted light
curve is shown in the bottom panel of Fig. 2. For clarity in the visibility of
pulsations the first segment of the light curve of Fig. 2 is also shown in
Fig. 3. Pulsations of X-rays from the source are distinctly visible in the
light curve from both figures.

Employing the FTOOL subroutine “powspec” in HEASOFT, the one-second binned
power density spectrum (PDS) of the source was generated using the light curve
of the source given by the LAXPC20 detector. We divide the light curve into
stretches of 1024 bins per interval. The final PDS was created by averaging
the PDSs from all the segments. The generated PDS has a very significant peak
at $\sim$ 0.005 Hz corresponding to the source’s spin period of $\sim$ 200 s
(see Fig. 4). The PDS shows no quasi-periodic oscillation (QPO) [5] peaks
since no additional nearby significant peaks have been observed except few
harmonics in the spectrum. Applying the standard $\chi^{2}$ maximization
approach with the FTOOLS job “efsearch” a more accurate estimation of the
pulse period can be obtained. The period was searched around 200 s as
approximated in the PDS generated from the light curve. It was found to be
207.93 s (see Fig. 5) confirming the pulse period detected by the NICER
mission [1].

Figure 2: (a) Light curve of the X-ray source RX J0440.9+4431 including the
background count rates. (b) Background light curve. (c) Background subtracted
light curve of the source. The gaps in the light curve is due to the passage
of the satellite through the South Atlantic Anomaly regions. In these plots,
10 s binning is used in the energy range of 3-80 keV. Figure 3:  Time scale
amplified background-subtracted light curve of the source RX J0440.9+4431 in
3-80 keV energy range obtained by using the 10 s bins. Figure 4: The power
density spectrum of RX J0440.9+4431 generated by using the LAXPC20 data from
10 s binned barycenter corrected light curve in 3-80 keV energy band. A strong
pulsation with a peak at $\sim$ 0.005 Hz was found in the PDS. Figure 5: The
pulse period of the pulsar RX J0440.9+4431 estimated by using the FTOOLS
subroutine “efsearch” from the LAXPC20 data’s light curve. The period was
found to be around 207.932 s.

To investigate how intensity fluctuations depend on energy, light curves were
produced within five unique energy bands: 3-12 keV, 12-20 keV, 20-30 keV,
30-50 keV and 50-80 keV for the LAXPC20 detector. Only Layer 1 data were
utilised for the construction of the light curves in the energy range from 3
to 20 keV. This selection was considered due to the fact that most incident
photons ($\sim$ 90 %) with energy $<$ 20 keV are likely to be absorbed in
Layer 1 [42]. Similar to this, light curves in the 20 to 50 keV range were
produced by summing data from Layers 1 and 2, while those in the 50 to 80 keV
range used data from all five layers. The count rates for the LAXPC20 detector
within these five energy bands are shown in Table 1. Fig. 6 shows that the
higher amplitudes of oscillations are noticeable in the energy bands of 3-12
keV, 12-20 keV and 20-30 keV than in the 30-50 keV and 50-80 keV bands.

To analyse the spectral evolution during the intensity oscillations, hardness
ratios (HRs) were computed. These HRs are defined as the count rates in a
higher energy band divided by the count rates in a lower energy band. We
generate four hardness ratio diagrams (background subtracted) for 12-20
keV/3-12 keV, 20-30 keV/12-20 keV, 30-50 keV/20-30 keV and 50-80 keV/30-50
keV. In Fig. 7, it is noteworthy to notice that the source’s intensity shows
no correlation with the HR. It indicates that the intensity oscillations are
not directly related to spectral changes within the detected energy ranges.

Figure 6: Background subtracted light curves for (a) 3–12 keV, (b) 12–20 keV, (c) 20-30 keV, (d) 30–50 keV and (e) 50–80 keV for LAXPC20 data of the pulsar RX J0440.9+4431. Table 1: Background subtracted LAXPC20 count rates in five energy bands of the source RX J0440.9+4431. Energy Range | Average Count Rate
---|---
(keV) | (counts s-1)
3-12 | 240.41
12-20 | 53.57
20-30 | 17.75
30-50 | 18.88
50-80 | 16.73

Figure 7: Hardness ratios for 12-20 keV/7-12 keV (top left panel), 20-30
keV/12-20 keV (top right panel), 30-50 keV/20-30 keV (bottom left panel) and
50-80 keV/30-50 keV (bottom right panel). The bottom section of each panel
(black color) represents the hardness ratio of the corresponding energy band.

###  III.2 Spectral Analysis

Since the spectral response matrix of the LAXPC20 detector is well determined,
we only conduct a thorough spectral analysis on LAXPC20 data. The spectral
fitting and statistical analysis are done using the XSPEC version 12.12.1.
Data from all the eight orbits observed by LAXPC20 were merged to derive the
spectrum of the source in the energy band ranging from 3 keV to 80 keV. To
maximize the inclusion of X-ray photons and to minimize the background, the
spectrum was generated only from the top layer of LAXPC20. The background
spectrum was found to be dominating above 30 keV. Therefore, the photons below
3 keV (ignored channel: 1) and above 30 keV (ignored channels: 30-77) for
LAXPC are ignored to avoid larger systematic errors. The LAXPC software has a
single routine to generate the source spectra, the background spectra and the
response matrix files. The spectra were fitted with a combination of models
“TBabs*(cutoffpl + bbodyrad + gaussian)” which are built-in models found in
XSPEC.

To model the spectrum, we first consider a single-component model, the power-
law model with an exponential cutoff (cutoffpl). Even after using a systematic
error of 0.02 (i.e. 2%) in data, it gives a poor fit with $\chi^{2}$/dof value
of 5056/25. Next, we add a blackbody radiation model (bbodyrad) to the power-
law model. With this addition, we gain a noticeable improvement in fitting
parameters ($\chi^{2}$/dof = 2723.11/23). However, the combined model left a
residual at $\sim$ 6.4 keV. Therefore a Gaussian line at 6.4 keV (due to iron
K$\alpha$ line) was included which significantly improved the fit with
$\chi^{2}$/dof = 270.98/20. Finally, we include the multiplicative model,
TBabs [51, 52] for interstellar absorption. Here the column density value has
been fixed at $N_{H}=$ 1.0 $\times$ 1022 cm-2. It gives a very good fit of
$\chi^{2}$/dof = 22.29/20 (i.e. $\chi^{2}_{red}=1.11$). Table 2 lists the
spectral parameters obtained from the best fit of our combined model. The
best-fitted spectrum of the source is shown in Fig. 8.

Table 2: Best fitting spectral parameters obtained from the model: TBabs*(cutoffpl + bbodyrad + Gaussian). |  Model |  Parameter | Value |
---|---|---|---|---
|  TBabs |  $N_{H}$ ($10^{22}$ cm${}^{-}2$) | 1.0000 |
|  cutoffpl |  PhoIndex | 0.9495 |
|  |  HighECut (keV) | 19.5063 |
|  |  Norm | 0.1032 |
|  bbodyrad |  kT (keV) | 0.2818 |
|  |  Norm ($\times 10^{5}$) | 1.9452 |
|  Gaussian |  LineE ($\times 10^{-15}$ keV) | 9.2099 |
|  |  Sigma (KeV) | 4.4204 |
|  |  Norm | 0.1855 |
Figure 8: Fitted LAXPC spectrum of the source RX J0440.9+4431. The best fit is
obtained by using the model TBabs*(cutoffpl + bbodyrad + Gaussian). The fitted
model spectrum is shown by the red solid line along with the observed one in
black colour. The residuals between the data and the corresponding model
predictions are shown in the lower section of the plot.

##  IV Results and Discussion

The AstroSat observation of RX J0440.9+4431 provides us with an opportunity to
investigate the behaviour of the pulsar at X-ray energies 3-80 keV during the
enormous outburst in 2023. _Swift_ /BAT observations show that the huge
outburst reached a record-high flux of 2.3 Crab [1]. During the outburst of
the source, cleaned event data were used to create a 10-second binned light
curves in various energy ranges to observe the periodicity in the data.
According to Reig & Roche in 1999 [15], the source RX J0440.9+4431 has been
classified as a persistent BeXB due to its longer spin period and less X-ray
variability than typical transient pulsars. In such cases, the neutron star
and the Be star are in a wide orbit. The neutron star orbits through the outer
portion of the Be circumstellar disc, where matter density is low. However, RX
J0440.9+4431 displayed an eruption which was contrary to what was anticipated.
This suggests that the source was in transition to a supercritical regime [16,
23].

The light curves of the source clearly exhibit the prominent intensity
oscillations of a period of approximately 200 seconds. These oscillations are
readily visible across all of the light curve data. The $\sim$ 200 s X-ray
pulsations are clearly visible also in the energy-resolved light curves in
3-12 keV, 12-20 keV, 20-30 keV, 30-50 keV and 50-80 keV bands. To search for
the periodic signals, the barycentre corrected event data were employed and
efsearch task in FTOOLS was executed. This task folded the light curve over a
trial period and identified the most likely period. From this analysis, we
obtain a measurement of the pulse period $\approx$ 207.9 s which confirms the
pulse period as described by NICER observation for this source [1]. The
evolution of the pulse period could not be studied due to AstroSat’s
observation for a very short period of time. However, Reig & Roche in 1999
[15] found its period to be $\sim 202.5$ s. According to LaPalombara (2012)
[21], the pulse period was $\sim 204.96$ s. Another study showed its period to
be $\sim 205$ s [23]. Thus, our measured period of 207.9 s in the recent
AstroSat observation is notably longer than the one recorded in 1999. This
discrepancy suggests that over the previous ten years, the neutron star has
been slowing down. This deceleration indicates that the neutron star’s
rotation rate will gradually slow down over time.

In energy-resolved light curves, the spectral changes can be seen clearly. It
is noticed that the source is softer, i.e., it emits more low energy photons,
primarily in the 3-12 keV band ($\sim$ 240 counts s-1 on average). This may
refer to cold disk accretion in our Be/X-ray binary. Be/X-ray binaries are
thought to be persistent systems. This hypothesis supports that the long-term,
low-level X-ray activity of these sources is probably due to the accretion of
matter onto the neutron star from the winds of the star. BeXBs sometimes show
a periodic increase in X-ray activity rather than being persistent X-ray
generators. This is thought to be caused by changes in the size of the
equatorial disc of the Be star [5]. During the outburst of such systems, the
compact object, i.e., the neutron star and the companion star’s equatorial
disk comes into proximity (periastron passage). Thus we get quiescent emission
from the source [53].

During the modelling of the RX J0440.9+4431 spectrum, it does not show any
presence of narrow absorption features which can be identified as a cyclotron
line as reported in earlier studies [14, 19]. At this luminosity of 2 $\times$
1037 erg s-1 (as observed by _Swift_ /BAT) during Type-II outburst in 2023,
opting for a power-law model with an exponential cutoff was found to be a
feasible choice [14, 54]. It is a characteristic of high accretion rates [14,
55, 56]. The Fe K$\alpha$ line found at 6.4 keV may be generated through the
interaction of the hard X-ray radiation with relatively cooler matter. This
line may be associated with iron atoms in a nearly neutral state (composite of
Fe II and Fe XVIII) [1, 57, 58, 59].

##  V Conclusion

Here, using the AstroSat LAXPC data, we present the findings of the brief
investigation of the temporal and spectral properties of the BeXB system RX
J0440.9+4431 during a giant outburst observed at the beginning of 2023. A
shift from the subcritical to supercritical accretion regime can be inferred
from the change in the intensity and flux hardness during the outburst. The
pulse period of the source was also estimated and it was found to be prominent
at 207.9 seconds. A longer pulse time has been observed as compared to the
previous measurements, which suggests that the rotation of the neutron star
has slowed down during the past ten years. A study of broadband spectra near
the outburst was also carried out. The spectra of the source can be modelled
using interstellar absorption along with high energy cutoff power-law and
blackbody radiation which shows the presence of no narrow absorption features,
such as cyclotron lines. The iron line at 6.4 keV shows the presence of nearly
neutral iron atoms in the interaction of hard X-ray with colder matter.

Overall, our investigation of the outburst of RX J0440.9+4431 reveals
important information on the complicated dynamics of the Be/X-ray binary
during severe events and delivers insightful knowledge about its behaviour and
spectral properties.

##  VI Acknowledgemnet

We express our gratitude to the LAXPC team for their contribution to the
development of the LAXPC instrument. In this study, we used the data obtained
through the HEASARC Online Service, which was made available by NASA/GSFC, in
support of NASA High Energy Astrophysics Programs. This paper also makes use
of data from the AstroSat mission of the Indian Space Research Organisation
(ISRO), which is archived at the Indian Space Science Data Centre (ISSDC). UDG
is thankful to the Inter-University Centre for Astronomy and Astrophysics
(IUCAA), Pune, India for awarding the Visiting Associateship of the institute.

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