# The Laplacian spectral moments of power hypergraphs

Jueru Liu  Lixiang Chen  Changjiang Bu<EMAIL_ADDRESS>College of
Mathematical Sciences, Harbin Engineering University, Harbin 150001, PR China

###### Abstract

The $d$-th order Laplacian spectral moment of a $k$-uniform hypergraph is the
sum of the $d$-th powers of all eigenvalues of its Laplacian tensor. In this
paper, we obtain some expressions of the Laplacian spectral moments for
$k$-uniform power hypergraphs, and these expressions can be represented by
some parameters of graphs. And we show that some graphs can be determined by
their high-order Laplacian spectrum by using the Laplacian spectral moments of
power hypergraphs.

###### keywords:

hypergraph, spectral moment, Laplacian tensor, trace.
_AMS classification (2020):_ 05C50, 05C65, 15A69.

##  1 Introduction

For a $k$-uniform hypergraph $\mathcal{H}$, the $d$-th order (Laplacian)
spectral moment of $\mathcal{H}$ is equal to the sum of the $d$-th powers of
all eigenvalues of its adjacency (Laplacian) tensor. Since the $d$-th order
trace of a tensor is equal to the sum of the $d$-th powers of its all
eigenvalues [1], the $d$-th order (Laplacian) spectral moment of $\mathcal{H}$
is equal to the $d$-th order trace of its adjacency (Laplacian) tensor.

In 2013, Shao et al. [2] gave a formula for the trace of tensors in terms of
some graph parameters. The coefficients of characteristic polynomial and
topological index of hypergraphs can be studied by spectral moments of
hypergraphs [3, 4, 5, 6]. In 2021, Clark and Cooper [4] expressed the spectral
moments of hypergraph by the number of Veblen multi-hypergraphs and obtained
the Harary-Sachs coefficient theorem for hypergraph. A formula for the
spectral moment of a hypertree was given in terms of the number of some
subhypertrees [5], and some high-order cospectral invariants of trees were
given by the spectral moments of hypertrees [7]. In [6], the Estrada index and
subgraph centrality of uniform hypergraphs were studied, which are closely
related to the traces of the adjacency tensor.

For Laplacian spectral moments of hypergraphs, the expressions of the first
$k$ orders traces of the Laplacian tensors were given by the degree sequence
of $k$-uniform hypergraphs [3]. And an expression of the $k+1$-st order trace
of Laplacian tensor of $k$-uniform hypergraphs was given in [8].

In this paper, we study Laplacian spectral moments of power hypergraphs. The
expressions of the first $2k$ orders Laplacian spectral moments of $k$-uniform
power hypergraphs are given, which can be represented by some parameters of
graphs. And we show that some graphs, which are not determined by (signless)
Laplacian spectrum, can be determined by their high-order (signless) Laplacian
spectrum by considering the (signless) Laplacian spectral moments of power
hypergraphs.

##  2 Preliminaries

Next, we introduce some notations and concepts for tensors and hypergraphs.
For a positive integer $n$, let $[n]=\\{1,2,\ldots,n\\}$ and
$[n]^{k}=\\{i_{1}i_{2}\cdots i_{k}|i_{j}\in[n],j=1,\ldots,k\\}$. A $k$-order
$n$-dimension complex tensor $\mathcal{T}=(t_{i\alpha})$ is a multi-
dimensional array with $n^{k}$ entries on complex number field $\mathbb{C}$,
where $i\in[n]$ and $\alpha\in[n]^{k-1}$.

A hypergraph $\mathcal{H}=(V,E)$ consists of vertex set $V=\\{1,2,\ldots,n\\}$
and edge set $E=\\{e_{1},e_{2},\ldots,e_{m}\\}$, where $e_{j}\subseteq
V(\mathcal{H})$ for $j\in[m]$. If $|e_{j}|=k$ for each $j\in[m]$ and $k\geq
2$, then $\mathcal{H}$ is called a k-uniform hypergraph. For a $k$-uniform
hypergraph $\mathcal{H}$ with $n$ vertices, its adjacency tensor
$\mathcal{A}_{\mathcal{H}}=(a_{i\alpha})$ is a $k$-order $n$-dimension tensor
has entries

$a_{i\alpha}=\left\\{\begin{array}[]{ll}\frac{1}{(k-1)!},&\textnormal{if}\
\\{i,i_{2},\ldots,i_{k}\\}\in E(\mathcal{H})\ \textnormal{for}\
\alpha=i_{2}\cdots i_{k},\\\ 0,&\textnormal{otherwise}.\end{array}\right.$

The spectrum and eigenvalues of $\mathcal{A}_{\mathcal{H}}$ is called the
spectrum and eigenvalues of $\mathcal{H}$, repectively [9]. For a vertex $i\in
V(\mathcal{H})$, the degree of $i$ is the number of edges of $\mathcal{H}$
containing the vertex $i$, denoted by $d_{i}$. The degree tensor
$\mathcal{D}_{\mathcal{H}}=\textnormal{diag}(d_{1},\ldots,d_{n})$ of
$\mathcal{H}$ is a $k$-order $n$-dimension diagonal tensor. And tensor
$\mathcal{L}_{\mathcal{H}}=\mathcal{D}_{\mathcal{H}}-\mathcal{A}_{\mathcal{H}}$
is the Laplacian tensor of $\mathcal{H}$ [10].

In 2005, Lim [11] and Qi [12] introduced the eigenvalues of tensors
independently. Denote the set of $n$-dimension complex vectors and the set of
$k$-order $n$-dimension complex tensors by $\mathbb{C}^{n}$ and
$\mathbb{C}^{[k,n]}$, respectively. For a tensor
$\mathcal{T}=(t_{i\alpha})\in\mathbb{C}^{[k,n]}$ and
$x=(x_{1},\ldots,x_{n})^{\mathsf{T}}\in\mathbb{C}^{n}$, $\mathcal{T}x^{k-1}$
is a vector in $\mathbb{C}^{n}$ whose $i$-th component is

$(\mathcal{T}x^{k-1})_{i}=\sum\limits_{\alpha\in[n]^{k-1}}t_{i\alpha}x^{\alpha},$

where $x^{\alpha}=x_{i_{1}}\cdots x_{i_{k-1}}$ if $\alpha=i_{2}\cdots
i_{k-1}$. For a complex number $\lambda\in\mathbb{C}$, if there is a vector
$x\in\mathbb{C}^{n}\setminus\\{0\\}$ such that

$\mathcal{T}x^{k-1}=\lambda x^{[k-1]},$

then $\lambda$ is called an eigenvalue of $\mathcal{T}$ and $x$ is an
eigenvector of $\mathcal{T}$ associated with $\lambda$, where
$x^{[k-1]}=(x_{1}^{k-1},\ldots,x_{n}^{k-1})^{\mathsf{T}}$. The multi-set of
all eigenvalues of tensor $\mathcal{T}$ is the spectrum of $\mathcal{T}$,
denoted by $\sigma(\mathcal{T})$.

In [13], an expression of $d$-th order trace for tensors is given. And Hu et
al. [1] proved that the $d$-th order trace of a $k$-order $n$-dimension tensor
$\mathcal{T}$ is equal to the sum of the $d$-th powers of its all eigenvalues,
that is,
$\textnormal{Tr}_{d}(\mathcal{T})=\sum\nolimits_{\lambda\in\sigma(\mathcal{T})}\lambda^{d}$.

In 2013, Shao et al. [2] gave a formula for
$\textnormal{Tr}_{d}(\mathcal{T})$. Next, we introduce some related notations.
For a positive integer $d$, let

$\mathcal{F}_{d}=\\{(i_{1}\alpha_{1},\ldots,i_{d}\alpha_{d})|\ 1\leq
i_{1}\leq\cdots\leq i_{d}\leq n;\alpha_{1},\ldots,\alpha_{d}\in[n]^{k-1}\\}.$

For $f=(i_{1}\alpha_{1},\ldots,i_{d}\alpha_{d})\in\mathcal{F}_{d}$ and a
$k$-order $n$-dimension tensor $\mathcal{T}=(t_{i\alpha})$, let
$\pi_{f}(\mathcal{T})=\prod\nolimits_{j=1}^{d}t_{i_{j}\alpha_{j}}$. Suppose
$i_{j}\alpha_{j}=i_{j}v_{1}^{(j)}\cdots v_{k-1}^{(j)}$, let
$E_{j}(f)=\\{(i_{j},v_{1}^{(j)}),\ldots,(i_{j},v_{k-1}^{(j)})\\}$ be the set
of arcs from $i_{j}$ to $v_{1}^{(j)},\ldots,v_{k-1}^{(j)}$ and
$E(f)=\bigcup\nolimits_{j=1}^{d}E_{j}(f)$ be an arc multi-set. Let
$V_{j}(f)=\\{i_{j},v_{1}^{(j)},\ldots,v_{k-1}^{(j)}\\}$ and
$V(f)=\bigcup\nolimits_{j=1}^{d}V_{j}(f)$ be a vertex set. Let multi-digraph
$D(f)=(V(f),E(f))$. Let $b(f)$ be the product of the factorials of the
multiplicities of all the arcs in $D(f)$. Let $c(f)$ be the product of the
factorials of the outdegrees of all the vertices in $D(f)$. Let $W(f)$ be the
set of all closed walks with the arc multi-set $E(f)$. In this paper, if a
multi-set $\mathrm{A}$ contains $m$ distinct elements $a_{1},\ldots,a_{m}$
with multiplicities $r_{1},\ldots,r_{m}$ respectively, then we write
$\mathrm{A}=\\{a_{1}^{r_{1}},\ldots,a_{m}^{r_{m}}\\}$.

The formula for the $d$-th order trace of tensors given by Shao et al. is
shown as follows.

######  Lemma 2.1.

[2] Let $\mathcal{T}=(t_{i\alpha})$ be a $k$-order $n$-dimension tensor. Then

$\textnormal{Tr}_{d}(\mathcal{T})=(k-1)^{n-1}\sum_{f\in\mathcal{F}_{d}}\frac{b(f)}{c(f)}\pi_{f}(\mathcal{T})|W(f)|.$
(2.1)

Since the $d$-th order Laplacian spectral moment of $\mathcal{H}$ is equal to
the $d$-th order trace of its Laplacian tensor, we study the Laplacian
spectral moment of uniform hypergraphs by considering the trace formula of
tensor given by Shao et al.

For a $k$-uniform hypergraph $\mathcal{H}$ with $n$ vertices, let
$\mathcal{L}_{\mathcal{H}}$ be the Laplacian tensor of $\mathcal{H}$. When
$\mathcal{T}=\mathcal{L}_{\mathcal{H}}$ in Eq.(2.1), the $d$-th order
Laplacian spectral moment of $\mathcal{H}$ is

$\textnormal{Tr}_{d}(\mathcal{L}_{\mathcal{H}})=(k-1)^{n-1}\sum_{f\in\mathcal{F}_{d}}\frac{b(f)}{c(f)}\pi_{f}(\mathcal{L}_{\mathcal{H}})|W(f)|.$
(2.2)

Next, we simplify Eq.(2.2) by classifying $f$ and introduce some related
concepts. For $i_{j}\alpha_{j}\in[n]^{k}$ and a $k$-order $n$-dimension tensor
$\mathcal{T}=(t_{i\alpha})$, the entry $t_{i_{j}\alpha_{j}}$ in tensor
$\mathcal{T}$ is called the corresponding entry of $i_{j}\alpha_{j}$. Suppose
$\alpha_{j}=v_{1}^{(j)}\cdots v_{k-1}^{(j)}$, for a $k$-uniform hypergraph
$\mathcal{H}$, $e=\\{i_{j},v_{1}^{(j)},\ldots,v_{k-1}^{(j)}\\}$ is called the
corresponding edge of tuple $i_{j}\alpha_{j}$ if the corresponding entry of
$i_{j}\alpha_{j}$ in its adjacency tensor is not equal to zero.

Let $\pi_{f}(\mathcal{L}_{\mathcal{H}})|W(F)|\neq 0$ for
$f=(i_{1}\alpha_{1},\ldots,i_{d}\alpha_{d})\in\mathcal{F}_{d}$. Since
$\pi_{f}(\mathcal{L}_{\mathcal{H}})=\prod\nolimits_{j=1}^{d}l_{i_{j}\alpha_{j}}\neq
0$, we know $l_{i_{j}\alpha_{j}}\neq 0$ for all $j\in[d]$. Then the tuple
$i_{j}\alpha_{j}(j=1,\ldots,d)$ in $f$ corresponds either to a diagonal entry
of $\mathcal{L}_{\mathcal{H}}$ or to an edge of $\mathcal{H}$. According to
the number of the tuples which correspond to the diagonal entries of
$\mathcal{L}_{\mathcal{H}}$, the set $\\{f\in\mathcal{F}_{d}|\
\pi_{f}(\mathcal{L}_{\mathcal{H}})\neq 0\\}$ can be represented as the union
of three disjoint sets, that is,

$\\{f\in\mathcal{F}_{d}|\ \pi_{f}(\mathcal{L}_{\mathcal{H}})\neq
0\\}=\mathcal{F}_{d}^{(1)}\cup\mathcal{F}_{d}^{(2)}\cup\mathcal{F}_{d}^{(3)},$
(2.3)

where $\mathcal{F}_{d}^{(1)}=\\{f\in\mathcal{F}_{d}|\textnormal{ all tuples
in}\ f\ \textnormal{correspond to diagonal entry of}\
\mathcal{L}_{\mathcal{H}}\\}$,
$\mathcal{F}_{d}^{(2)}=\\{f=(i_{1}\alpha_{1},\ldots,i_{d}\alpha_{d})\in\mathcal{F}_{d}|\
\alpha_{j}=v_{1}^{(j)}\cdots v_{k-1}^{(j)}\ \textnormal{and}\
\\{i_{j},v_{1}^{(j)},\ldots,v_{k-1}^{(j)}\\}\in E(\mathcal{H})\
\textnormal{for}\ j=1,\ldots,d\\}$,
$\mathcal{F}_{d}^{(3)}=\\{f\in\mathcal{F}_{d}|\
\pi_{f}(\mathcal{L}_{\mathcal{H}})\neq
0\\}\setminus(\mathcal{F}_{d}^{(1)}\cup\mathcal{F}_{d}^{(2)}).$

######  Lemma 2.2.

Let $\mathcal{H}$ be a $k$-uniform hypergraph with $n$ vertices. And the
degree sequence of $\mathcal{H}$ is $d_{1},d_{2},\ldots,d_{n}$. Then

$(k-1)^{n-1}\sum\limits_{f\in\mathcal{F}_{d}^{(1)}}\frac{b(f)}{c(f)}\pi_{f}(\mathcal{L}_{\mathcal{H}})|W(f)|=(k-1)^{n-1}\sum\limits_{i=1}^{n}d_{i}^{d},$
(2.4)
$(k-1)^{n-1}\sum\limits_{f\in\mathcal{F}_{d}^{(2)}}\frac{b(f)}{c(f)}\pi_{f}(\mathcal{L}_{\mathcal{H}})|W(f)|=(-1)^{d}\textnormal{Tr}_{d}(\mathcal{A}_{\mathcal{H}}).$
(2.5)

###### Proof.

For $f\in\mathcal{F}_{d}^{(1)}$, if $f=(i_{1}i_{1}\cdots
i_{1},\ldots,i_{d}i_{d}\cdots i_{d})$, since the arc multi-set $E(f)$ only
includes loops $(i_{j},i_{j})\ (j=1,\ldots,d)$, we know that $|W(f)|\neq 0$ if
and only if $i_{1}=\cdots=i_{d}$. Let $f_{i}=(ii\cdots i,\ldots,ii\cdots
i)\in\mathcal{F}_{d}(i=1,\ldots,n)$, then
$\mathcal{F}_{d}^{(1)}=\\{f_{1},\ldots,f_{n}\\}$. For
$f_{i}\in\mathcal{F}_{d}^{(1)}$, since $b(f_{i})=c(f_{i})=(d(k-1))!$,
$|W(f_{i})|=1$ and $\pi_{f_{i}}(\mathcal{L}_{\mathcal{H}})=l_{ii\cdots
i}^{d}=d_{i}^{d}$, Eq.(2.4) can be obtained directly.

For $f=(i_{1}\alpha_{1},\ldots,i_{d}\alpha_{d})\in\mathcal{F}_{d}^{(2)}$,
where $\alpha_{j}=v_{1}^{(j)}\cdots v_{k-1}^{(j)}$ for $j=1,\ldots,d$. Since
$\\{i_{j},v_{1}^{(j)},\ldots,v_{k-1}^{(j)}\\}\in E(\mathcal{H})$ for
$j=1,\ldots,d$, we have
$\pi_{f}(\mathcal{L}_{\mathcal{H}})=\prod\nolimits_{j=1}^{d}l_{i_{j}\alpha_{j}}=(-\frac{1}{(k-1)!})^{d}=(-1)^{d}\pi_{f}(\mathcal{A}_{\mathcal{H}}).$
And $\pi_{f}(\mathcal{A}_{\mathcal{H}})\neq 0$ if and only if
$\\{i_{j},v_{1}^{(j)},\ldots,v_{k-1}^{(j)}\\}\in E(\mathcal{H})$ for
$j=1,\ldots,d$, that is, $f\in\mathcal{F}_{d}^{(2)}$, then Eq.(2.5) can be
obtained. ∎

According to Lemma 2.2, in order to obtain the expressions of the first $2k$
orders Laplacian spectral moments for $k$-uniform power hypergraphs, we should
give some expressions of the spectral moments for $k$-uniform power
hypergraphs.

For a graph $G$ and a positive integer $k\geq 3$, the $k$-power hypergraph of
$G$, denoted by $G^{(k)}$, is a $k$-uniform hypergraph obtained by adding
$k-2$ new vertices whose degrees are $1$ to each edge of $G$ [14]. The
spectrum of a hypergraph is said to be $k$-symmetric, if its spectrum is
invariant under a rotation of an angle $2\pi/k$ in the complex plane. Shao et
al. [2] gave a characterization (in terms of the traces of the adjacency
tensors) of the $k$-uniform hypergraphs whose spectrum are $k$-symmetric, that
is, the spectrum of a $k$-uniform hypergraph $\mathcal{H}$ is $k$-symmetric if
and only if $\textnormal{Tr}_{d}(\mathcal{A}_{\mathcal{H}})=0$ for $k\nmid d$.
It is obvious that the spectrum of a $k$-uniform power hypergraph is
$k$-symmetric. Then, the $d$-th spectral moments of $G^{(k)}$ are equal to $0$
for $d=k+1,\ldots,2k-1$, that is,

$\textnormal{Tr}_{d}(\mathcal{A}_{G^{(k)}})=0\ \mathrm{for}\
d=k+1,\ldots,2k-1.$ (2.6)

And the expression of the $2k$-th order spectral moment of $G^{(k)}$ is given
as follows.

######  Lemma 2.3.

Let $G$ be a graph with $n$ vertices and $m$ edges. Let $d_{i}$ denote the
degree of vertex $i$ in $G$ ($i=1,\ldots,n$). Then the $2k$-th order spectral
moment of $G^{(k)}$ is

$\textnormal{Tr}_{2k}(\mathcal{A}_{G^{(k)}})=k^{k-1}(k-1)^{N-k}\big{(}1-2k^{k-3}(k-1)^{1-k}\big{)}m+k^{2k-3}(k-1)^{N-2k+1}\sum\limits_{i=1}^{n}d_{i}^{2},$
(2.7)

where $N=n+m(k-2)$.

###### Proof.

Let $\mathcal{G}=G^{(k)}$. Then $|V(\mathcal{G})|=n+m(k-2)=N$ and
$|E(\mathcal{G})|=m$. Let $N_{G}(P_{2})$ and $N_{\mathcal{G}}(P_{2}^{(k)})$
denote the number of paths with length $2$ in $G$ and $\mathcal{G}$,
respectively. Then
$N_{\mathcal{G}}(P_{2}^{(k)})=N_{G}(P_{2})=\sum\limits_{i=1}^{n}\dbinom{d_{i}}{2}$.
From Lemma 2.1, we get

$\textnormal{Tr}_{2k}(\mathcal{A}_{\mathcal{G}})=(k-1)^{N-1}\sum\limits_{f\in\mathcal{F}_{2k}}\frac{b(f)}{c(f)}\pi_{f}(\mathcal{A}_{\mathcal{G}})|W(f)|.$

For $f=(i_{1}\alpha_{1},\ldots,i_{2k}\alpha_{2k})\in\mathcal{F}_{2k}$, if
$\pi_{f}(\mathcal{A}_{\mathcal{G}})=\prod_{j=1}^{2k}a_{i_{j}\alpha_{j}}\neq
0$, then $a_{i_{j}\alpha_{j}}\neq 0$ for all $j\in[2k]$. For $|W(f)|\neq 0$,
there are the following two cases.

Case 1:
$f=(i_{1}\alpha_{1},i_{1}\beta_{1},\ldots,i_{k}\alpha_{k},i_{k}\beta_{k})=f_{e}\in\mathcal{F}_{2k}$,
where $\\{i_{1},\ldots,i_{k}\\}=e\in E(\mathcal{G})$ and
$\alpha_{j},\beta_{j}\in\big{(}\\{i_{1},\ldots,i_{k}\\}\setminus\\{i_{j}\\}\big{)}^{k-1}$
for $j=1,\ldots,k$. Then

$\displaystyle\sum_{e\in
E(\mathcal{G})}\frac{b(f_{e})}{c(f_{e})}\pi_{f_{e}}(\mathcal{A}_{\mathcal{G}})|W(f_{e})|$
$\displaystyle=$ $\displaystyle\sum_{e\in
E(\mathcal{G})}\frac{(2!)^{k(k-1)}}{\big{(}(2k-2)!\big{)}^{k}}\Big{(}\frac{1}{(k-1)!}\Big{)}^{2k}\frac{2k(k-1)}{(2!)^{k(k-1)}}2^{k-1}k^{k-2}\big{(}(2k-3)!\big{)}^{k}\big{(}(k-1)!\big{)}^{2k}$
$\displaystyle=$ $\displaystyle k^{k-1}(k-1)^{1-k}|E(\mathcal{G})|.$

Case 2:
$f=(i_{1}\alpha_{1},j_{1}\beta_{1},i_{2}\alpha_{2},\ldots,i_{k}\alpha_{k},j_{2}\beta_{2},\ldots,j_{k}\beta_{k})=f_{e_{1}e_{2}}\in\mathcal{F}_{2k}$,
where $i_{1}=j_{1}$, $\\{i_{1},i_{2},\ldots,i_{k}\\}=e_{1}\in E(\mathcal{G}),\
\\{i_{1},j_{2},\ldots,j_{k}\\}=e_{2}\in E(\mathcal{G})$ and
$\alpha_{l}\in\big{(}\\{i_{1},\ldots,i_{k}\\}\setminus\\{i_{l}\\}\big{)}^{k-1},\
\beta_{l}\in\big{(}\\{j_{1},\ldots,j_{k}\\}\setminus\\{j_{l}\\}\big{)}^{k-1}$
for $l=1,\ldots,k$. Then

$\displaystyle\sum_{e_{1}e_{2}\subset\mathcal{G}}\frac{b(f_{e_{1}e_{2}})}{c(f_{e_{1}e_{2}})}\pi_{f_{e_{1}e_{2}}}(\mathcal{A}_{\mathcal{G}})|W(f_{e_{1}e_{2}})|$
$\displaystyle=$
$\displaystyle\sum_{e_{1}e_{2}\subset\mathcal{G}}\frac{2k(k-1)(k^{k-2})^{2}(2k-3)!\big{(}(k-2)!\big{)}^{2k-2}}{\big{(}2(k-1)\big{)}!\big{(}(k-1)!\big{)}^{2k-2}}\Big{(}\frac{1}{(k-1)!}\Big{)}^{2k}2\big{(}(k-1)!\big{)}^{2k}$
$\displaystyle=$ $\displaystyle
2k^{2k-3}(k-1)^{2-2k}N_{\mathcal{G}}(P_{2}^{(k)}).$

Then

$\displaystyle\textnormal{Tr}_{2k}(\mathcal{A}_{\mathcal{G}})$
$\displaystyle=(k-1)^{N-1}\Big{(}k^{k-1}(k-1)^{1-k}|E(\mathcal{G})|+2k^{2k-3}(k-1)^{2-2k}N_{\mathcal{G}}(P_{2}^{(k)})\Big{)}$
$\displaystyle=k^{k-1}(k-1)^{N-k}\big{(}1-2k^{k-3}(k-1)^{1-k}\big{)}m+k^{2k-3}(k-1)^{N-2k+1}\sum\limits_{i=1}^{n}d_{i}^{2},$

where $N=n+m(k-2)$. ∎

##  3 Main results

In this section, we give an expression of the $d$-th order Laplacian spectral
moments for $k$-uniform hypergraphs. And the explicit expressions of the first
$2k$ orders Laplacian spectral moments for $k$-uniform power hypergraphs are
given.

Given two hypergraphs $\mathcal{H}=(V(\mathcal{H}),E(\mathcal{H}))$ and
$H=(V(H),E(H))$, if $V(H)\subseteq V(\mathcal{H})$ and $E(H)\subseteq
E(\mathcal{H})$, then $H$ is said to be a subhypergraph of $\mathcal{H}$. A
$k$-uniform multi-hypergraph $\mathcal{H}$ is a pair
$(V(\mathcal{H}),E(\mathcal{H}))$, where $E(\mathcal{H})$ is a multi-set of
subsets of $V(\mathcal{H})$ with cardinality $k$. A Veblen hypergraph is a
$k$-uniform, $k$-valent (i.e., the degree of every vertex is a multiple of
$k$) multi-hypergraph [4]. For a multi-hypergraph $H$, let $\underline{H}$ be
the simple $k$-uniform hypergraph formed by removing duplicate edges of $H$.
And $H$ is called a multi-subgraph of $\mathcal{H}$ if $\underline{H}$ is a
subhypergraph of $\mathcal{H}$. Let $\mathcal{V}_{d}(\mathcal{H})$ denote the
set of connected Veblen multi-subgraph of $\mathcal{H}$ with $d$ edges.

For $f=(i_{1}\alpha_{1},\ldots,i_{d}\alpha_{d})\in\mathcal{F}_{d}$ and a
$k$-uniform hypergraph $\mathcal{H}$ (where $\alpha_{j}=v_{1}^{(j)}\cdots
v_{k-1}^{(j)}$ for $j=1,\ldots,d$), the multi-subgraph induced by $f$, denoted
by $H(f)$, is the multi-hypergraph with the vertex set $V(f)\subseteq
V(\mathcal{H})$ and the edge multi-set
$E(H(f))=\\{\\{i_{j},v_{1}^{(j)},\ldots,v_{k-1}^{(j)}\\}|\
(\mathcal{A}_{\mathcal{H}})_{i_{j}\alpha_{j}}\neq 0,\ 1\leq j\leq d\\}$, and
$\underline{H}(f)$ is a subhypergraph of $\mathcal{H}$.

A walk ia a digraph $D$ is a non-empty alternating sequence
$v_{0}e_{0}v_{1}e_{1}\cdots v_{k}e_{k}$ of vertices and arcs in $D$ such that
$e_{i}=(v_{i},v_{i+1})$ for all $i<k$. A walk is closed if $v_{0}=v_{k}$. A
closed walk in a digraph is an Eulerian closed walk if it traverses each arc
of this digraph exactly once. A digraph $D$ is called Eulerian if $D$ has an
Eulerian closed walk. Let $d^{+}(v)$ and $d^{-}(v)$ be the outdegree and
indegree of the vertex $v\in V(D)$, respectively. The digraph $D$ is Eulerian
if and only if $d^{+}(v)=d^{-}(v)$ for all $v\in V(D)$ and $D$ is weakly
connected.

Since $W(f)$ is the set of all closed walks with the arc multi-set $E(f)$, we
know that $|W(f)|$ is equal to the number of Eulerian closed walks in the
multi-digraph $D(f)$. For $f\in\mathcal{F}_{d}^{(3)}$, we give the following
conclusion.

######  Lemma 3.4.

Let $\mathcal{H}$ be a $k$-uniform hypergraph with $n$ vertices. If
$f\in\mathcal{F}_{d}^{(3)}$ and $|W(f)|\neq 0$, then the multi-subgraph $H(f)$
induced by $f$ is a connected Veblen multi-subgraph of $\mathcal{H}$ with at
most $d-1$ edges.

###### Proof.

For $f=(i_{1}\alpha_{1},\ldots,i_{d}\alpha_{d})\in\mathcal{F}_{d}$ (where
$\alpha_{j}=v_{1}^{(j)}\cdots v_{k-1}^{(j)}$ for $j=1,\ldots,d$), if
$|W(f)|\neq 0$, then the multi-digraph $D(f)=(V(f),E(f))$ is Eulerian. For all
$v\in V(f)$, we have $d^{+}_{D(f)}(v)=d^{-}_{D(f)}(v)$ and

$\displaystyle d^{+}_{D(f)}(v)$ $\displaystyle=(k-1)|\\{i_{j}\alpha_{j}|\
i_{j}=v\\}|$ $\displaystyle=(k-1)\big{(}|\\{i_{j}\alpha_{j}|\ i_{j}=v\
\textnormal{and}\ \\{i_{j},v_{1}^{(j)}\cdots v_{k-1}^{(j)}\\}\in
E(\mathcal{H})\\}|+|\\{i_{j}\alpha_{j}|\ i_{j}\alpha_{j}=vv\cdots
v\\}|\big{)},$ $\displaystyle d^{-}_{D(f)}(v)$
$\displaystyle=|\\{i_{j}\alpha_{j}|\ i_{j}\neq v\ \textnormal{and}\ v\in
V_{j}(f)\\}|+(k-1)|\\{i_{j}\alpha_{j}|\ i_{i}\alpha_{j}=vv\cdots v\\}|.$

Then

$(k-1)|\\{i_{j}\alpha_{j}|\ i_{j}=v\ \textnormal{and}\
\\{i_{j},v_{1}^{(j)}\cdots v_{k-1}^{(j)}\\}\in
E(\mathcal{H})\\}|=|\\{i_{j}\alpha_{j}|\ i_{j}\neq v\ \textnormal{and}\ v\in
V_{j}(f)\\}|.$

Fix a vertex $v\in V(H(f))$. We have

$\displaystyle d_{H(f)}(v)$ $\displaystyle=|\\{i_{j}\alpha_{j}|\ i_{j}=v\
\textnormal{and}\ \\{i_{j},v_{1}^{(j)}\cdots v_{k-1}^{(j)}\\}\in
E(\mathcal{H})\\}|+|\\{i_{j}\alpha_{j}|\ i_{j}\neq v\ \textnormal{and}\ v\in
V_{j}(f)\\}|$ $\displaystyle=k|\\{i_{j}\alpha_{j}|\ i_{j}=v\ \textnormal{and}\
\\{i_{j},v_{1}^{(j)}\cdots v_{k-1}^{(j)}\\}\in E(\mathcal{H})\\}|.$

So $k|d_{H(f)}(v)$, it follows that $H(f)$ is a Veblen hypergraph by
definition. And $f\in\mathcal{F}_{d}^{(3)}$, then $H(f)$ has at most $d-1$
edges. ∎

For a connected Veblen multi-subgraph $H$ of $\mathcal{H}$ and
$f=(i_{1}\alpha_{1},\ldots,i_{d}\alpha_{d})\in\mathcal{F}_{d}$, $f$ is called
corresponding to $H$ if $f$ satisfy the following conditions:

(a) there is a integer $l(1\leq l\leq d-1)$, such that
$i_{j_{1}}\alpha_{j_{1}},\ldots,i_{j_{l}}\alpha_{j_{l}}$ are corresponding to
some edges of $H$;

(b) for every edge $e\in E(H)$, there exists $j\in[d]$ such that
$i_{j}\alpha_{j}$ is corresponding to $e$;

(c) and others in $f$ are $v\beta_{v}$ where $\beta_{v}=v\cdots v\in[n]^{k-1}$
for $v\in V(H)$.

Let $\mathcal{F}_{d}(H)=\\{f\in\mathcal{F}_{d}|\ f\ \textnormal{is
corresponding to}\ H\\}$. From Lemma 3.1, we have

$\\{f\in\mathcal{F}_{d}^{(3)}||W(f)|\neq
0\\}=\bigcup\limits_{z=1}^{d-1}\bigcup\limits_{H\in\mathcal{V}_{z}(\mathcal{H})}\mathcal{F}_{d}(H).$

For simplicity, $\tau(D(f))$ is abbreviated to $\tau(f)$, which is the number
of arborescences in multi-digraph $D(f)$. According to the above process, the
formula for the $d$-th order Laplacian spectral moment of $k$-uniform
hypergraphs is given as follows.

######  Theorem 3.5.

Let $\mathcal{H}$ be a $k$-uniform hypergraph with $n$ vertices. And the
degree sequence of $\mathcal{H}$ is $d_{1},d_{2},\ldots,d_{n}$. Then

$\displaystyle\textnormal{Tr}_{d}(\mathcal{L}_{\mathcal{H}})=(k-1)^{n-1}\sum\limits_{i=1}^{n}d_{i}^{d}+(-1)^{d}\textnormal{Tr}_{d}(\mathcal{A}_{\mathcal{H}})+d(k-1)^{n}\sum\limits_{z=1}^{d-1}\sum\limits_{H\in\mathcal{V}_{z}(\mathcal{H})}\sum\limits_{f\in\mathcal{F}_{d}(H)}\frac{\tau(f)\pi_{f}(\mathcal{L}_{\mathcal{H}})}{\prod\limits_{v\in
V(f)}d^{+}(v)}.$

###### Proof.

From Eq.(2.3), the $d$-th Laplacian spectral moments of $\mathcal{H}$ is

$\textnormal{Tr}_{d}(\mathcal{L}_{\mathcal{H}})=(k-1)^{n-1}\sum\limits_{j=1}^{3}\sum\limits_{f\in\mathcal{F}_{d}^{(j)}}\frac{b(f)}{c(f)}\pi_{f}(\mathcal{L}_{\mathcal{H}})|W(f)|.$
(3.1)

For $f\in\mathcal{F}_{d}^{(3)}$, let $\widetilde{D}(f)$ be the digraph
obtained by removing all repeated arcs of $D(f)$. Then
$c(f)=\prod\nolimits_{v\in V(f)}d^{+}(v)!$,
$b(f)=\prod_{e\in\widetilde{D}(f)}w(e)!$ and $|E(f)|=d(k-1)$. From Theorem 6
in [15], the number of Eulerian closed walks in $D(f)$ is

$|W(f)|=\frac{|E(f)|}{b(f)}|\mathfrak{E}(f)|,$ (3.2)

where $|\mathfrak{E}(f)|$ is the number of the Eulerian circuits in $D(f)$.

From BEST Theorem [16, 17], the number of the Eulerian circuits in $D(f)$ is

$|\mathfrak{E}(f)|=\tau(f)\prod\limits_{v\in V(f)}(d^{+}(v)-1)!.$ (3.3)

According to Eq.(3.2) and Eq.(3.3), we have

$\displaystyle(k-1)^{n-1}\sum\limits_{f\in\mathcal{F}_{d}^{(3)}}\frac{b(f)}{c(f)}\pi_{f}(\mathcal{L}_{\mathcal{H}})|W(f)|$
(3.4) $\displaystyle=$
$\displaystyle(k-1)^{n-1}\sum\limits_{z=1}^{d-1}\sum\limits_{H\in\mathcal{V}_{z}(\mathcal{H})}\sum\limits_{f\in\mathcal{F}_{d}(H)}\frac{b(f)}{c(f)}\pi_{f}(\mathcal{L}_{\mathcal{H}})|W(f)|$
$\displaystyle=$ $\displaystyle
d(k-1)^{n}\sum\limits_{z=1}^{d-1}\sum\limits_{H\in\mathcal{V}_{z}(\mathcal{H})}\sum\limits_{f\in\mathcal{F}_{d}(H)}\frac{\tau(f)}{\prod\limits_{v\in
V(f)}d^{+}(v)}\pi_{f}(\mathcal{L}_{\mathcal{H}}).$

Then we can obtain the expression for the $d$-th order Laplacian spectral
moment of $\mathcal{H}$ by substituting Eq.(2.4), Eq.(2.5) and Eq.(3.4) into
Eq.(3.1). ∎

######  Remark 3.6.

Since $\textnormal{Tr}_{d}(\mathcal{A}_{\mathcal{H}})=0\ (d=1,\ldots,k-1)$ and
$\textnormal{Tr}_{k}(\mathcal{A}_{\mathcal{H}})=k^{k-1}(k-1)^{n-k}|E(\mathcal{H})|$
[9], $\mathcal{V}_{d}(\mathcal{H})=\emptyset$ for $d=1,\ldots,k-1$. The
expressions of the first $k$ orders Laplacian spectral moments of a
$k$-uniform hypergraph $\mathcal{H}$ can be obtained directly by using the
formulas given in Theorem 3.1. And these expressions have been given in [3].

######  Remark 3.7.

Since
$\textnormal{Tr}_{k+1}(\mathcal{A}_{\mathcal{H}})=(k+1)(k-1)^{n-k}C_{k}(\\#\
\textnormal{of simplices in}\ \mathcal{H})$ [9], and
$\mathcal{V}_{k}(\mathcal{H})=\\{ke|\ e\in E(\mathcal{H})\\}$. The expressions
of the $k+1$-st orders Laplacian spectral moments of a $k$-uniform hypergraph
$\mathcal{H}$ can be obtained by using the formulas given in Theorem 3.1. And
these expressions have been given in [8].

If the $k$-uniform hypergraph is the $k$-power hypergraph $G^{(k)}$ of a graph
$G$ in Remark 3.1 and Remark 3.2, the expressions of the first $k+1$ order
Laplacian spectral moments for $G^{(k)}$ can be obtained.

For a graph $G$, the expressions of the first $2k$ orders Laplacian spectral
moments of its $k$-power hypergraph $G^{(k)}$ can be given by considering the
formulas shown in Theorem 3.1, and these expressions are represented by some
parameters of $G$.

######  Theorem 3.8.

Let $G$ be a graph with $n$ vertices and $m$ edges. Let $d_{i}$ denote the
degree of vertex $i$ in $G$ ($i=1,\ldots,n$). For the $k$-power hypergraph
$G^{(k)}$ of $G$, then

$\displaystyle\textnormal{Tr}_{d}(\mathcal{L}_{G^{(k)}})$
$\displaystyle=(k-1)^{N-1}\sum\limits_{i=1}^{n}d_{i}^{d}+(-1)^{k}dk^{k-2}(k-1)^{N-k}\Big{(}\sum\limits_{i-1}^{n}d_{i}^{d-k+1}+\sum\limits_{\\{i,j\\}\in
E(G)}N_{d-k}(d_{i},d_{j})\Big{)}$
$\displaystyle+(k-1)^{N-k}\big{(}(k-1)^{k-1}+(-1)^{k}dk\big{)}(k-2)m,$

for $d=k+1,\ldots,2k-1$, and

$\displaystyle\textnormal{Tr}_{2k}(\mathcal{L}_{G^{(k)}})$
$\displaystyle=(k-1)^{N-1}\sum\limits_{i=1}^{n}d_{i}^{2k}+(-1)^{k}2k^{k-1}(k-1)^{N-k}\Big{(}\sum\limits_{i-1}^{n}d_{i}^{k+1}+\sum\limits_{\\{i,j\\}\in
E(G)}N_{k}(d_{i},d_{j})\Big{)}$
$\displaystyle+k^{2k-3}(k-1)^{N-2k+1}\sum\limits_{i=1}^{n}d_{i}^{2}+\ell m,$

where $N=n+m(k-2)$,
$N_{s}(d_{i},d_{j})=\sum\nolimits_{\begin{subarray}{c}1\leq c_{i}+c_{j}\leq
s\\\ 0\leq
c_{i},c_{j}<s\end{subarray}}d_{i}^{c_{i}}d_{j}^{c_{j}}\big{(}s=1,\ldots,k\big{)}$,
$\ell=(k-1)^{N-k}\big{(}(k-1)^{k-1}(k-2)+(-1)^{k}2k^{k-1}(k-2)+k^{k-1}-2k^{2k-3}(k-1)^{1-k}\big{)}$.

###### Proof.

Let $\mathcal{G}=G^{(k)}$, then $|V(\mathcal{G})|=N=n+m(k-2)$ and
$|E(\mathcal{G})|=m$.

Since $\mathcal{G}$ is the $k$-power hypergraph of $G$, from the definition of
Veblen hypergraph, we know that $\mathcal{V}_{d}(\mathcal{G})=\emptyset$ for
$k\nmid d$ and $d\in[2k]$. For a Veblen multi-subgraph
$H\in\mathcal{V}_{k}(\mathcal{G})$, it is easy to see that $\underline{H}\in
E(\mathcal{G})$. For convenience, let $ke$ denote the connected Veblen multi-
subgraph with $k$ edges and $\underline{ke}=e\in E(\mathcal{G})$. Then,
$\mathcal{V}_{k}(\mathcal{G})=\\{ke|\ e\in E(\mathcal{G})\\}$. For
$d=k+1,\ldots,2k$, we have

$\displaystyle\textnormal{Tr}_{d}(\mathcal{L}_{\mathcal{G}})=$
$\displaystyle(k-1)^{N-1}\Big{(}m(k-2)+\sum\limits_{i=1}^{n}d_{i}^{d}\Big{)}+(-1)^{d}\textnormal{Tr}_{d}(\mathcal{A}_{\mathcal{G}})$
$\displaystyle+d(k-1)^{N}\sum\limits_{e\in
E(\mathcal{G})}\sum\limits_{f\in\mathcal{F}_{d}(ke)}\frac{\tau(f)}{\prod\limits_{v\in
V(f)}d^{+}(v)}\pi_{f}(\mathcal{L}_{\mathcal{G}}).$

For $f\in\mathcal{F}_{d}(ke)\ (\textnormal{where}\
e=\\{i_{1},i_{2},\ldots,i_{k}\\}\in E(\mathcal{G}))$, let

$f=((i_{1}\beta_{1})^{c_{1}},i_{1}\alpha_{1},(i_{2}\beta_{2})^{c_{2}},i_{2}\alpha_{2},\ldots,(i_{k}\beta_{k})^{c_{k}},i_{k}\alpha_{k}),$

where $i_{1}<i_{2}<\cdots<i_{k}$. For any $j\in[k]$,
$\alpha_{j}\in(\\{i_{1},\ldots,i_{k}\\}\setminus\\{i_{j}\\})^{k-1}$,
$\beta_{j}=i_{j}\cdots i_{j}\in[N]^{k-1}$, $c_{j}\geq 0$ is the total number
of times that $i_{j}\beta_{j}$ appears in $f$, and
$\sum\nolimits_{j=1}^{k}c_{j}=d-k$. Next, we consider the following two cases
for $f\in\mathcal{F}_{d}(ke)$.

Case 1: If there exists $j\in[k]$ such that $c_{j}=d-k$, then

$f=f_{e,i_{j}}=(i_{1}\alpha_{1},\ldots,i_{j-1}\alpha_{j-1},(i_{j}\beta_{j})^{d-k},i_{j}\alpha_{j},\ldots,i_{k}\alpha_{k})\in\mathcal{F}_{d}(ke).$

We have

$\tau(f)=k^{k-2},\ \prod\limits_{v\in V(f)}d^{+}(v)=(d-k+1)(k-1)^{k},$

and there are $(d-k+1)((k-1)!)^{k}$ elements in $\mathcal{F}_{d}$ which share
the same arc multi-set as $f_{e,i_{j}}$, then

$\sum\limits_{j=1}^{k}\frac{\tau(f)}{\prod\limits_{v\in
V(f)}d^{+}(v)}\pi_{f}(\mathcal{L}_{\mathcal{G}})=(-1)^{k}k^{k-2}(k-1)^{-k}\sum\limits_{j=1}^{k}d_{i_{j}}^{d-k}.$

Case 2: If for any $j\in[k]$, $0\leq c_{j}<d-k$, then

$f=f_{e,\\{c_{1},c_{2},\ldots,c_{k}\\}}=((i_{1}\beta_{1})^{c_{1}},i_{1}\alpha_{1},(i_{2}\beta_{2})^{c_{2}},i_{2}\alpha_{2},\ldots,(i_{k}\beta_{k})^{c_{k}},i_{k}\alpha_{k})\in\mathcal{F}_{d}(ke).$

We have

$\tau(f)=k^{k-2},\ \prod\limits_{v\in
V(f)}d^{+}(v)=(k-1)^{k}\prod\limits_{j=1}^{k}(c_{j}+1),$

and there are $((k-1)!)^{k}\prod\limits_{j=1}^{k}(c_{j}+1)$ elements in
$\mathcal{F}_{d}$ which share the same arc multi-set as
$f_{e,\\{c_{1},c_{2},\ldots,c_{k}\\}}$, then

$\sum\limits_{\begin{subarray}{c}c_{1}+\cdots+c_{k}=d-k\\\ \forall
j\in[k],0\leq c_{j}<d-k\end{subarray}}\frac{\tau(f)}{\prod\limits_{v\in
V(f)}d^{+}(v)}\pi_{f}(\mathcal{L}_{\mathcal{G}})=(-1)^{k}k^{k-2}(k-1)^{-k}\sum\limits_{\begin{subarray}{c}c_{1}+\cdots+c_{k}=d-k\\\
\forall j\in[k],0\leq
c_{j}<d-k\end{subarray}}\prod\limits_{j=1}^{k}d_{i_{j}}^{c_{j}}.$

Then

$\displaystyle\sum\limits_{e\in
E(\mathcal{G})}\sum\limits_{f\in\mathcal{F}_{d}(ke)}\frac{\tau(f)}{\prod\limits_{v\in
V(f)}d^{+}(v)}\pi_{f}(\mathcal{L}_{\mathcal{G}})$ $\displaystyle=$
$\displaystyle(-1)^{k}k^{k-2}(k-1)^{-k}\sum\limits_{\\{i_{1},\ldots,i_{k}\\}\in
E(\mathcal{G})}\bigg{(}\sum\limits_{j=1}^{k}d_{i_{j}}^{d-k}+\sum\limits_{\begin{subarray}{c}c_{1}+\cdots+c_{k}=d-k\\\
\forall j\in[k],0\leq
c_{j}<d-k\end{subarray}}\prod\limits_{j=1}^{k}d_{i_{j}}^{c_{j}}\bigg{)}$
$\displaystyle=$
$\displaystyle(-1)^{k}k^{k-2}(k-1)^{-k}\bigg{(}\sum\limits_{i=1}^{N}\sum\limits_{e\in
E_{i}}d_{i}^{d-k}+\sum\limits_{\\{i_{1},\ldots,i_{k}\\}\in
E(\mathcal{G})}\sum\limits_{\begin{subarray}{c}c_{1}+\cdots+c_{k}=d-k\\\
\forall j\in[k],0\leq
c_{j}<d-k\end{subarray}}\prod\limits_{j=1}^{k}d_{i_{j}}^{c_{j}}\bigg{)}$
$\displaystyle=$
$\displaystyle(-1)^{k}k^{k-2}(k-1)^{-k}\bigg{(}\sum\limits_{i=1}^{N}d_{i}^{d-k+1}+\sum\limits_{\\{i_{1},\ldots,i_{k}\\}\in
E(\mathcal{G})}\sum\limits_{\begin{subarray}{c}c_{1}+\cdots+c_{k}=d-k\\\
\forall j\in[k],0\leq
c_{j}<d-k\end{subarray}}\prod\limits_{j=1}^{k}d_{i_{j}}^{c_{j}}\bigg{)}.$

For $e=\\{i,j\\}\in E(G)$, let
$e^{(k)}=\\{i,j\\}^{(k)}=\\{i,j,v_{e,1},\ldots,v_{e,k-2}\\}\in E(G^{(k)})$,
where $v_{e,l}$ are cored vertex (the vertex whose degree is 1 [14]), then
$d_{e,l}=1(l=1,\ldots,k-2)$ and $1\leq
c_{i}+c_{j}=d-k-\sum\nolimits_{l=1}^{k-2}c_{e,l}\leq d-k$. Then, for
$d=k+1,\ldots,2k$, the $d$-th order Laplacian spectral moment of $G^{(k)}$ is

$\displaystyle\textnormal{Tr}_{d}(\mathcal{L}_{G^{(k)}})=$
$\displaystyle(-1)^{d}\textnormal{Tr}_{d}(\mathcal{A}_{G^{(k)}})+(k-1)^{N-1}\Big{(}m(k-2)+\sum\limits_{i=1}^{n}d_{i}^{d}\Big{)}$
$\displaystyle+$
$\displaystyle(-1)^{k}dk^{k-2}(k-1)^{N-k}\bigg{(}\sum\limits_{i=1}^{N}d_{i}^{d-k+1}+\sum\limits_{\\{i,j\\}\in
E(G)}\sum\limits_{\begin{subarray}{c}1\leq c_{i}+c_{j}\leq d-k\\\ 0\leq
c_{i},c_{j}<d-k\end{subarray}}d_{i}^{c_{i}}d_{j}^{c_{j}}\bigg{)}.$

By substituting Eq.(2.6) and Eq.(2.7) into the above equation, the expressions
of $\textnormal{Tr}_{d}(\mathcal{L}_{G^{(k)}})(d=k+1,\ldots,2k)$ can be
obtained. ∎

Let $\sum\nolimits_{i=s}^{t}a_{i}=0$ if $t<s$. Let $G=(V(G),E(G))$ be a finite
simple graph. Let $d_{v}$ denote the degree of the vertex $v$ in $G$. The
first and second Zagreb indices were introduced in [18, 19], which are
$M_{1}(G)=\sum\nolimits_{v\in V(G)}d_{v}^{2}=\sum\nolimits_{\\{u,v\\}\in
E(G)}\big{(}d_{u}+d_{v}\big{)}$ and $M_{2}(G)=\sum\nolimits_{\\{u,v\\}\in
E(G)}d_{u}d_{v}$, respectively. The first and second variable Zagreb indices
were introduced in [20, 21], which are $M_{1}^{(r)}(G)=\sum\nolimits_{v\in
V(G)}d_{v}^{r}=\sum\nolimits_{\\{u,v\\}\in
E(G)}\big{(}d_{u}^{r-1}+d_{v}^{r-1}\big{)}$ and
$M_{2}^{(r)}(G)=\sum\nolimits_{\\{u,v\\}\in E(G)}\big{(}d_{u}d_{v}\big{)}^{r}$
(where $r$ is a variable parameter), respectively. And the generalized Zagreb
index $M_{\\{r,s\\}}(G)=\sum\nolimits_{\\{u,v\\}\in
E(G)}\big{(}d_{u}^{r}d_{v}^{s}+d_{u}^{s}d_{v}^{r}\big{)}$ (where $r$ and $s$
are variable parameters) was introduced in [22]. Then the expressions of
Laplacian spectral moments of power hypergraphs given in Theorem 3.2 can be
represented by the Zagreb indices of graphs.

######  Remark 3.9.

Let $G$ be a graph with $n$ vertices and $m$ edges. Let $d_{i}$ denote the
degree of vertex $i$ in $G$ ($i=1,\ldots,n$). Then

$\displaystyle\textnormal{Tr}_{d}(\mathcal{L}_{G^{(k)}})$
$\displaystyle=(k-1)^{N-k}\big{(}(k-1)^{k-1}+(-1)^{k}dk\big{)}(k-2)m+(k-1)^{N-1}M_{1}^{(d)}(G)$
$\displaystyle+(-1)^{k}dk^{k-2}(k-1)^{N-k}\Bigg{(}\sum\limits_{r=2}^{d-k+1}M_{1}^{(r)}(G)+\sum\limits_{r=1}^{\lfloor\frac{d-k}{2}\rfloor}M_{2}^{(r)}(G)+\sum\limits_{r=1}^{\lfloor\frac{d-k}{2}\rfloor}\sum\limits_{s=r+1}^{d-k-r}M_{\\{r,s\\}}(G)\Bigg{)},$

for $d=k+1,\ldots,2k-1$, and

$\displaystyle\textnormal{Tr}_{2k}(\mathcal{L}_{G^{(k)}})$ $\displaystyle=\ell
m+(k-1)^{N-1}M_{1}^{(2k)}(G)+k^{2k-3}(k-1)^{N-2k+1}M_{1}(G)$
$\displaystyle+(-1)^{k}2k^{k-1}(k-1)^{N-k}\Bigg{(}\sum\limits_{r=2}^{k+1}M_{1}^{(r)}(G)+\sum\limits_{r=1}^{\lfloor\frac{k}{2}\rfloor}M_{2}^{(r)}(G)+\sum\limits_{r=1}^{\lfloor\frac{k}{2}\rfloor}\sum\limits_{s=r+1}^{k-r}M_{\\{r,s\\}}(G)\Bigg{)},$

where $N=n+m(k-2)$ and
$\ell=(k-1)^{N-k}\big{(}(k-1)^{k-1}(k-2)+(-1)^{k}2k^{k-1}(k-2)+k^{k-1}-2k^{2k-3}(k-1)^{1-k}\big{)}$.

###### Proof.

For the terms related to degree of vertex, we have

$\displaystyle\sum\limits_{i=1}^{N}d_{i}^{d-k+1}+\sum\limits_{\\{i,j\\}\in
E(G)}\sum\limits_{\begin{subarray}{c}1\leq c_{i}+c_{j}\leq d-k\\\ 0\leq
c_{i},c_{j}<d-k\end{subarray}}d_{i}^{c_{i}}d_{j}^{c_{j}}$ $\displaystyle=$
$\displaystyle\sum\limits_{\\{i,j\\}\in
E(G)}\bigg{(}d_{i}^{d-k}+d_{j}^{d-k}+\sum\limits_{\begin{subarray}{c}1\leq
c_{i}+c_{j}\leq d-k\\\ 0\leq
c_{i},c_{j}<d-k\end{subarray}}d_{i}^{c_{i}}d_{j}^{c_{j}}\bigg{)}$
$\displaystyle=$ $\displaystyle\sum\limits_{\\{i,j\\}\in
E(G)}\bigg{(}\sum\limits_{r=1}^{d-k}\big{(}d_{i}^{r}+d_{j}^{r}\big{)}+\sum\limits_{r=1}^{\lfloor\frac{d-k}{2}\rfloor}(d_{i}d_{j})^{r}+\sum\limits_{r=1}^{\lfloor\frac{d-k}{2}\rfloor}\sum\limits_{s=r+1}^{d-k-r}\big{(}d_{i}^{r}d_{j}^{s}+d_{i}^{s}d_{j}^{r}\big{)}\bigg{)}$
$\displaystyle=$
$\displaystyle\sum\limits_{r=1}^{d-k}\sum\limits_{\\{i,j\\}\in
E(G)}\big{(}d_{i}^{r}+d_{j}^{r}\big{)}+\sum\limits_{r=1}^{\lfloor\frac{d-k}{2}\rfloor}\sum\limits_{\\{i,j\\}\in
E(G)}(d_{i}d_{j})^{r}+\sum\limits_{r=1}^{\lfloor\frac{d-k}{2}\rfloor}\sum\limits_{s=r+1}^{d-k-r}\sum\limits_{\\{i,j\\}\in
E(G)}\big{(}d_{i}^{r}d_{j}^{s}+d_{i}^{s}d_{j}^{r}\big{)}$ $\displaystyle=$
$\displaystyle\sum\limits_{r=2}^{d-k+1}M_{1}^{(r)}(G)+\sum\limits_{r=1}^{\lfloor\frac{d-k}{2}\rfloor}M_{2}^{(r)}(G)+\sum\limits_{r=1}^{\lfloor\frac{d-k}{2}\rfloor}\sum\limits_{s=r+1}^{d-k-r}M_{\\{r,s\\}}(G),\
\mathrm{for}\ d=k+1,\ldots,2k.$

Then the expressions shown in Theorem 3.2 can be represented by the Zagreb
indices of graphs. ∎

Given a $k$-uniform hypergraph $\mathcal{H}$, the signless Laplacian tensor of
$\mathcal{H}$ is
$\mathcal{Q}_{\mathcal{H}}=\mathcal{D}_{\mathcal{H}}+\mathcal{A}_{\mathcal{H}}$.
And the $d$-th order signless Laplacian spectral moment of $\mathcal{H}$ is
equal to the $d$-th order trace of $\mathcal{Q}_{\mathcal{H}}$. For the
signless Laplacian spectral moments of hypergraphs, similar conclusions as
Theorem 3.1 and Theorem 3.2 can be obtained by the same method, which is shown
as follows.

######  Theorem 3.10.

Let $\mathcal{H}$ be a $k$-uniform hypergraph with $n$ vertices. And the
degree sequence of $\mathcal{H}$ is $d_{1},d_{2},\ldots,d_{n}$. Then

$\displaystyle\textnormal{Tr}_{d}(\mathcal{Q}_{\mathcal{H}})=(k-1)^{n-1}\sum\limits_{i=1}^{n}d_{i}^{d}+\textnormal{Tr}_{d}(\mathcal{A}_{\mathcal{H}})+d(k-1)^{n}\sum\limits_{z=1}^{d-1}\sum\limits_{H\in\mathcal{V}_{z}(\mathcal{H})}\sum\limits_{f\in\mathcal{F}_{d}(H)}\frac{\tau(f)\pi_{f}(\mathcal{Q}_{\mathcal{H}})}{\prod\limits_{v\in
V(f)}d^{+}(v)}.$

######  Theorem 3.11.

Let $G$ be a graph with $n$ vertices and $m$ edges. Let $d_{i}$ denote the
degree of vertex $i$ in $G$ ($i=1,\ldots,n$). For the $k$-power hypergraph
$G^{(k)}$ of $G$, then

$\displaystyle\textnormal{Tr}_{d}(\mathcal{Q}_{G^{(k)}})$
$\displaystyle=(k-1)^{N-1}\sum\limits_{i=1}^{n}d_{i}^{d}+dk^{k-2}(k-1)^{N-k}\Big{(}\sum\limits_{i-1}^{n}d_{i}^{d-k+1}+\sum\limits_{\\{i,j\\}\in
E(G)}N_{d-k}(d_{i},d_{j})\Big{)}$
$\displaystyle+(k-1)^{N-k}\big{(}(k-1)^{k-1}+dk\big{)}(k-2)m,\ \mathrm{for}\
d=k+1,\ldots,2k-1,$ $\displaystyle\textnormal{Tr}_{2k}(\mathcal{Q}_{G^{(k)}})$
$\displaystyle=(k-1)^{N-1}\sum\limits_{i=1}^{n}d_{i}^{2k}+2k^{k-1}(k-1)^{N-k}\Big{(}\sum\limits_{i-1}^{n}d_{i}^{k+1}+\sum\limits_{\\{i,j\\}\in
E(G)}N_{k}(d_{i},d_{j})\Big{)}$
$\displaystyle+k^{2k-3}(k-1)^{N-2k+1}\sum\limits_{i=1}^{n}d_{i}^{2}+qm,$

where $N=n+m(k-2)$,
$N_{s}(d_{i},d_{j})=\sum\nolimits_{\begin{subarray}{c}1\leq c_{i}+c_{j}\leq
s\\\ 0\leq
c_{i},c_{j}<s\end{subarray}}d_{i}^{c_{i}}d_{j}^{c_{j}}\big{(}s=1,\ldots,k\big{)}$,
$q=(k-1)^{N-k}\big{(}(k-1)^{k-1}(k-2)+2k^{k-1}(k-2)+k^{k-1}-2k^{2k-3}(k-1)^{1-k}\big{)}$.

And the signless Laplacian spectral moments of $k$-power hypergraph $G^{(k)}$
can also be represented by Zagreb indices of $G$.

Next, we introduce some concepts for the high-order (signless) Laplacian
spectrum of graphs. For a graph $G$ and an integer $k\geq 2$, the (signless)
Laplacian spectrum of $G^{(k)}$ is called the $k$-th order (signless)
Laplacian spectrum of $G$. A graph $G$ is said to be determined by its high-
order (signless) Laplacian spectrum, if there does not exist other non-
isomorphic graph $H$ such that $H$ has the same $k$-th order (signless)
Laplacian spectrum as $G$ for all $k\geq 2$. And we give the following
examples to show that some (signless) Laplacian cospectral graphs can be
determined by their high-oeder (signless) Laplacian spectrum.

Figure 1: non-isomorphic Laplacian cospectral graph Figure 2: non-isomorphic
signless Laplacian cospectral graph

######  Remark 3.12.

The graphs shown in Figure 1 are non-isomorphic Laplacian cospectral graph. By
the $3$-th order Laplacian spectral moments of $3$-power hypergraphs, we have
$\textnormal{Tr}_{3}(\mathcal{L}_{(G_{1})^{(3)}})\neq\textnormal{Tr}_{3}(\mathcal{L}_{(G_{2})^{(3)}})$,
then $(G_{1})^{(3)}$ and $(G_{2})^{(3)}$ have different Laplacian spectrum. So
$G_{1}$ and $G_{2}$ can be distinguished by their high-order Laplacian
spectrum.

The graphs shown in Figure 1 are non-isomorphic signless Laplacian cospectral
graph. By the $3$-th order signless Laplacian spectral moments of $3$-power
hypergraphs, we have $\textnormal{Tr}_{3}(\mathcal{Q}_{(K_{3}\cup
K_{1})^{(3)}})\neq\textnormal{Tr}_{3}(\mathcal{Q}_{(K_{1,3})^{(3)}})$, then
$(K_{3}\cup K_{1})^{(3)}$ and $(K_{1,3})^{(3)}$ have different signless
Laplacian spectrum. So $K_{3}\cup K_{1}$ and $K_{1,3}$ can be distinguished by
their high-order signless Laplacian spectrum.

Acknowledgements

This work is supported by the National Natural Science Foundation of China
(No. 11801115, No. 12071097, No. 12042103, No. 12242105 and No. 12371344), the
Natural Science Foundation of the Heilongjiang Province (No. QC2018002) and
the Fundamental Research Funds for the Central Universities.

## References

## References

  * [1] S. Hu, Z. Huang, C. Ling, and L. Qi.  On determinants and eigenvalue theory of tensors.  J. Symbolic Comput., 50:508–531, 2013.
  * [2] J. Shao, L. Qi, and S. Hu.  Some new trace formulas of tensors with applications in spectral hypergraph theory.  Linear Multilinear Algebra, 63(5):971–992, 2015.
  * [3] J. Zhou, L. Sun, and C. Bu.  Some spectral properties of uniform hypergraph.  Electron. J. Combin., 4(21), 2014.  Article ID #P4.24.
  * [4] G. Clark and J. Cooper.  A Harary-Sachs theorem for hypergraphs.  J. Combin. Theory Ser. B, 149:1–15, 2021.
  * [5] L. Chen, C. Bu, and J. Zhou.  Spectral moments of hypertrees and their applications.  Linear Multilinear Algebra, 70(21):6297–6311, 2022.
  * [6] H. Zhou, L. Sun, and C. Bu.  Estrada index and subgraph centrality of hypergraphs via tensors.  Discrete Appl. Math., (341):120–129, 2023.
  * [7] L. Chen, L. Sun, and C. Bu.  High-order spectral characterizations of graphs.  arXiv:2111.03877.
  * [8] C. Duan, E.R. van Dam, and L. Wang.  Signless Laplacian Estrada index and Laplacian Estrada index of uniform hypergraphs.  arXiv:2206.001109.
  * [9] J. Cooper and A. Dutle.  Spectra of uniform hypergraphs.  Linear Algebra Appl., 436(9):3268–3292, 2012.
  * [10] L. Qi.  ${H}^{+}$-Eigenvalues of Laplacian and signless Laplacian tensors.  Commun. Math. Sci., 12:1045–1064, 2014.
  * [11] L. Lim.  Singular values and eigenvalues of tensors: a variational approach.  In 1st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, pages 129–132. IEEE, 2005.
  * [12] L. Qi.  Eigenvalues of a real supersymmetric tensor.  J. Symbolic Comput., 40(6):1302–1324, 2005.
  * [13] A. Morozov and S. Shakirov.  Analogue of the identity Log Det = Trace Log for resultants.  J. Geom. Phys., 61(3):708–726, 2011.
  * [14] S. Hu, L. Qi, and J. Shao.  Cored hypergraphs, power hypergraphs and their Laplacian H-eigenvalues.  Linear Algebra Appl., (439):2980–2998, 2013.
  * [15] M. Farrell and L. Levine.  Multi-Eulerian tours of directed graphs.  Electron. J. Combin., 23(2), 2016.  Article ID #P2.21.
  * [16] T. van Aardenne-Ehrenfest and N.G. de Bruijin.  Circuits and trees in oriented linear graphs.  Master’s thesis, Birkh$\rm\ddot{a}$user Boston, 1987.
  * [17] W.T. Tutte and C.A.B. Smith.  On unicursal paths in a network of degree 4.  Amer. Math. Monthly, 48(4):233–237, 1941.
  * [18] I. Gutman and N. Trinajstic.  Graph theory and molecular orbitals. total $\pi$-electron energy of alternant hydrocarbons.  Chem. Phys. Lett., 4(17):535–538, 1972.
  * [19] I. Gutman, B. Ruscic, N. Trinajstic, and C.F. Wilcox.  Graph theory and molecular orbitals. xii. Acyclic polyenes.  J. Chem. Phys., (62):3399–3405, 1976.
  * [20] X. Li and H. Zhao.  Trees with the first three smallest and largest generalized topological indices.  MATCH Commun. Math. Comput. Chem., (50):57–62, 2004.
  * [21] X. Li and J. Zheng.  A unified approach to the extremal trees for different indices.  MATCH Commun. Math. Comput. Chem., (54):195–207, 2005.
  * [22] M. Azari and A. Iranmanesh.  Generalized Zagreb index of graphs.  Studia Univ. Babes-Bolyai., 3(56):59–70, 2011.


11institutetext:  INFN, Sezione di Torino and Osservatorio Astrofisico di
Torino (INAF), Torino, Italy 22institutetext: Av. San Martín Norte 304, 5613
Malargüe, Argentina

# AugerPrime: the Pierre Auger Observatory Upgrade

Antonella Castellina for the Pierre Auger Collaboration Authors list:
http://www.auger.org/archive/authors$\\_$2018$\\_$10.html11
<EMAIL_ADDRESS>22<EMAIL_ADDRESS>
###### Abstract

The world largest exposure to ultra-high energy cosmic rays accumulated by the
Pierre Auger Observatory led to major advances in our understanding of their
properties, but the many unknowns about the nature and distribution of the
sources, the primary composition and the underlying hadronic interactions
prevent the emergence of a uniquely consistent picture. The new perspectives
opened by the current results call for an upgrade of the Observatory, whose
main aim is the collection of new information about the primary mass of the
highest energy cosmic rays on a shower-by-shower basis. The evaluation of the
fraction of light primaries in the region of suppression of the flux will open
the window to charged particle astronomy, allowing for composition-selected
anisotropy searches. In addition, the properties of multiparticle production
will be studied at energies not covered by man-made accelerators and new or
unexpected changes of hadronic interactions will be searched for. After a
discussion of the motivations for upgrading the Pierre Auger Observatory, a
description of the detector upgrade is provided. We then discuss the expected
performances and the improved physics sensitivity of the upgraded detectors
and present the first data collected with the already running Engineering
Array.

##  1 Introduction

Although we still lack a comprehensive theory capable of explaining where the
ultra-high energy cosmic rays (UHECRs) are produced, what is the acceleration
mechanism driving nuclei to such extreme energies and their chemical
composition, a large amount of information was gained in the recent years by
exploiting the data taken with the Pierre Auger Observatory PAO1 .

The suppression of the cosmic ray flux for $E>4\times 10^{19}$ eV has been
unambiguously established; the current limits on the photon and neutrino
fluxes at ultra-high energy strongly limit the possibility that the observed
flux is produced in non standard “top-down” processes unger .
A dipolar large scale anisotropy has been observed in cosmic rays with $E>8$
EeV, proving that they are indeed of extragalactic origin and opening a window
on the study of the properties of extragalactic magnetic fields dipole .
Strong hints of correlation with the direction of bright, nearby starburst
galaxies have been obtained in intermediate-scale anisotropy searches, while
an excess of events have been seen for $E>5.8\times 10^{19}$ eV in the
direction of the Centaurus A radio galaxy SBG .

In the standard astrophysical Scenario, the UHECRs are mainly extra-galactic
protons and the observed suppression of the flux is ascribed to the energy
losses suffered by the particles during propagation, after having been
accelerated with a power-law injection spectrum $E^{-2}$ in sources
distributed as the matter in the Universe (“photo-disintegration Scenario”).
However, the energy at which the integral flux measured at Auger drops by a
factor of two below what would be expected without suppression, $E_{1/2}$ =
(23$\pm$1(stat.)$\pm$4(syst.)) EeV, is quite different from the $E_{1/2}=53$
EeV predicted for this scenario. Furthermore, this model is challenged by the
results obtained on the primary composition comp1 ; comp2 ; comp3 . Indeed,
the studies of the evolution with energy of the shower development in the
atmosphere point to a mass composition becoming progressively heavier with
increasing energy. The heaviest (iron) component appears almost unnecessary,
while the protons add up to not more than 10% at the highest energies.

The best fit to the spectrum and composition as measured at the Pierre Auger
Observatory is obtained for a model in which the sources accelerate particles
to maximum energies proportional to their charge, with a hard injection index
$\gamma\sim 1$. The observed suppression in the flux would then be ascribed to
a cutoff in the energy spectra of the sources (“maximum rigidity scenario”)
scenarios .

The interpretation of the results in terms of composition is made even more
difficult by the large uncertainties in predicting hadronic multiparticle
production at energies beyond those accessible in the laboratory. Different
studies aiming at the evaluation of the muon content in air showers muon1 ;
muon2 ; muon3 , exploiting the study of hybrid events muon4 , or analysing the
azimuthal asymmetry in the risetime of signals in SD muon5 , found
inconsistencies in the modeling of hadronic interactions and muon production
as implemented in the simulations .

Figure 1: Left: One station of the AugerPrime Engineering Array, featuring the
Surface Scintillator Detector (SSD) on top the WCD. Top-right: layout of the
SSD. Bottom-right: details of the fibers in the router.

An upgrade of the Pierre Auger Observatory, dubbed “AugerPrime” PrimeArch ;
PrimeGen , has been planned to address the following open questions :

$\bullet$ understand the origin of the flux suppression, by differentiating
between a cut-off due to propagation effects and the maximum energy reached in
the sources. This will allow us to provide fundamental constraints on the
sources of the ultra high energy cosmic rays and their properties.

$\bullet$ evaluate the possible existence of a fraction of protons at the
highest energies. The answer to this fundamental question will set the stage
for future experiments and assess the feasibility of charged particle
astronomy.

$\bullet$ add information about hadronic interactions in a range of energy and
in a kinematic region which is not accessible in man-made accelerators. Non-
standard mechanisms, such as Lorentz invariance violation or extra dimensions
will also be searched for.

To reach these goals, it is essential to obtain additional measurements of
composition-sensitive observables on a high statistics sample of ultra-high
energy events, thus exploiting the Surface Detector of the Pierre Auger
Observatory.

##  2 Elements of AugerPrime

The key elements of the upgrade consist of new plastic scintillator detectors
on top of the water-Cherenkov stations (WCD) of the surface array (SD), an
additional small photomultiplier (PMT) installed in the WCD for the extension
of the dynamic range, and new SD electronics.
Additionally an underground muon detector is going to be deployed aside each
WCD in the infill region of the Observatory, to provide direct muon
measurements. The upgrade will also be complemented by extending the
measurements of the existing Fluorescence Detectors (FD) into periods of
higher night-sky background, to increase their duty cycle. Finally, based on
the Radio Engineering Array (AERA) experience and results, a new project for
adding a radio detector to AugerPrime is now on-going.

###  2.1 The Scintillator Detectors

Complementing the water-Cherenkov detectors of the surface array with
scintillator detectors will provide a way to sample shower particles with
detectors having very different responses to the muonic and electromagnetic
components. Indeed, the ratio of the electromagnetic to muonic components of
the showers is more than a factor of two higher in a thin scintillator with
respect to the WCD over a very wide range of distances to the core.
A Surface Scintillator Detector (SSD) unit PrimeSSD is composed of a box of
(3.8 $\times$ 1.3) m2, housing two scintillator sub-modules, each covering an
area of 1.9 m2. The scintillator planes are protected by light-tight,
waterproof enclosures and are mounted on top of the existing WCD with a strong
support frame, as shown in the left panel of Fig.1. A double roof, with the
upper layer being corrugated aluminum, is used to reduce the temperature
variations. The scintillation light is collected with wavelength-shifting
fibers inserted into straight extruded holes in the scintillator planes (right
panels of Fig.1). The fibers, Kuraray Y11(300)M S-type, are positioned
following the grooves of the routers at both ends, in a “U” configuration that
maximises the light yield. They are bundled in a cookie, acting also as a
diffuser, whose PMMA front window is connected to a single photomultiplier
(PMT) tube. The PMT is a bi-alkali Hamamatsu R9420, 1.5" diameter, with
$\sim$18% quantum efficiency at a wavelength of 500 nm; the power supply is
based on a custom made design manufactured by ISEG company.
After production, the SSD modules are tested using muon tower facilities,
exploiting atmospheric muon tomography to measure the minimum ionizing
particle (MIP) as well as the uniformity for each SSD unit. From these
measurements, the number of photoelectrons per vertical MIP is found to be
(30$\pm$2). Due to the U-turn of the fibers and their length outside the
scintillator bars, the uniformity of the measured signals is $\pm$5% along the
bars and $\pm$10% between bars. The attenuation length of the light in the
fibers is $\lambda=(312\pm 3)$ cm.
Each SSD module is equipped with two sensors to measure temperature and
humidity; by measurements in the field, it has been shown that the temperature
inside the SSD closely follows the air temperature evolution measured in the
shade, almost never exceeding 40∘C.

###  2.2 The dynamic range of the Surface Detector

The SD data quality will be improved by extending the acquisition dynamic
range in both detectors, thus allowing us to measure non-saturated signals at
distances as close as 250 m from the shower core PrimeSPMT . In order to
benefit the most from the combined information of SSD and WCD, the dynamic
range of both detectors should be similar. Significant signals are expected
both close to the shower core and at intermediate distances, where
simultaneous measurements will be most important for the separation of the
different components of the shower, furthermore allowing a direct cross-check
of the two detectors.

To increase the dynamic range, a fourth small PMT (Hamamatsu R8619, 1"
diameter) is inserted in the WCD, exploiting an hitherto unused and easily
accessible 30 mm window on the Tyvek bag containing the ultra-pure water. With
an active area of about 1/80 with respect to the large WCD PMTs, it
potentially allows for an equivalent dynamic range extension. The required
range up to about 20000 VEM can be obtained by adjusting the gain in such a
way that the ratio of the large to the small PMT signals be limited to a value
of 32.
The linearity of the small PMT is guaranteed in the required range for a large
set of voltages, down to 800 V. The power supply is provided by an external
custom made HVPS manufactured by CAEN company.

For consistency with the associated WCD, the dynamic range in the SSD must
span from the signal of a single particle, as needed for calibration, to large
signals, up to $\sim 2\times 10^{4}$ MIP. Indeed, the SSD PMT has been chosen
also based on its excellent linear response when operated at low gain, being
linear within 5% for peak currents up to 160 mA (for a gain of $8\times
10^{4}$). To reach the required dynamic range, the anode signal is filtered
and split in two in the electronics front-end; then one of the two signals is
attenuated by a factor of 4, while the other is amplified by a factor of 32.

Figure 2: Left: Charge spectrum in one of the upgraded WCD stations, as
measured by the large PMTs (pink: unsaturated, green:saturated) and by the
small PMT (brown: unsaturated, blue=saturated). Right: Linearity behaviour of
one of the SSD-PMT. Figure 3: Left: Correlation of signals in SSD vs WCD. The
large PMTs are used up to the saturation (black dots); the measurements are
extended further by means of the small PMT (red dots). Right: Lateral
distribution of an event close to the Engineering Array

###  2.3 The Electronics of the Surface Detector

Implemented in a single board, called the upgraded unified board (UUB), the
electronics of the SD is used to record the PMT signals, make local triggering
decisions, send time-stamps to the central data acquisition system and store
event data for retrieval upon a global trigger condition PrimeSDEU . The
upgraded electronics will allow us to read the four photomultipliers of the
WCD and the SSD one, processing the signals at 120 MHz, corresponding to an
increase of the current rate of a factor of 3.
The anode channel of the three large XP1805 PMTs of the WCD is split and
amplified to reach a gain ratio of 32. The signals are filtered and digitized
by commercial 12 bit 120 MHz flash ADC (FADC); as the XP1805 pulse response in
terms of bandwidth is $\sim$70 MHz, this is well matched to a 120 MHz FADC and
associated 60 MHz Nyquist filter.
The small PMT signal is also digitized with 12 bits at 120 MHz in a separate
channel.
The anode of the SSD PMT is split in two different channels, the first one
amplified to have a gain ratio of 32 and the other attenuated by a factor of
4. This yields a total gain ratio of 128. The signals are filtered and
digitized similarly to the WCD PMT signals.
The local trigger and processing capabilities are increased by using a more
powerful local-station processor and FPGA (field-programmable gate array),
thus allowing the implementation of new triggers, e.g. combined SSD and WCD
triggers, in addition to the current ones.
The recorded traces can be digitally filtered and downsampled to 40 MHz to
emulate the current triggers in addition to any new ones, thus providing
backwards-compatibility with the current dataset. Furthermore, this will allow
us to deploy the new electronics without stopping or disturbing the current
data taking.
The timing accuracy is improved to 2 nanoseconds thanks to the implementation
of a new GPS receiver, the I-Lotus M12M, providing the same pulse per second
timing output with serial control and data as the currently used Motorola
Oncore UT+ one, as such providing compatibility with the running electronics.
The UUB is equipped with a micro-controller (MSP430) for the control and
monitoring of the PMT high voltage, the supervision of the various supply
voltages and reset functionality. More than 90 monitoring variables will be
managed by the slow-control software.

The VEM (vertical equivalent muon) is the reference calibration unit of the
WCD high gain channels. It is derived using a test station with an external
hodoscope triggering on atmospheric muons and corresponds on average to 95
photoelectrons at the PMT cathode; in the UUB, this is equivalent to about
1800 integrated ADC counts above pedestal.
Due to its dimensions, no direct calibration of the small PMT with muons is
feasible. By selecting small shower events, the small PMT can be cross-
calibrated with the large PMTs in such a way that the signal spectrum is
correctly reconstructed up to about 20,000 VEM.
The SSD calibration is based on the signal of a minimum-ionising particle
going through the detector. A cross-trigger with the WCD can be used to remove
all of the low-energy background. About 40% of the calibration triggers of the
WCD produce a MIP in the SSD.
For both the WCD and the SSD, the cross-calibration between high gain and low
gain channels is defined in the electronics, by a simple ratio between the
amplified and non amplified anode signals.

###  2.4 Performances of the upgraded detectors

Started in 2016, an Engineering Array of twelve AugerPrime stations has been
taking data in the field, with the aim of testing and monitoring the
performances of the upgraded detectors.

The logarithm of the charge spectrum in one of the upgraded AugerPrime WCD
stations is shown in Fig. 2(left). The dynamic range is extended to more than
20000 VEM by means of the small PMT, as one can see by comparing the
unsaturated spectrum measured by the large PMTs (magenta histogram) to the one
obtained with the small PMT (brown curve). The reconstruction resolution
obtained with the small PMT is better than 5% above 3000 VEM, which correspond
to the signal produced at 250 m from the core in showers of 10 EeV.
The very good linearity behaviour of the photomultipliers of the SSD
guarantees the possibility to measure signals up to about 20000 MIP. As shown
in the right panel of the same figure for one SSD-PMT, the non-linearity stays
below 5% up to the highest densities.
The correlation between the signals in one of the WCD and in the corresponding
scintillator is shown in the left panel of Fig. 3, where both scales are
expressed in the corresponding physical units (VEM for the WCD PMTs and MIP
for the SSD PMT). The signals in the WCD are measured up to saturation ($\sim$
650 VEM) by the large PMTs (black dots). In the superposition region and above
the large PMT saturation, they are derived from the small PMT (red dots). The
correlation obtained demonstrates the validity of the independent calibration
procedures and shows that the required dynamic range in the upgraded Surface
Detector is nicely covered up to the highest particle densities.
The reconstructed lateral distribution function for one of the events
collected in the region of the Engineering Array is shown in the right panel
of Fig. 3. The signal recorded in the upgraded WCD (red dot) is in very good
agreement with expectations, following the lateral distribution reconstructed
by means of the standard stations. The SSD signal, in blue, is relatively
lower than the WCD ones, as expected due to the different sensitivity of the
two detectors to the electromagnetic component of the showers at different
core distances and to the relatively smaller area of the SSD with respect to
the WCD.

###  2.5 The Underground Detector

As part of the AMIGA enhancement (Auger Muons and Infill for the Ground
Array), an array of underground detectors is being deployed beside the 61 WCD
of the Infill area of the Observatory, on a 750 m triangular grid. In each
position, 3 scintillator modules of 10 m2 area are buried 2.3 m underground,
to minimise the contamination from electromagnetic shower particles. The
scintillation light is collected using wavelength shifting fibers read by
silicon photo sensors AMIGA . Two modules in the assembly hall are shown in
the top panel of Fig.4.
The segmented structure of the counters allows direct counting of muons; a
channel for signal integration will be used to extract the muon information
close to the shower core, in the saturation region.
These detectors will provide a direct measure of the muon content of the
extensive air showers, with the aim of studying the primary composition and
hadronic interactions in a region of energy corresponding to the transition of
cosmic rays from a Galactic to an extra-galactic origin.
Furthermore, the underground modules will allow us to cross-check and fine-
tune the methods used to derive the muon information from the upgraded SD.

Figure 4: Top: Two AMIGA modules during assembly. Bottom: the Moon and Jupiter
above one of the Fluorescence Detectors in the Pierre Auger Observatory.

###  2.6 The extended duty cycle of the Fluorescence Detector

The Auger Fluorescence Detector (FD) is composed of 27 telescopes overlooking
the SD area (Fig.4, bottom panel). They are used to observe the longitudinal
development of air showers in the atmosphere, thus providing mass sensitive
observables and a model independent energy reconstruction.
The FD duty cycle is currently limited to about 15 %, being operations limited
to dark and moonless nights. The current criteria for FD measurements require
the Sun to be more than 18∘ below the horizon, the Moon to be below the
horizon for more than 3 hours and the illuminated fraction of the Moon to be
below 70%.
A significant increase of about 50% in the duty cycle can be obtained by
extending the FD operations to times with larger night sky background,
relaxing the second and third requirements. By reducing the supplied high
voltage, the PMT gains can indeed be reduced by a factor of 10.
Preliminary tests show that the PMTs operated at reduced gain satisfy the
criteria required for the FD performance (such as linearity, stability and
lifetime), avoiding too high anode currents that could result in a
deterioration of the tubes.

###  2.7 The Radio Upgrade

An engineering array of radio detectors (AERA) has been running in the Pierre
Auger Observatory for a few years. With the collected data, it was possible to
demonstrate the feasibility of radio detection with a grid of antennas at 1500
m mutual distances, and measure horizontal showers
($\theta=60^{\circ}-84^{\circ}$), which illuminate an area of several km2 on
the ground radioHor .
Based on these results, a full radio upgrade of the Pierre Auger Observatory
has been proposed radio , where each WCD of the SD will be equipped with a
radio antenna mounted on its top surface, as shown in Fig.5.
The new detectors will operate together with the upgraded SD, forming a unique
setup to measure the properties of cosmic rays above $10^{17.5}$ eV. For
horizontal showers, they will nicely complement the information on the
particle type we will get from the WCD+SSD, thereby increasing the exposure
for mass-sensitive investigations.

Figure 5: Sketch of the radio antenna mounted atop the WCD+SSD.

Figure 6: Left: Reconstruction bias (full symbols) and resolution (empty ones)
for the muonic signal derived with the matrix inversion method in individual
stations, as a function of distance to the shower core, for protons (circles)
and iron nuclei (squares). Right: Relative difference between the number of
muons reconstructed in individual showers using shower universality and the
true (Monte Carlo) ones as a function of energy for different primary species.
Error bars represent the RMS of the distributions

##  3 Expected physics performances

As discussed above, the main aim of AugerPrime is that of providing event-by-
event composition-sensitive measurements by increasing the ability to
recognize and separate the different components of air showers. The goodness
of the separation between two different primaries i and j can be quantified by
defining a merit factor

$f_{MF}=\frac{|S_{i}-S_{j}|}{\sqrt{\sigma^{2}(S_{i})+\sigma^{2}(S_{j})}}$ (1)

The most direct technique to evaluate the muon component exploits the
different responses of the WCD and SSD at station level, relating the total
signals $S_{WCD}$(VEM) and $S_{SSD}$(MIP) to the electromagnetic and muonic
fluxes at ground. Employing a matrix inversion method ALS , the muonic signal
in the WCD can be derived as

$S_{\mu,WCD}={\it a}S_{WCD}+{\it b}S_{SSD}$ (2)

where the coefficients a and b, although derived from simulations, show only a
very weak dependence on the mass and arrival direction of the primary
particles and on the hadronic interaction models used in the Monte Carlo.
The muon signals so obtained are subject to large fluctuations, being
restricted to individual stations.

Figure 7: Mean depth of shower maximum $X_{max}$ (left), its fluctuations
(center) and the relative muon number at $38^{\circ}$ (right), evaluated for
the two scenarios (1=maximum rigidity model; 2=photo-disintegration model)
using simulated SD data only, for pure proton and iron compositions.

In Fig.6, left, the reconstruction bias (solid symbols) and the resolution
(open symbols) of the muonic signal contribution for individual detector
stations are shown for two different primary compositions: at 1000 m from the
core, a resolution of 20 to 30% can be achieved.
The performances of the method can be improved when the reconstructed signals
at station level are used to evaluate the lateral distribution function. The
matrix inversion method is then applied to evaluate the muon signal at 800 m
from the core. Using $S_{\mu}(800)$ as estimator of the composition, a merit
factor $f_{MF}>1.5$ can be obtained in separating proton and iron primaries at
small zenith angles for energies larger than $10^{19.5}$ eV.

A more general approach based on the properties of universality of air showers
univ or multivariate analyses would allow us to take advantage of all the
information provided by AugerPrime, not only by correlating the detector
signals at different distances to the core but also considering their temporal
structure (arrival times of the particles at the stations). In this case, an
event-by-event reconstruction allows us to derive not only the energy, but
also the depth of maximum shower development in atmosphere ($X_{max}$) and the
number of muons ($N_{\mu}$) using the information from the scintillators
combined with the WCD data. The bias and resolution in the evaluation of
$N_{\mu}$ are shown in the right panel of Fig.6, as a function of composition
and energy. As an example, a merit factor of about 1.7 (2.1) can be obtained
for the separation of proton from iron primaries at 60 EeV, for zenith angles
of 38∘ (56∘) respectively. The reconstruction bias for $X_{max}$ is less than
15 g/cm2 with the resolution improving from 40 g/cm2 at $10^{19}$ eV to 25
g/cm2 at $10^{20}$ eV.

###  3.1 The interpretation of the suppression in the flux

To explore the power of AugerPrime with respect to meeting the scientific
goals, we simulated two sets of mock data corresponding to seven years of data
taking in AugerPrime and to an exposure of 50000 km2 sr yr, assuming the two
different simplified models described in Sect.1: a maximum rigidity (“Scenario
1”) and a photo-disintegration one (“Scenario 2”). These two benchmark models
approximately reproduce the energy spectrum and $X_{max}$ distributions as
measured in Auger above $10^{18.7}$ eV, as resulting from a numerical fit
following the method described in scenarios .
The first two moments of the $X_{max}$ distributions and the mean muon number
as predicted based on the two scenarios are shown in Fig.7 as a function of
energy for two extreme nuclear masses. Note that the RMS($X_{max}$) includes
both the shower-to-shower fluctuations and the detector resolution.
To get rid of the dependencies of the total muon number on energy and zenith
angle (due to the shower intrinsic fluctuations and to the fact that the muon
path-length varies with $\theta$), the total muon number is shown relative to
that expected from a mix of equal fractions of primaries. Although the
expected values are quite similar below $10^{19.2}$ eV, a range already
covered by the FD, the models predict quite different results in the highest
energy region, showing that the two scenarios can be distinguished with high
statistics and significance in 7 years of measurements.

###  3.2 Is there a proton fraction at UHE?

In the first scenario, no protons can reach us from above 50 EeV, while on the
contrary we would see a large fraction of them in the second one.
To explore the feasibility of detection of a small fraction of protons with
AugerPrime in the framework of Scenario 1, we artificially added a
contribution of 10% protons to it (“Scenario 1p”). The proton-rich sample was
produced by assigning to these events a proton-like $X_{max}$, i.e. an
$X_{max}>700$ g/cm2 at 10 EeV, adjusted to the event energy with an elongation
rate of 55 g/cm2 per decade.
The difference $\sigma$ between the means of the distributions as obtained
from the two scenarios (“1” and “1p”) divided by the statistical uncertainty
is shown in Fig. 8 for each observable. As shown in the figure by the black
line, the best separation power is obtained by constructing a combined
significance using all observables at our disposal:

$\displaystyle\sigma^{2}$
$\displaystyle=\sigma^{2}(<X_{max}>)+\sigma^{2}(RMS(X_{max}))$
$\displaystyle+\sigma^{2}(<R_{\mu,38^{\circ}}>)+\sigma^{2}(RMS(R_{\mu,38^{\circ}}))$
(3)

We conclude that using this combination and integrating above 10 EeV we might
be able to distinguish between Scenario 1 and 1p (i.e. to observe a fraction
of protons as low as 10%) at the highest energies with more than 5$\sigma$
after 5 years of operation of AugerPrime.

Figure 8: Significance of distinguishing Scenarios 1p and 1 (with and without
at least 10% protons respectively) as a function of the operation time of
AugerPrime. Figure 9: Angular correlation of Auger data modified according to
Scenario “1” and “1p” with the AGNs of the “70 Months Swift-BAT catalog” (see
text for details). The relative excesses of pairs of events as a function of
their angular separation is shown for the complete data set (left), the
selection deprived of light elements (center) and the proton-enriched one
(right panel). Figure 10: Expected sensitivity on the flux of photons (left)
and neutrinos (right) phot ; neut . For photons, full black lines show more
conservative estimates based only on the increase of statistics.

###  3.3 Composition-wise anisotropy studies

To study the gain in sensitivity with AugerPrime in the search for the sources
of ultra-high energy cosmic rays, we used the arrival directions of 454 events
with E $>4\times 10^{19}$ eV measured at $\theta<60^{\circ}$ ev454 , assigning
them random $X_{max}$ values according to Scenario 1. A 10% enhanced proton
fraction was then implemented as in the previous study (Scenario 1p).
Furthermore, we randomly assigned to 50% of these protons an arrival direction
corresponding within 3∘ to one of the AGNs of the Swift-BAT catalog, chosen
among those with distances within 100 Mpc Swbat .
The angular correlation is shown in Fig. 9. It is not significant when
considering the complete data set of 454 events without composition
information (left panel of the figure), but with the proton-enriched sample a
correlation in excess of 3$\sigma$ is clearly evident (right panel). As
expected, no excess of pairs is found with the AGNs for the proton-deprived
selection (central plot), as in this case the angular correlation is just due
to statistical fluctuations.

###  3.4 Neutral particles limits

Auger observations provided the currently most stringent limits on the flux of
photons at energies above $10^{19}$ eV and severe constraints on cosmogenic
neutrinos phot ; neut .
Even better limits are expected to be reached with AugerPrime: five more years
of data will provide us with a large increase in statistics, the new
electronics will allow the introduction of more efficient triggers, a
reduction of the hadronic background will be possible thanks to an improved
muon discrimination.
In Fig.10, the current limits for both photons and neutrinos are shown
together with preliminary calculations of the expected sensitivities from
AugerPrime (dashed lines). This ideal case assumes complete rejection of the
hadronic background. In the photon case, the introduction of the new triggers
allows us to extend the measurement with SD below 10 EeV.

###  3.5 Hadronic interactions and fundamental physics studies

AugerPrime can shed light on whether the inconsistencies between air shower
simulations and experimental data are due to fundamental shortcomings in our
understanding of hadronic multiparticle production.
Since the electromagnetic energy deposit in the atmosphere is mainly due to
the first high energy interactions, while on the other hand muons at ground
come as products of low energy interactions, an event-by-event correlation
between the depth of shower maximum and the muon content in showers generated
by ultra-high energy cosmic rays can bring strong constraints on hadronic
interaction models GF . As an example, the muon density as obtained by
different modifications of hadronic interaction models, relative to that
predicted by QGSJetII-03 for protons, is shown in Fig.11 in correlation with
the depth of shower maximum, for various compositions.

A high resolution measurement of the muon number such as the one expected from
AugerPrime can prove to be a very powerful observable also to look for exotic
interaction Scenarios. As an example, if Lorentz invariance violation effects
exist, the absorption and energy losses of UHECRs during propagation would be
modified. Furthermore, strong changes in the development of shower produced by
the interaction of UHECRs with nuclei in the atmosphere may also be visible
exotic .

##  4 Conclusion

The quest for the sources of UHECRs will be pursued by AugerPrime, the Pierre
Auger Observatory upgrade, by providing a high statistics sample of events
with mass information.

Composition-wise anisotropy searches will allow us to understand the origin of
the flux suppression and explore the anisotropy dependencies on the particle
rigidities. AugerPrime will be able to measure the presence of even a small
fraction of protons at the highest energies, as such assessing the feasibility
of future projects willing to detect ultra high energy neutrinos.

A much improved measurement of the muon content in air showers will help in
the study of hadronic multiparticle production and the search for exotic
interactions. Furthermore, once a better tuning of the models is reached, we
will have the possibility to reanalyse also the pre-upgrade data set and study
the composition with much lower uncertainties.

While the main aim of the upgrade is the study of the highest energy region,
above $10^{19.5}$ eV, the composition sensitive information will also allow to
gain more information on the region of transition from a Galactic to an extra-
galactic contribution to the cosmic ray flux, at $E=10^{17}-10^{19}$ eV.

An Engineering Array of twelve upgraded stations has been taking data in the
field since 2016; as of October 2018, thirty more fully upgraded stations have
been deployed. The data collected so far demonstrate the quality of the new
detectors and the physics potential of the upgrade project. The production of
scintillators and small PMT systems is proceeding at full pace: a large number
of them is already available in Malargüe, while some units of the upgraded
electronics boards are under test in the field. The production and deployment
of the AugerPrime detectors and electronics will be completed within 2019.
Operations and full data taking are planned from 2020 to 2025.

Figure 11: Relative number of muons at 1000 m from the shower core wrt that
predicted by QGSJETII-03 as a function of $X_{max}$. Different modifications
of hadronic interactions are considered GF . Auger data are also shown as
derived from showers of $10^{19}$ eV for vertical and horizontal events.

## References

  * (1) A. Aab et al. (Pierre Auger Collaboration), Nucl. Instr. Meth. A 798, 172 (2015).
  * (2) M. Unger (Pierre Auger Collaboration), Highlight talk in Proc. 35th Int. Cosmic Ray Conf., Busan (Korea), PoS(ICRC2017) 1102 (2017).
  * (3) A. Aab et al. (Pierre Auger Collaboration), Science 357, 1266 (2017).
  * (4) A. Aab et al. (Pierre Auger Collaboration), Astrophys. J. 853, L29 (2018).
  * (5) A. Aab et al. (Pierre Auger Collaboration), Phys. Rev. D 90, 122006 (2014).
  * (6) A. Aab et al. (Pierre Auger Collaboration), Phys. Lett. B 762, 288 (2016).
  * (7) A. Aab et al. (Pierre Auger Collaboration), Phys. Rev. D 96, 122003 (2017).
  * (8) A. Aab et al. (Pierre Auger Collaboration), JCAP 04, 038 (2017).
  * (9) A. Aab et al. (Pierre Auger Collaboration), Phys. Rev. D 91, 032003 (2015).
  * (10) A. Aab et al. (Pierre Auger Collaboration), Phys. Rev. D 90, 012012 (2014). Erratum-ibid: 92 (2015) 019903.
  * (11) M. Mallamaci (Pierre Auger Collaboration), Proc. 35th Int. Cosmic Ray Conf., Busan (Korea), PoS(ICRC2017) 509 (2017).
  * (12) A. Aab et al. (Pierre Auger Collaboration), Phys. Rev. Lett. 117, 192001 (2016).
  * (13) A. Aab et al. (Pierre Auger Collaboration), Phys. Rev. D 93, 072006 (2016).
  * (14) The Pierre Auger Collaboration, _The Pierre Auger Observatory Upgrade: AugerPrime. Preliminary Design Report_ , arXiv:1604.03637 (2016).
  * (15) D. Martello (Pierre Auger Collaboration), Proc. 35th Int. Cosmic Ray Conf., Busan (Korea), PoS(ICRC2017) 383 (2017).
  * (16) R. Smida (Pierre Auger Collaboration), Proc. 35th Int. Cosmic Ray Conf., Busan (Korea), PoS(ICRC2017) 390 (2017).
  * (17) A. Castellina (Pierre Auger Collaboration), Proc. 35th Int. Cosmic Ray Conf., Busan (Korea), PoS(ICRC2017) 397 (2017).
  * (18) T. Suomijarvi (Pierre Auger Collaboration), Proc. 35th Int. Cosmic Ray Conf., Busan (Korea), PoS(ICRC2017) 450 (2017).
  * (19) A. Aab et al. (Pierre Auger Collaboration), JINST 12, P03002 (2017).
  * (20) A. Aab et al. (Pierre Auger Collaboration), JCAP 10, 026 (2018).
  * (21) J. Hörandel, these Proceedings.
  * (22) A. Letessier-Selvon et al., Nucl. Instrum. Meth. A 767, 41 (2014).
  * (23) D. Maurel et al., Proc. of 33rd Int. Cosmic Ray Conf., Rio de Janeiro, Brazil, 600 (2013).
  * (24) A. Aab et al. (Pierre Auger Collaboration), Astrophys. J. 804, 15 (2015).
  * (25) W. Baumgartner, J. Tueller, C. Markwardt, G. Skinner, S. Barthelmy et al., Astrophys. J. Suppl. 207, 19 (2013).
  * (26) M. Niechciol (Pierre Auger Collaboration), Proc. 35th Int. Cosmic Ray Conf., Busan (Korea), PoS(ICRC2017) 517 (2017).
  * (27) E. Zas (Pierre Auger Collaboration), Proc. 35th Int. Cosmic Ray Conf., Busan (Korea), PoS(ICRC2017) 972 (2017).
  * (28) G.R. Farrar, J.D. Allen, Proc. 33rd Int. Cosmic Ray Conf., Rio de Janeiro (Brasil), arXiv:1307.7131 (2013).
  * (29) R. Aloisio et al., Frascati Phys. Ser. 58, 274 (2014).