# Dyson Maps and Unitary Evolution for Maxwell Equations in Tensor Dielectric
Media

Efstratios Koukoutsis  School of Electrical and Computer Engineering, National
Technical University of Athens, Zographou 15780, Greece  Kyriakos Hizanidis
School of Electrical and Computer Engineering, National Technical University
of Athens, Zographou 15780, Greece  Abhay K. Ram  Plasma Science and Fusion
Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139,
USA  George Vahala  Department of Physics, William & Mary, Williamsburg, VA
23187, USA

###### Abstract

A reformulation of Maxwell equations for an inhomogeneous, anisotropic,
passive and non-dispersive medium results in a quantum-like Schrodinger
equation that admits unitary time evolution. In contrast to other approaches,
there is no a-priori introduction of the Riemann-Silberstein-Weber (RSW)
vector but the Maxwell equations are considered in their standard fields, with
given constitutive relations. From the electromagnetic conservation quantities
a pseudo-Hermitian dynamics is found together with a Dyson map that recovers
the full Hermicity of the dynamics in an extended Hilbert space that describes
the physical notion of unitary evolution. As an example, a uniaxial tensor
dielectric medium is considered, with the explicit Dyson map yielding an
optimal representation in a set of generalized RSW vectors. In this newly
discovered form, a Quantum Computing (QC) implementation for simulation of
electromagnetic wave propagation in complex media can be made, with further
extension into plasmas.

##  1 Introduction

Quantum Computing (QC), plays a major role in the computational studies of
physical systems that require a quantum description [1, 2, 3]. Specifically,
quantum computers show enormous promise for solving classes of problems for
which a quantum algorithm exhibits exponential speedup over the classical
counterpart [4].

However, QC is also being explored for applications in classical branches of
physics as well as in mathematics, where computational speed and
parallelization is an issue [5, 6, 7, 8, 9, 10, 11, 12, 13].
In a top-down approach that considers ideal quantum computers, the main task
for a QC implementation of a classical problem would be a representation of
the classical dynamics in terms of a Schrodinger equation

$i\partialderivative{\ket{\boldsymbol{\psi}}_{c}}{t}=\hat{D}\ket{\boldsymbol{\psi}}_{c}.$
(1)

The classical state vector $\ket{\boldsymbol{\psi}}_{c}$ belongs to a suitable
Hilbert state $\mathcal{H}$ in which the generator of the dynamics $\hat{D}$
is Hermitian. Hermicity ensures that the evolution of the dynamics is unitary.
In the case of time-independent $\hat{D}=\hat{D}(\boldsymbol{r})$, the unitary
evolution propagator $\hat{\mathcal{U}}$ has the simple form

$\ket{\boldsymbol{\psi}}_{c}=\hat{\mathcal{U}}\ket{\boldsymbol{\psi}}_{c0}=\exp{-it\hat{D}}\ket{\boldsymbol{\psi}}_{c0},$
(2)

where $\ket{\boldsymbol{\psi}}_{c0}$ is the initial state vector. Quantum
simulation of the dynamics of Eq. 1 demands the existence of a sequence of
unitary gates $\hat{\mathcal{U}}_{1},...,\hat{\mathcal{U}}_{N}$ that
approximate the evolution operator $\hat{\mathcal{U}}$ to an error
$\varepsilon>0$,

$\norm{\hat{\mathcal{U}}_{N}\hat{\mathcal{U}}_{N-1}...\hat{\mathcal{U}}_{2}\hat{\mathcal{U}}_{1}-\hat{\mathcal{U}}}<\varepsilon.$
(3)

In Eq. 3, the $\norm{\cdot}$ is the Cartesian norm in Hilbert space
$\mathcal{H}$. The construction of such a gate sequence 3 is generally labeled
as Quantum Hamiltonian Simulation (QHS). The unitary nature of gates
$\hat{\mathcal{U}}_{N}$ is essential for the conservation of information [14].

In magnetic fusion, a quantum mechanical description seems irrelevant since
the typical length scales in such plasmas are extremely large compared with
the de-Broglie wavelength. Thus, QC cannot be trivially applied until a
reformulation of plasma dynamics can be constructed, see Eq. 1. Nonetheless,
because of the inherent parallelization of such schemes, QC has been attempted
- to varying levels of success - in the fields of MHD-Burgers turbulence [15,
16], electromagnetic wave propagation in scalar inhomogeneous dielectric media
[17, 18, 19, 20, 21], linear kinetic waves [22] and some cold plasma waves
[23, 24].

Here, a Hermitian electromagnetic Schrodinger representation for complex media
is developed that serves a double purpose. First, it clarifies the connection
between the Maxwell equations and various quantum mechanical isomorphisms
(such as the Dirac equation). Second, it forms a point of reference for future
QC, suitable for modeling the propagation and scattering of electromagnetic
waves and wave-packets in the core of an inhomogeneous plasma as well as in
the edge of magnetically confined plasmas [25, 26, 27]. These studies play an
important role in plasma confinement studies.

In this present paper we restrict ourselves to a quantum description of
Maxwell equations in passive, non-disperive and tensor dielectric media.

##  2 Quantum Representation of Maxwell Equations

Starting from a historical retrospective in the quantum-like reformulations of
Maxwell equations, a general electromagnetic Schrodinger equation is obtained
for a tensor, inhomogeneous and non-dispersive medium. From that point on,
taking into consideration the passivity characteristics of the medium, the
Schrodinger equation is transformed to its Hermitian counterpart in a new
suitable (physical) Hilbert space. In that way the dynamics of our
reformulation are unitary and can be simulated in the framework of QC.

###  2.1 The RSW vector Formulation

Attempts to connect the relativistically-invariant Maxwell equations to the
Dirac equation spans almost 90 years. For a vacuum (i.e., homogeneous scalar
media), Laporte-Uhlenbeck [28] and Oppenheimer [29] applied spinor calculus to
the full set of Maxwell equations in order to obtain an isomorphim to the
Dirac equation. Later related works of Good [30] and Moses [31] investigated
the spectral properties of the generator of dynamics in analogy with the
standard procedure of construction solutions of Schrodinger equation. Our
overlap with these earlier attempts lies in the introduction of the Riemann-
Silberstein-Weber (RSW) vector $\boldsymbol{F}^{\pm}$

$\boldsymbol{F}^{\pm}=\frac{1}{\sqrt{2}}\Big{(}\sqrt{\epsilon}\boldsymbol{E}\pm{i}\frac{\boldsymbol{B}}{\sqrt{\mu}}\Big{)}$
(4)

for dielectric (scalar) electric permittivity $\epsilon$ and magnetic
permeability $\mu$. More recent research has focused on topics like the
connection between the RSW vector and the photon wave function [32, 33, 34],
the quantum representation in scalar inhomogeneous and time varying media
[35], quantum optics [36] and the coupling of Maxwell equations to various
field theories [37].

The time-dependent Maxwell equations in source-free scalar and homogeneous
media

$\partialderivative{\boldsymbol{B}}{t}=-\curl{\boldsymbol{E}},\quad\partialderivative{\boldsymbol{D}}{t}=\curl{\boldsymbol{H}}$
(5)

(in a vacuum, $\epsilon=\epsilon_{0}$ and $\mu=\mu_{0}$) can be written in
compact form in terms of the RSW vector $\boldsymbol{F}^{+}$[30, 33],

$i\partialderivative{\boldsymbol{F}^{+}}{t}=v(\boldsymbol{S}\cdot\hat{\boldsymbol{p}})\boldsymbol{F}^{+}$
(6)

with the normalized velocity $v=1/\sqrt{\epsilon\mu}$. Here, in equation 6,
the $\boldsymbol{S}$ vector consists of the three spin 1 matrices

$\boldsymbol{S}=(S_{x},S_{y},S_{z}),\quad{S}_{x}=\begin{bmatrix}0&0&0\\\
0&0&-i\\\ 0&i&0\end{bmatrix}\quad{S_{y}}=\begin{bmatrix}0&0&i\\\ 0&0&0\\\
-i&0&0\end{bmatrix}\quad{S_{z}}=\begin{bmatrix}0&-i&0\\\ i&0&0\\\
0&0&0\end{bmatrix}$ (7)

that obey the commutator relation $[S_{a},S_{b}]=i\epsilon_{abc}S_{c}$ and
$\hat{\boldsymbol{p}}$ is the standard quantum momentum operator
$\hat{\boldsymbol{p}}=-i\hbar\nabla$. Equation 6 has been viewed as a quantum
representation of classical electrodynamics while the RSW vector plays the
role of the photon wave function [32, 33] \- not in a strict mathematical
sense but rather as an analogy.
In the standard square-integrable Hilbert space
$\mathcal{H}=L^{2}(\mathbb{R}^{3},\mathbb{C})$, the Hermitian Hamiltonian
operator in Eq. 6

$\hat{H}=v(\boldsymbol{S}\cdot\hat{\boldsymbol{p}})$ (8)

has eigenvalues $E=\hbar\omega$, reflecting the monochromatic energy of a
photon. The norm of the RSW vector $\boldsymbol{F}^{+}$ corresponds to the
electromagnetic energy of macroscopic electromagnetic field

$\innerproduct{\boldsymbol{F}^{+}}{\boldsymbol{F}^{+}}=\norm{\boldsymbol{F}^{+}}^{2}=\int_{\Omega}\boldsymbol{F}^{+\dagger}\boldsymbol{F^{+}}d\,\boldsymbol{r}=\frac{1}{2}\int_{\Omega}\Big{(}\epsilon\boldsymbol{E}^{2}+\frac{\boldsymbol{B}^{2}}{\mu}\Big{)}d\,\boldsymbol{r}.$
(9)

Equation 6 is the analog of the Weyl equation for spin 1/2 massless particles

$i\partialderivative{\boldsymbol{\phi}}{t}=c(\boldsymbol{\sigma}\cdot\hat{\boldsymbol{p}})\boldsymbol{\phi},$
(10)

where the $\boldsymbol{\sigma}$ Pauli matrices (appropriate for spin 1/2
particles) have replaced the spin-1 $\boldsymbol{S}$ matrices.

The importance of the RSW vector is highlighted in the 6-vector formulation of
the time-dependent Maxwell equations [38]

$i\partialderivative{\boldsymbol{u}}{t}=W^{-1}\hat{M}\boldsymbol{u}=\hat{D}\boldsymbol{u},\quad\boldsymbol{u}=(\boldsymbol{E},\boldsymbol{H})^{T}$
(11)

with

$\hat{M}=i\begin{bmatrix}0&\curl\\\
-\curl&0\end{bmatrix},\quad{W}=\begin{bmatrix}\epsilon{I}_{3\times
3}&0_{3\times 3}\\\ 0_{3\times 3}&\mu{I}_{3\times 3}\end{bmatrix}.$ (12)

The invertible and Hermitian $W$-matrix embodies the constitutive relation
connecting $\boldsymbol{u}=(\boldsymbol{E},\boldsymbol{H})^{T}$ to the
$(\boldsymbol{D},\boldsymbol{B})^{T}$ fields and $I_{3\times 3}$ is the
$3\times 3$ identity matrix. The $\hat{M}$ operator is called the Maxwell
operator and is Hermitian in $L^{2}(\mathbb{R}^{3},\mathbb{C})$ under
appropriate boundary conditions. For example, for electromagnetic propagation
in a bounded domain $\Omega\subset\mathbb{R}^{3}$ (interior problem) then one
must impose appropriate Dirichlet boundary condition on $\partial\Omega$ of
$\Omega$. For homogeneous media, (e.g., for a vacuum $\epsilon=\epsilon_{0}$
and $\mu=\mu_{0}$), the generator of the evolution operator $\hat{D}$ in 11 is
Hermitian since the individual Hermitian oeprators commute:
$W^{-1}\hat{M}=\hat{M}W^{-1}$. Applying the $W^{1/2}$ transformation to Eq. 11
we obtain

$i\partialderivative{\boldsymbol{U}}{t}=(-\sigma_{y}\otimes{v}\boldsymbol{S}\cdot\hat{\boldsymbol{p}})\boldsymbol{U}=\hat{D}_{\rho}\boldsymbol{U},\quad\boldsymbol{U}=W^{1/2}\boldsymbol{u}.$
(13)

Equation 13 is the analog of the massless Dirac equation for spin 1 photons
with Hermitian operator $\hat{D}_{\rho}$. Using the unitary transformation
$L:\boldsymbol{U}\to\boldsymbol{F}$ with

$L=\frac{1}{\sqrt{2}}\begin{bmatrix}I_{3\times 3}&i_{3\times 3}\\\ I_{3\times
3}&-i_{3\times 3}\end{bmatrix}$ (14)

Eq. 13 takes the block diagonal form

$i\partialderivative{t}\begin{bmatrix}\boldsymbol{F^{+}}\\\
\boldsymbol{F^{-}}\end{bmatrix}=\begin{bmatrix}v\boldsymbol{S}\cdot\hat{\boldsymbol{p}}&0\\\
0&-v\boldsymbol{S}\cdot\hat{\boldsymbol{p}}\end{bmatrix}\begin{bmatrix}\boldsymbol{F^{+}}\\\
\boldsymbol{F^{-}}\end{bmatrix}.$ (15)

The evolution of the RSW vectors $\boldsymbol{F^{+}}$ and $\boldsymbol{F^{-}}$
are uncoupled since a homogeneous medium does not couple field helicities. In
other words, the RSW transformation is a Weyl representation of Dirac equation
13. Notice how the evolution of $\boldsymbol{F^{+}}$ in Eq. 13 is nothing but
Eq. 6. Finally, the representations Eq. 13 and Eq. 15 are equivalent due to
the unitary nature of transformation Eq. 14.

###  2.2 Maxwell Equations in Inhomogeneous Passive Media

For source-free problems, the time-dependent Maxwell equations 5 read

$i\partialderivative{\boldsymbol{d}}{t}=\hat{M}\boldsymbol{u}.$ (16)

The electromagnetic flux densities
$\boldsymbol{d}(\boldsymbol{r},t)=(\boldsymbol{D},\boldsymbol{B})^{T}$ are
related to the electromagnetic fields
$\boldsymbol{u}(\boldsymbol{r},t)=(\boldsymbol{E},\boldsymbol{H})^{T}$ by some
given constitutive operator $\hat{\mathcal{L}}$

$\boldsymbol{d}=\boldsymbol{d}(\boldsymbol{u})\Rightarrow\boldsymbol{d}=\hat{\mathcal{L}}\boldsymbol{u}$
(17)

with initial conditions on the fields

$\divergence\boldsymbol{d}_{0}=\divergence(\hat{\mathcal{L}}\boldsymbol{u})_{0}=0$
(18)

From Eq. 16 one immediately notices that the divergence Maxwell equations
$\divergence{\boldsymbol{d}}=0$ will be satisfied for all times, if they are
satisfied initially at $t=0$, hence the choice of initial conditions
$\boldsymbol{u}(\boldsymbol{r},0)=\boldsymbol{u}_{0}$ in Eq. 18. For boundary
conditions we will impose perfect boundaries so that

$\hat{\boldsymbol{n}}(\boldsymbol{r})\times\boldsymbol{u}_{1}=0\,\,in\,\,boundary\,\,\partial\Omega,$
(19)

where $\hat{\boldsymbol{n}}(\boldsymbol{r})$ denotes the unit outward normal
to the boundary surface. The selection of boundary conditions is critical in
both energy considerations and in the Hermicity of the Maxwell operator
$\hat{M}$.
Relations 16-19 pose the complete mathematical problem under investigation in
Hilbert state space
$\mathcal{H}=L^{2}(\Omega,\mathbb{R}^{6})\ni\boldsymbol{u}$ equipped the inner
product [39]

$\innerproduct{\boldsymbol{v}}{\boldsymbol{u}}=\int_{\Omega}\boldsymbol{v}^{\dagger}(\boldsymbol{r},t)\boldsymbol{u}(\boldsymbol{r},t)d\,\boldsymbol{r},\quad\Omega\subseteq\mathbb{R}^{3},\,\,t\in\mathcal{T}=[0,T].$
(20)

A general form of the linear constitutive equations $\hat{\mathcal{L}}$ that
obeys physical postulates as described in [39], is

$\boldsymbol{d}(\boldsymbol{r},t)=\hat{\mathcal{L}}\boldsymbol{u}(\boldsymbol{r},t)=W(\boldsymbol{r})\boldsymbol{u}(\boldsymbol{r},t)+\int_{0}^{t}G(\boldsymbol{r},t-\tau)\boldsymbol{u}(\boldsymbol{r},\tau)d\,\tau$
(21)

where the first term in Eq. 21 is the optical (instantaneous) response of the
medium while the integral term incorporates a dispersive response with memory
effects ($G$ is the susceptibility kernel). Here, we shall restrict ourselves
to anisotropic non-dispersive media so that Eq. 21 reduces to
$\boldsymbol{d}(\boldsymbol{r},t)=W(\boldsymbol{r})\boldsymbol{u}(\boldsymbol{r},t)$
with the $6\times 6$ matrix $W(\boldsymbol{r})$

$W=\begin{bmatrix}\epsilon(\boldsymbol{r})&0_{3\times 3}\\\ 0_{3\times
3}&\mu(\boldsymbol{r})\end{bmatrix}.$ (22)

In general, constitutive relations of the form
$\boldsymbol{d}=\hat{\mathcal{L}}\boldsymbol{u}=W(\boldsymbol{r})\boldsymbol{u}$
can only be seen as an approximation of the real electromagnetic response of a
medium [40], there is always a non-local in time response function just like
the full version of Eq. 21. Since $W$ is invertible [39], the Maxwell Eq. 16
for the electromagnetic field
$\boldsymbol{u}=(\boldsymbol{E},\boldsymbol{H})^{T}$ is

$i\partialderivative{\boldsymbol{u}}{t}=W^{-1}(\boldsymbol{r})\hat{M}\boldsymbol{u}=\hat{D}\boldsymbol{u}.$
(23)

In the absence of free sources the Poynting theorem [39] reads

$\divergence{\boldsymbol{S}}+\boldsymbol{u}^{\dagger}\partialderivative{\boldsymbol{d}}{t}=0,$
(24)

with $\boldsymbol{S}=\boldsymbol{E}\times\boldsymbol{H}$ the Poynting vector.
Following [41], the electromagnetic energy density $e_{opt}$ in the optical
response region is

$e_{opt}(\boldsymbol{r},t)=\int_{0}^{t}\boldsymbol{u}^{\dagger}\partialderivative{\boldsymbol{d}(\boldsymbol{r},\tau)}{\tau}d\,\tau=\frac{1}{2}\boldsymbol{u}^{\dagger}{W}\boldsymbol{u}+\int_{0}^{t}\boldsymbol{u}^{\dagger}(\boldsymbol{r},\tau)W^{A}\partialderivative{\boldsymbol{u}(\boldsymbol{r},\tau)}{\tau}d\,\tau,$
(25)

with $W^{A}=\frac{W-W^{\dagger}}{2}$ the anti-Hermitian part of matrix $W$.
A medium is characterized as passive (dissipative) in optical response region
when [39, 41]

$e_{opt}(\boldsymbol{r},t)\geq 0,\quad\forall\boldsymbol{r}\in\Omega.$ (26)

Thus, from relation 25, we deduce that the optical $W$ matrix for passive
media must be Hermitian and semi-positive definite

$W\geq 0\quad{and}\quad{W}=W^{\dagger}$ (27)

and the total integrated stored electromagnetic energy $E$ in volume $\Omega$
is

$E(t)=\frac{1}{2}\int_{\Omega}\boldsymbol{u}^{\dagger}{W}\boldsymbol{u}d\,\boldsymbol{r}\geq
0.$ (28)

The above equation reflects that no net production of electromagnetic energy
is possible in $\Omega$, and therefore the medium is passive [39, 41].
Finally, integrating Eq. 24 over the volume $\Omega$ and using divergence
theorem we obtain

$\partialderivative{E(t)}{t}+\int_{\partial\Omega}(\boldsymbol{u}_{1}\times\boldsymbol{u}_{2})\cdot\hat{\boldsymbol{n}}(\boldsymbol{r})d\,a=0.$
(29)

###  2.3 Pseudo-Hermitian Operators

The central objective of this part is to examine, whether our Maxwell-
Schrodinger representation 23 admits the standard quantum mechanical
mathematical treatment, focusing on unitary evolution which is crucial for
Quantum Algorithm implementation.

An operator $\hat{A}$ acting on a Hilbert space $\mathcal{H}$,
$\hat{A}:\mathcal{H}\to\mathcal{H}$ equipped with inner product
$\innerproduct{\cdot}{\cdot}$ is Hermitian when,

$\innerproduct{\cdot}{\hat{A}\cdot}=\innerproduct{\cdot\hat{A}}{\cdot}\Leftrightarrow\hat{A}=\hat{A}^{\dagger}.$
(30)

Using the inner product expression Eq. 20 in Hilbert space
$\mathcal{H}=L^{2}(\Omega,\mathbb{R}^{6})$ for the Maxwell operator $\hat{M}$
provided in Eq. 12, we obtain

$\displaystyle\innerproduct{\boldsymbol{v}}{\hat{M}\boldsymbol{u}}$
$\displaystyle=i\int_{\Omega}\Big{(}\boldsymbol{v^{*}_{1}}\cdot(\curl{\boldsymbol{u}_{2}})-\boldsymbol{v^{*}_{2}}\cdot(\curl{\boldsymbol{u}_{1}})\Big{)}d\,\boldsymbol{r}=i\int_{\Omega}\boldsymbol{u}_{2}\cdot(\curl{\boldsymbol{v^{*}_{1}}})-\boldsymbol{u_{1}}\cdot(\curl{\boldsymbol{v^{*}_{2}}})d\,\boldsymbol{r}+$
(31)
$\displaystyle+i\int_{\partial\Omega}\Big{(}\boldsymbol{u}_{2}\cdot(\boldsymbol{v}_{1}\times\hat{\boldsymbol{n}})+\boldsymbol{v}_{2}\cdot(\hat{\boldsymbol{n}}\times\boldsymbol{u}_{1})\Big{)}d\,a,\quad\boldsymbol{v},\boldsymbol{u}\in\mathcal{H}=L^{2}(\Omega,\mathbb{R}^{6}).$

Deriving Eq. 31 has been made use of Divergence theorem, commutation of mixed
product, and vector identity
$\divergence{(\boldsymbol{a}\times\boldsymbol{b})}=\boldsymbol{b}\cdot(\curl{\boldsymbol{a}})-\boldsymbol{a}\cdot(\curl{\boldsymbol{b}})$.
Dictation of boundary condition Eq. 19 sets the surface integral terms in 31
equal to zero, rendering the Maxwell operator $\hat{M}$ Hermitian

$\innerproduct{\boldsymbol{v}}{\hat{M}\boldsymbol{u}}=i\int_{\Omega}\boldsymbol{u}_{2}\cdot(\curl{\boldsymbol{v^{*}_{1}}})-\boldsymbol{u_{1}}\cdot(\curl{\boldsymbol{v^{*}_{2}}})d\,\boldsymbol{r}=\innerproduct{\boldsymbol{v}\hat{M}}{\boldsymbol{u}}\Rightarrow\hat{M}=\hat{M}^{\dagger},$
(32)

according to definition 30.

For a passive medium it is already shown that the constitutive matrix
$W(\boldsymbol{r})$ is Hermitian 27, thus $W^{-1}=(W^{-1})^{\dagger}$.
Nonetheless, the product structure of operator $\hat{D}=W^{-1}\hat{M}$ demands
commutation of operators, $[W^{-1},\hat{M}]=0$, in order to be Hermitian in
Hilbert space $L^{2}(\Omega,\mathbb{R}^{6})$ which in general does not apply
to inhomogeneous media. Subsequently, the generator operator of evolution
$\hat{D}$ is non-Hermitian and the evolution of electromagnetic Schrodinger
equation 23 is non-unitary.

From Poynting theorem Eq. 29 and the boundary condition Eq. 19, we obtain
energy conservation

$\partialderivative{E}{t}=0\Leftrightarrow{E}(t)=E(t=0)=\int_{\Omega}\boldsymbol{u}_{0}^{\dagger}{W}(\boldsymbol{r})\boldsymbol{u}_{0}d\,\boldsymbol{r}=constant.$
(33)

so that an underlying symmetry must be present. What shall be explicitly shown
is that this will permit us to define a new inner product with its Hilbert
space $\mathcal{H}_{phys}$ that will represent the physics and the symmetries
of our problem [42]. In other words, in $\mathcal{H}_{phys}$, we can assign to
unitary evolution the notion of energy conservation 33 similar to QC where
unitary evolution means probability conservation. The existence of such an
inner product as a bi-linear symmetry reflects the fact that the operator
$\hat{D}$ is a very special kind of non-Hermitian operators, called pseudo-
Hermitian [43].

Non-Hermitian quantum mechanics has aroused great interest since the
pioneering works of Bender on the real eigenvalue spectrum of $PT$-symmetric
Hamiltonians [44, 45]. $PT$-symmetric Hamiltonians are a subset of a more
general theory of Pseudo-Hermitian Hamiltonians [46, 47] that reflects the
different representation of the same physical theory. In this sense, there
exists an appropriate Hilbert space of states, a generator of dynamics and
observables [48, 49, 50] associated with this new representation.

A linear operator $\hat{D}$ acting in a Hilbert space $\mathcal{H}$,
$\hat{D}:\mathcal{H}\to\mathcal{H}$ is said to be Pseudo-Hermitian [47] if
there exists an invertible Hermitian linear operator $\eta$, such that

$\hat{D}^{\dagger}=\eta\hat{D}\eta^{-1}.$ (34)

In addition, if $\eta$ is positive definite (i.e.,
$\matrixelement{\boldsymbol{u}}{\eta}{\boldsymbol{u}}>0$ for all nonzero
states $\boldsymbol{u}$) then it is called a metric operator and we can endow
the Hilbert space $\mathcal{H}$ with the following inner product

$\innerproduct{\cdot}{\cdot}_{\eta}=\innerproduct{\cdot}{\eta\cdot},$ (35)

to generate the new weighted Hilbert space $\mathcal{H}_{\eta}$. It is
critical to realize that in $\mathcal{H}_{\eta}$ the operator $\hat{D}$ is
Hermitian with respect to the weighted inner product 35. The new, weighted
Hilbert space $\mathcal{H}_{\eta}$ represents the physical space
$\mathcal{H}_{phys}$ of the electromagnetic problem in which we obtain the
Hermitian generator of the dynamics and a proper physical correspondence of
unitary to conservative quantities [42, 48, 49].

In our Maxwell equations problem, it is possible to determine explicitly the
electromagnetic metric operator $\eta$ by exploiting the product structure of
$\hat{D}=W^{-1}\hat{M}$ [51] : indeed, we find $\eta=W$, since

$\hat{D}^{\dagger}=\hat{M}W^{-1}=W\hat{D}W^{-1}\Leftrightarrow\eta=W.$ (36)

Immediately, it is explicitly seen that $\hat{D}$ is Hermitian in the weighted
Hilbert space $\mathcal{H}_{\eta}$:

$\displaystyle\innerproduct{\boldsymbol{v}}{\hat{D}\boldsymbol{u}}_{W}$
$\displaystyle=\innerproduct{\boldsymbol{v}}{\hat{M}\boldsymbol{u}}=\innerproduct{\boldsymbol{v}\hat{M}}{\boldsymbol{u}}=\innerproduct{\boldsymbol{v}\hat{M}W^{-1}W}{\boldsymbol{u}}=\innerproduct{\boldsymbol{v}\hat{M}W^{-1}}{W\boldsymbol{u}}$
(37)
$\displaystyle=\innerproduct{\boldsymbol{v}\hat{M}W^{-1}}{\boldsymbol{u}}_{W}=\innerproduct{\boldsymbol{v}\hat{D}}{\boldsymbol{u}}_{W}\Rightarrow\hat{D}=\hat{D}^{\dagger}$

Thus, for a passive non-dispersive medium with the optical response matrix $W$
(i.e., with $\boldsymbol{d}=W\boldsymbol{u}$) the generator of the dynamics
$\hat{D}$ is Hermitian in $\mathcal{H}_{\eta}=H_{phys}$ with the inner product
[51]

$\innerproduct{\boldsymbol{v}}{\boldsymbol{u}}_{W}=\int_{\Omega}\boldsymbol{v}^{\dagger}(\boldsymbol{r},t)W(\boldsymbol{r})\boldsymbol{u}(\boldsymbol{r},t)d\,\boldsymbol{r}=\innerproduct{\boldsymbol{v}}{W\boldsymbol{u}}<\infty,\quad\boldsymbol{v},\boldsymbol{u}\in\mathcal{H}$
(38)

and the evolution is unitary with

$\innerproduct{\boldsymbol{u}}{\boldsymbol{u}}_{W}=\norm{\boldsymbol{u}}^{2}_{W}=\matrixelement{\boldsymbol{u}}{W}{\boldsymbol{u}}=\int_{\Omega}\boldsymbol{u}^{\dagger}(\boldsymbol{r},t)W(\boldsymbol{r})\boldsymbol{u}(\boldsymbol{r},t)d\,\boldsymbol{r}=2E=constant$
(39)

as expected from 33.

The Pseudo-Hermitian structure of wave propagation has also been studied in
[52], where the spectral properties of the kernel of the Helmholtz equation
are considered. Here, the pseudo-Hermitian structure of Maxwell equations has
been studied instead.

###  2.4 The Dyson Map for the Maxwell Equations

Whereas we have succeed to ensure Hermicity of operator $\hat{D}$ in the new
physical Hilbert space
$\mathcal{H}_{phys}=\\{\boldsymbol{u}\in\mathcal{H}:\innerproduct{\boldsymbol{u}}{\boldsymbol{u}}_{W}<\infty\\}$
under the weighted inner product 38, nothing has explicitly changed in the
form of Maxwell-Schrodinger equation 23 except the fact that now,
$\boldsymbol{u}\in\mathcal{H}_{phys}$. Reconnecting the original Hilbert space
$\mathcal{H}$, in which $\hat{D}$ is not Hermitian to the new weighted Hilbert
space $\mathcal{H}_{phys}$ under whose inner product $\hat{D}$ is Hermitian
demands the construction of a transformation that preserves the inner product
structure between those two Hilbert spaces. As such, this transformation must
be an isometry. In the context of quantum mechanics such a invertible
transformation $\rho(\boldsymbol{r})$ between equivalent descriptions of a
quantum system in different Hilbert spaces,
$\rho(\boldsymbol{r}):\mathcal{H}_{phys}\to\mathcal{H}$ is called Dyson map
[42, 48, 47].

The construction of $\rho$ map is based on the factorization of the metric
operator $\eta$ in the form

$\eta(\boldsymbol{r})=\rho^{\dagger}(\boldsymbol{r})\rho(\boldsymbol{r}).$
(40)

This preserves the inner product structure between $\mathcal{H}$ and
$\mathcal{H}_{phys}$ since

$\innerproduct{\boldsymbol{v}}{\boldsymbol{u}}_{\eta}=\innerproduct{\boldsymbol{v}}{\rho^{\dagger}\rho\boldsymbol{u}}=\innerproduct{\boldsymbol{v}\rho}{\rho\boldsymbol{u}}=\innerproduct{\boldsymbol{\phi}}{\boldsymbol{\psi}},\quad\boldsymbol{v},\boldsymbol{u}\in\mathcal{H}_{phys}\,\,and\,\,\boldsymbol{\phi},\boldsymbol{\psi}\in\mathcal{H}.$
(41)

For the Maxwell equations, the Hermitian and positive definite electromagnetic
metric $\eta=W$ can have three different decomposition Eq, 40 schemes [53].
$\bullet$ Spectral decomposition:

$W(\boldsymbol{r})=U^{\dagger}{D}(\boldsymbol{r})U=U^{\dagger}\sqrt{D(\boldsymbol{r})}\sqrt{D(\boldsymbol{r)}}U=\rho^{\dagger}(\boldsymbol{r})\rho(\boldsymbol{r}),$
(42)

giving the Dyson map

$\rho(\boldsymbol{r})=\sqrt{D(\boldsymbol{r})}U.$ (43)

. $\bullet$ Square root decomposition:

$W(\boldsymbol{r})=W^{1/2}(\boldsymbol{r})W^{1/2}(\boldsymbol{r})=\rho^{\dagger}(\boldsymbol{r})\rho(\boldsymbol{r}),$
(44)

giving the Dyson map

$\quad\rho(\boldsymbol{r})=W^{1/2}(\boldsymbol{r}).$ (45)

$\bullet$ Cholesky decomposition:

$W(\boldsymbol{r})=L^{\dagger}(\boldsymbol{r}){L}(\boldsymbol{r})=\rho^{\dagger}(\boldsymbol{r})\rho(\boldsymbol{r}),$
(46)

giving the Dyson map

$\quad\rho(\boldsymbol{r})=L(\boldsymbol{r}).$ (47)

In spectral decomposition Eq. 42,
$D(\boldsymbol{r})=\lambda_{i}(\boldsymbol{r})\delta_{ij}$ with
$\lambda(\boldsymbol{r})>0$ and
$\sqrt{D(\boldsymbol{r})}=\sqrt{\lambda_{i}(\boldsymbol{r})}\delta_{ij}$,
whereas in the Cholesky decomposition the $L$ matrix is upper triangular with
positive diagonal elements. Typically, one will choose the factorization that
leads to a sparse final Dyson map $\rho$.

Consequently, the Hermitian counterpart of Eq. 23 in our original
$\mathcal{H}$ Hilbert space takes the form

$i\partialderivative{\boldsymbol{\psi}}{t}=\rho(\boldsymbol{r})\hat{D}\rho^{-1}(\boldsymbol{r})\boldsymbol{\psi}=\hat{D}_{\rho}\boldsymbol{U},\quad\boldsymbol{\psi}=\rho\boldsymbol{u}\in\mathcal{H}.$
(48)

The generator of dynamics $\hat{D}_{\rho}$ in now Hermitian in Hilbert space
$\mathcal{H}$ and yields unitary evolution (energy conserving). Specifically,
the evolution is given by

$\boldsymbol{\psi}(\boldsymbol{r},t)=\exp{-it\hat{D}_{\rho}}\boldsymbol{\psi}_{0}=\rho(\boldsymbol{r})\exp{-it\hat{D}}\rho^{-1}(\boldsymbol{r})\boldsymbol{\psi}_{0}.$
(49)

Every operator $\hat{A}_{\eta}:\mathcal{H}_{phys}\to\mathcal{H}_{phys}$ is
related to its $\hat{A}$ counterpart in $\mathcal{H}$ through the similarity
transformation

$\hat{A}=\rho\hat{A}_{\eta}\rho^{-1}.$ (50)

A schematic of the Dyson map $\rho$ is shown below. It is the catalyst
connecting the different representation basis for the initial Hilbert space
$\mathcal{H}$, $\rho:\mathcal{H}\to\mathcal{H}$

${\boldsymbol{u}\in\mathcal{H}_{phys}}$${\boldsymbol{\psi}\in\mathcal{H}}$${\boldsymbol{u}\in\mathcal{H}}$$\scriptstyle{\rho}$$\scriptstyle{\text{definition
of}\,\,\mathcal{H}_{phys}}$$\scriptstyle{\rho^{-1}}$

This composition of spaces proposes, similarly with the three-Hilbert space
formulation of quantum mechanics [54], that the optimal workspace is the one
containing the elements $\boldsymbol{\psi}\in\mathcal{H}$. This in turn
reinforces the approach of Eq. 48 for QC.

###  2.5 Application to Uniaxially Dielectric Media

Let us now examine the form of the Hermitian operator $\hat{D}$, 49, for the
important case of a uniaxial dielectric and non-magnetic medium. The tensor
constitutive relations are:

$\epsilon(\boldsymbol{r})=\begin{bmatrix}\epsilon_{x}(\boldsymbol{r})&0&0\\\
0&\epsilon_{x}(\boldsymbol{r})&0\\\
0&0&\epsilon_{z}(\boldsymbol{r})\end{bmatrix},\quad\mu=\mu_{0}I_{3\times 3}$
(51)

then, the obvious choice (and best in terms of sparsity) for the Dyson map
$\rho$ is

$\rho=W^{1/2}=\begin{bmatrix}\epsilon^{1/2}&0_{3\times 3}\\\ 0_{3\times
3}&\sqrt{\mu_{0}}I_{3\times 3}\end{bmatrix},\quad
where\quad\epsilon^{1/2}=\begin{bmatrix}\sqrt{\epsilon_{x}(\boldsymbol{r})}&0&0\\\
0&\sqrt{\epsilon_{x}(\boldsymbol{r})}&0\\\
0&0&\sqrt{\epsilon_{z}(\boldsymbol{r})},\end{bmatrix}$ (52)

and the Hermitian generator of dynamics $\hat{D}_{\rho}$ has the familiar form

$\hat{D}_{\rho}=\begin{bmatrix}0_{3\times
3}&ic\boldsymbol{Z}\cdot\hat{\boldsymbol{p}}\\\
-ic\hat{\boldsymbol{p}}\cdot\boldsymbol{Z}^{\dagger}&0_{3\times
3}\end{bmatrix}.$ (53)

The components of the $\boldsymbol{Z}=(Z_{x},Z_{y},Z_{z})$ matrix vector, in
terms of the refractive index
$n_{i}(\boldsymbol{r})=\sqrt{\frac{\epsilon_{i}(\boldsymbol{r})}{\epsilon_{0}}}$,
are

$Z_{x}=\begin{bmatrix}0&0&0\\\ 0&0&-\frac{i}{n_{x}}\\\
0&\frac{i}{n_{z}}&0\end{bmatrix},\quad{Z}_{y}=\begin{bmatrix}0&0&\frac{i}{n_{x}}\\\
0&0&0\\\
-\frac{i}{n_{z}}&0&0\end{bmatrix},\quad{Z}_{z}=\begin{bmatrix}0&-\frac{i}{n_{x}}&0\\\
\frac{i}{n_{x}}&0&0\\\ 0&0&0\end{bmatrix}.$ (54)

Clearly, Eqs. 53 and 54 are the generalization of those for the homogeneous
(isotropic) scalar medium. Setting
$n_{x}(\boldsymbol{r})=n_{z}(\boldsymbol{r})=n=constant$ then we find

$\boldsymbol{Z}\to\frac{1}{n}\boldsymbol{S},\,\,{and}\,\,\hat{D}_{\rho}\to\begin{bmatrix}0_{3\times
3}&iv\boldsymbol{S}\cdot\hat{\boldsymbol{p}}\\\
-iv\boldsymbol{S}\cdot\hat{\boldsymbol{p}}&0_{3\times 3}\end{bmatrix}.$ (55)

where $\boldsymbol{S}$ vector is the spin-1 matrices, Eq. 7. For homogeneous
scalar media $[\boldsymbol{S},\hat{\boldsymbol{p}}]=0$, while in the
inhomogeneous case, both for isotropic and anisotropic media,
$[\boldsymbol{Z},\hat{\boldsymbol{p}}]\neq 0$. Finally in the inhomogeneous
isotropic (scalar) case the $\boldsymbol{Z}$ operator is Hermitian similar to
$\boldsymbol{S}$ operator.

Acting with the $L$ operator Eq. 14 in Eq. 53 we obtain the generalized RSW
vector $\boldsymbol{F}_{gen}^{\pm}$,

$\boldsymbol{F}_{gen}^{\pm}=L\boldsymbol{\psi}=LW^{1/2}\boldsymbol{u}=\frac{1}{\sqrt{2}}\Big{(}\epsilon^{1/2}(\boldsymbol{r})\boldsymbol{E}\pm{i}\frac{\boldsymbol{B}}{\mu_{0}}\Big{)}$
(56)

and the respective unitary evolution equation

$i\partialderivative{t}\begin{bmatrix}\boldsymbol{F}_{gen}^{+}\\\
\boldsymbol{F}_{gen}^{-}\end{bmatrix}=c\begin{bmatrix}(\boldsymbol{Z}\cdot\hat{\boldsymbol{p}})^{H}&-(\boldsymbol{Z}\cdot\hat{\boldsymbol{p}})^{A}\\\
(\boldsymbol{Z}\cdot\hat{\boldsymbol{p}})^{A}&-(\boldsymbol{Z}\cdot\hat{\boldsymbol{p}})^{H}\end{bmatrix}\begin{bmatrix}\boldsymbol{F}_{gen}^{+}\\\
\boldsymbol{F}_{gen}^{-}\end{bmatrix}$ (57)

where the superscripts $H$ and $A$ refer to Hermitian and anti-Hermitian part
respectively. Decomposition of equation 57 produces two Weyl equations coupled
with an effective term similar to the Weyl representation [32, 33] of Dirac
spinor for the electron.

$\displaystyle i\partialderivative{\boldsymbol{F}^{+}_{gen}}{t}$
$\displaystyle=c(\boldsymbol{Z}\cdot\hat{\boldsymbol{p}})^{H}\boldsymbol{F}^{+}_{gen}-c(\boldsymbol{Z}\cdot\hat{\boldsymbol{p}})^{A}\boldsymbol{F}^{-}_{gen}$
(58) $\displaystyle i\partialderivative{\boldsymbol{F}^{-}_{gen}}{t}$
$\displaystyle=-c(\boldsymbol{Z}\cdot\hat{\boldsymbol{p}})^{H}\boldsymbol{F}^{-}_{gen}+c(\boldsymbol{Z}\cdot\hat{\boldsymbol{p}})^{A}\boldsymbol{F}^{+}_{gen}$

In Eq. 58 note how the anti-Hermitian part of the operator
$\boldsymbol{Z}\cdot\hat{\boldsymbol{p}}$ couples the two Weyl equations
playing the role of an effective mass operator. When the operator
$\boldsymbol{Z}\cdot\hat{\boldsymbol{p}}$ is Hermitian, i.e:
$(\boldsymbol{Z}\cdot\hat{\boldsymbol{p}})^{A}=0$ we obtain the analogous
expressions of homogeneous space Eq. 15.

##  3 Connection with QC

This section focus on future possible implementation of electromagnetic
Schrodinger equation 48 into a quantum algorithmic scheme. Since the main
interest is on electromagnetic propagation in an inhomogeneous medium in the
temporal domain, evaluattion of the time evolution operator Eq. 49 is
necessary. The procedure of evolution operator simulation is called Quantum
Hamiltonian Simulation (QHS) because the generator of the dynamics (the
Hamiltonian in standard quantum mechanics) dictates the complexity of the
simulation. While we will leave any explicit quantum simulations for future
publications, here we outline some basic ideas and consequences that apply to
our proposed Maxwell solvers. The current status of the qubit lattice
algorithm (QLA) approach to Maxwell equations os also discussed. QLA, in its
optimal form, consists of an interleaved sequence of unitary collision and
streaming operators which perturbatively recover the Maxwell system. QLA has
been extremely successful in simulations of soliton collisions in the
nonlinear Schrodinger and KdV equations.

###  3.1 Quantum Hamiltonian Simulation (QHS) for Uniaxial Dielectric Media

QHS is basically an approximation procedure Eq. 3 of unitary evolution
operator Eq. 49 to an error $\varepsilon$. This approximation can be achieved
through Trotter-Suzuki [55, 56] product formula [57, 58, 59], Taylor expansion
[60] or modern signal processing [61, 62].
Following [58], the operator $\hat{D}_{\rho}$ in Eq. 53 for the uniaxial case
is written as

$\hat{D}_{\rho}=\sum_{\gamma=1}^{\Gamma}\hat{D}_{\gamma},\quad\hat{D}_{\gamma}=\hat{D}^{\dagger}_{\gamma}.$
(59)

If there is a $p$th order product formula approximation $\hat{S}$ of the
unitary evolution operator $\hat{\mathcal{U}}=e^{-it\hat{D}_{\rho}}$ then,

$\norm{\hat{S}-e^{-it\hat{D}_{\rho}}}=\mathcal{O}\Big{(}\Big{(}\sum_{\gamma=1}^{\Gamma}\norm{\hat{D}_{\gamma}}t)^{p+1}\Big{)}\Big{)}.$
(60)

The above error bound in Eq. 60 is valid only for sufficiently small time
parameter $t$. In order to simulate the evolution of larger time we need to
divide the evolution into $r$ time steps and apply the product formula in each
step. Then, the error of the overall simulation is

$\norm{\hat{S}^{r}(t/r)-e^{it\hat{D}_{\rho}}}=\mathcal{O}\Big{(}\frac{(\sum_{\gamma=1}^{\Gamma}\norm{D_{\gamma}}t)^{p+1}}{r^{p}}\Big{)}.$
(61)

As a result, demanding overall simulation accuracy of order $\varepsilon$, the
number $r$ of time steps is given from

$r=\mathcal{O}\Big{(}\frac{(\sum_{\gamma=1}^{\Gamma}\norm{D_{\gamma}}t)^{1+1/p}}{\epsilon^{1/p}}\Big{)}.$
(62)

While the uniaxial operator $\hat{D}_{\rho}$ is a $4$-sparse operational
matrix with $66,7\%$ sparsity, a fact that favors the quantum simulation [59],
it contains the unbounded operator $\hat{p}$ that leads to an unbounded number
of $r$ steps according to Eq. 62. This issue, has been stated in [63] where
the author has assumed the existence of extra structure in the evolution
operator in order to produce a bounded number $r$ of applied gates. This extra
structure is that the evolution operator must not only be unitary but special
unitary, $\det{e^{-it\hat{D}_{\rho}}}=1$. In our case the uni-axial operator
$\hat{D}_{\rho}$ as can be seen from Eq. 53 is a trace-less Hermitian operator
belonging to a Lie algebra. As a result a QHS implementation is possible under
the findings of [63].

Another way to simulate the wave dynamics in an inhomogeneous uniaxial medium
is through the Quantum Fourier Transform (QFT) [14]. QFT is a quantum
algorithm that acts as a sub-routine in many important quantum algorithms such
as Phase Estimation [14] and Amplitude Estimation [64].

Starting now from Eq. 57, the Hermitian operator $\hat{D}_{\rho}$ can be
decomposed into a convenient tensor product sum of similar Hermitian operators

$\hat{D}_{\rho}=\sigma_{z}\otimes\hat{D}_{1}+\sigma_{y}\otimes\hat{D_{2}},\quad\hat{D}_{1}=(\boldsymbol{Z}\cdot\hat{\boldsymbol{p}})^{H}\,\,and\,\,\hat{D}_{2}=-i(\boldsymbol{Z}\cdot\hat{\boldsymbol{p}})^{A}.$
(63)

Operators $\hat{D}_{1}$ and $\hat{D}_{2}$ are not only Hermitian but also
linearly dependent on the momentum operator,
$\hat{D}_{1,2}=\hat{D}_{1,2}(\hat{\boldsymbol{p}})$. Through this paper our
representations are based on the configuration space
$\Omega\subset\mathbb{R}^{3}$ so exploiting the conjugate nature between
position $\hat{\boldsymbol{r}}$ and momentum $\hat{\boldsymbol{p}}$ operators,
we act with a double sided QFT in 63 to obtain the unitary evolution operator

$\hat{\mathcal{U}}=\Big{[}(QFT)\exp{-it(\sigma_{z}\otimes\hat{D}_{1}(\hat{\boldsymbol{r}})}(QFT)^{\dagger}\Big{]}\Big{[}(QFT)\exp{-it(\sigma_{y}\otimes\hat{D}_{2}(\hat{\boldsymbol{r}})}(QFT)^{\dagger}\Big{]}+\varepsilon$
(64)

with
$\hat{D}_{1}(\hat{\boldsymbol{r}})=(\boldsymbol{Z}\cdot\hat{\boldsymbol{r}})^{H}$
and
$\hat{D}_{2}(\hat{\boldsymbol{r}})=(\boldsymbol{Z}\cdot\hat{\boldsymbol{r}})^{A}$.
In a descretization scheme of the position space, the produced error
$\varepsilon$ in Eq. 64 is now bounded.

In QC, under a measurement we are able to identify only the expected value of
the Hermitian operators with physical meaning (observables). For example,
starting from an initial state $\ket{\boldsymbol{\psi}}_{0}$ to a future state
$\ket{\boldsymbol{\psi}}(t=t_{1}>0)$ the physical observable quantities can be
the electromagnetic energy, the electric energy and the magnetic energy of the
system in $t=t_{1}$ corresponding to the following measurement operators:

$\displaystyle E(t_{1})$
$\displaystyle=\matrixelement{\boldsymbol{\psi}}{\frac{1}{2}I_{6\times
6}}{\boldsymbol{\psi}}=\frac{1}{2}\int_{\Omega}\Big{(}\epsilon(\boldsymbol{r})\boldsymbol{E}^{2}(\boldsymbol{r},t_{1})+\frac{\boldsymbol{B}^{2}(\boldsymbol{r},t_{1})}{\mu_{0}}\Big{)}d\,\boldsymbol{r}$
(65) $\displaystyle E_{elec}(t_{1})$
$\displaystyle=\matrixelement{\boldsymbol{\psi}}{\hat{P}_{\boldsymbol{E}}}{\boldsymbol{\psi}}=\frac{1}{2}\int_{\Omega}\epsilon(\boldsymbol{r})\boldsymbol{E}^{2}(\boldsymbol{r},t_{1})d\,\boldsymbol{r},\quad\hat{P}_{\boldsymbol{E}}=\begin{bmatrix}I_{3\times
3}&0_{3\times 3}\\\ 0_{3\times 3}&0_{3\times 3}\end{bmatrix}$ $\displaystyle
E_{magn}(t_{1})$
$\displaystyle=\matrixelement{\boldsymbol{\psi}}{\hat{P}_{\boldsymbol{B}}}{\boldsymbol{\psi}}=\frac{1}{2}\int_{\Omega}\frac{\boldsymbol{B}^{2}(\boldsymbol{r},t_{1})}{\mu_{0}}d\,\boldsymbol{r},\quad\hat{P}_{\boldsymbol{B}}=\begin{bmatrix}0_{3\times
3}&0_{3\times 3}\\\ 0_{3\times 3}&I_{3\times 3}\end{bmatrix}$

The measurement operators
$\hat{P}_{\boldsymbol{E}}$,$\hat{P}_{\boldsymbol{B}}$ are projection
operators. In order to access the information on the amplitudes of the wave
function $\ket{\boldsymbol{\psi}}(t_{1})$ we have to perform with Amplitude
Estimation [64].

###  3.2 Qubit Lattice Algorithm (QLA) For Uniaxial Dielectric Media

The Qubit Lattice Algorithm (QLA) is an attempt to model perturbatively the
time evolution of a given partial differential equation system at a mesoscopic
level using an appropriately chosen sequence of interleaved non-commuting
collide-stream unitary operators acting on a set of qubits [65].

Following Khan representation [35] of the full Maxwell equations in scalar
dielectric media by the RSW vectors, we devised QLAs for these systems [17,
18, 19, 20, 21]. Our current QLAs are second order accurate schemes, but for
inhomogheneous media (whether scalar or uniaxial) the coupling of one of the
Weyl operators that can represent mass-terms is currently modeled by a
Hermitian, rather than a unitary, operator. All the other operators are
unitary. Current work is underway to determine a fully unitary QLA.

QLA was also bench-marked against exact theoretical solutions to the one-
dimensional nonlinear Schrodinger equation and its generalizations to
inelastic vector solitons [12]. It was then applied to quantum turbulence
simulations in spinor Bose-Einstein condensates and non-Abelian vortex
reconnections [66, 67]. These QLAs were fully unitary, but direct application
to quantum computers awaits the solution of how nonlinear systems can be well
approximated by algorithms that are intrinsically linear.

Interestingly, QLA is massively parallelized on classical supercomputers and
seems to outperform standard CFD algorithms for quantum turbulence.

##  4 Conclusions

Here, for the first time, it has been shown how to construct unitary evolution
dynamics for the continuous time dependent Maxwell equations in passive,
inhomogeneous and non-dispersive media. Our derivation is self-consistent and
generalizes all the previous formulations for the scalar dielectric media in
the optical response region. Incorporating methods from the theory of pseudo-
Hermitian operators that plays an important role in the study of non-Hermitian
quantum mechanics we derive the proper isomorphism between Maxwell equations
and Dirac equation for the macroscopic electromagnetic fields. This
isomorphism is based on the correspondence of energy conservation (classical
system) to probability conservation (quantum system).

In addition, for the simple case of uniaxial medium the generalized RSW
vectors form a suitable change of basis $L\rho$ in the Hilbert space
$\mathcal{H}$ to obtain Hermitian generator of dynamics. This change of basis
is directly connected with the Dyson map $\rho=W^{1/2}$. These considerations
apply directly to bi-axial media.

The strength of our formulation lies on the exploitation of conservative
quantities in the pursuit of the corresponding symmetries (existence of
suitable inner product) that can be used to transform from non-Hermitian to
pseudo-Hermitian dynamics. The intermediate connection of the two
representations provided by the Dyson map, enables us to study the Hermitian
counterpart in the initial space. This kind of procedure helps us to study the
construction of unitary evolution dynamics for more complex electromagnetic
media, such as plasmas, where the dispersion operator in Eq. 21 cannot be
ignored. While it is possible to represent the dynamics of cold linear waves
in plasma within a Hermitian Schrodinger picture [23, 24], such formulations
lack on the evaluation of the induced electromagnetic fields in plasma due to
particle motion. Therefore, the self consistent picture of the electromagnetic
wave propagation in plasma consists of Maxwell equations along with
constitutive relation 21 in temporal domain. This is the set of equations
together with proper initial and boundary conditions (similar to Eqs. 16-19)
that we seek unitary time evolution. This present study on the electromagnetic
wave propagation in tensor non-dispersive medium is the first step in that
direction.

We believe that our new formulation should have direct bearing on QC. While
our algorithms are initial value solutions, they should be directly
implementable on stable qubit systems and yield quantum simulations of
electromagnetic wave propagation in complex media, extending to plasma
physics.

## Acknowledgments

This work has been carried out partially within the framework of the
EUROfusion Consortium. E.K has received funding from the Euratom research and
training programme WPEDU under grant agreement no. 101052200 as well as from
the National Programme for Controlled Thermonuclear Fusion, Hellenic Republic.
K.H is supported by the National Programme for Controlled Thermonuclear
Fusion, Hellenic Republic. The views and opinions expressed herein do not
necessarily reflect those of the European Commission. A.K.R is supported by
the Department of Energy under Grant Nos. DE-SC0021647 and DE-FG02-91ER-54109.
G.V is supported by the Department of Energy under Grant Nos. DE-SC0021651.

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Nordhaus-Guddum type results for the Steiner Gutman index of graphs

Zhao Wang1, Yaping Mao2,3, Kinkar Chandra Das${}^{4,*},$ Yilun Shang5,∗

1College of Science, China Jiliang University,
Hangzhou 310018, China
e-mail<EMAIL_ADDRESS>

2Department of Mathematics, Qinghai Normal University,
Xining, Qinghai 810008, China

3Center for Mathematics and Interdisciplinary Sciences of Qinghai Province,
Xining, Qinghai 810008, China
e-mail<EMAIL_ADDRESS>

4Department of Mathematics, Sungkyunkwan University,
Suwon 16419, Republic of Korea

e-mail<EMAIL_ADDRESS>

4Department of Computer and Information Sciences, Northumbria University,
Newcastle NE1 8ST, UK

e-mail<EMAIL_ADDRESS>
(Received September 16, 2020)

###### Abstract

Building upon the notion of Gutman index $\operatorname{SGut}(G)$, Mao and Das
recently introduced the Steiner Gutman index by incorporating Steiner distance
for a connected graph $G$. The _Steiner Gutman $k$-index_
$\operatorname{SGut}_{k}(G)$ of $G$ is defined by $\operatorname{SGut}_{k}(G)$
$=\sum_{S\subseteq V(G),\ |S|=k}\left(\prod_{v\in
S}deg_{G}(v)\right)d_{G}(S)$, in which $d_{G}(S)$ is the Steiner distance of
$S$ and $deg_{G}(v)$ is the degree of $v$ in $G$. In this paper, we derive new
sharp upper and lower bounds on $\operatorname{SGut}_{k}$, and then
investigate the Nordhaus-Gaddum-type results for the parameter
$\operatorname{SGut}_{k}$. We obtain sharp upper and lower bounds of
$\operatorname{SGut}_{k}(G)+\operatorname{SGut}_{k}(\overline{G})$ and
$\operatorname{SGut}_{k}(G)\cdot\operatorname{SGut}_{k}(\overline{G})$ for a
connected graph $G$ of order $n$, $m$ edges and maximum degree $\Delta$,
minimum degree $\delta$.
Keywords: Distance; Steiner distance; Gutman index; Steiner Gutman $k$-index
AMS subject classification 2010: 05C05; 05C12; 05C35

##  1 Introduction

We consider simple, undirected graphs in this paper. The standard graph-
theoretical terminology and notations not defined here follow [3]. For a graph
$G$, let $V(G)$ and $E(G)$ represent its sets of vertices and edges,
respectively. Let $|E(G)|=m$ be the size of $G$. The complement of $G$ is
conventionally denoted by $\overline{G}$. For a vertex $v\in V(G)$,
$deg_{G}(v)$ is the degree of $v$. The maximum and minimum degrees are,
respectively, denoted by $\Delta$ and $\delta$. Like degrees, distance is a
fundamental concept of graph theory [4]. For two vertices $u,v\in V(G)$ with
connected $G$, the _distance_ $d(u,v)=d_{G}(u,v)$ between these two vertices
is defined as the length of a shortest path connecting them. An excellent
survey paper on this subject can be found in [12].

The above classical graph distance has been extended by Chartrand et al. in
1989 to the Steiner distance, which since then has become an essential concept
of graph theory. Given a graph $G(V,E)$ and a vertex set $S\subseteq V(G)$
containing no less than two vertices, _an $S$-Steiner tree_ (or _an $S$-tree_,
_a Steiner tree connecting $S$_) is defined as a subgraph
$T(V^{\prime},E^{\prime})$ of $G$ which is a subtree satisfying $S\subseteq
V^{\prime}$. If $G$ is connected with order no less than $2$ and $S\subseteq
V$ is nonempty, the _Steiner distance_ $d(S)$ among the vertices of $S$
(sometimes simply put as the distance of $S$) is the minimum size of connected
subgraph whose vertex sets contain the set $S$. Clearly, for a connected
subgraph $H\subseteq G$ with $S\subseteq V(H)$ and $|E(H)|=d(S)$, $H$ is a
tree. When $T$ is subtree of $G$, we have $d(S)=\min\\{|E(T)|\,\,,\,S\subseteq
V(T)\\}$. For $S=\\{u,v\\}$, $d(S)=d(u,v)$ reduces to the classical distance
between the two vertices $u$ and $v$. Another basic observation is that if
$|S|=k$, $d(S)\geq k-1$. For more results regarding varied properties of the
Steiner distance, we refer to the reader to [1, 5, 6, 10, 12, 24].

In [16], Li et al. generalized the concept of Wiener index through
incorporating the Steiner distance. The _Steiner $k$-Wiener index_
$\operatorname{SW}_{k}(G)$ of $G$ is defined by

$\operatorname{SW}_{k}(G)=\sum_{\overset{S\subseteq V(G)}{|S|=k}}d(S)\,.$

For $k=2$, it is easy to see the Steiner Wiener index coincides with the
ordinary Wiener index. The interesting range of the Steiner $k$-Wiener index
$\operatorname{SW}_{k}$ resides in $2\leq k\leq n-1$, and the two trivial
cases give $\operatorname{SW}_{1}(G)=0$ and $\operatorname{SW}_{n}(G)=n-1$.

Gutman [13] studied the Steiner degree distance, which is a generalization of
ordinary degree distance. Formally, the _$k$ -center Steiner degree distance_
$\operatorname{SDD}_{k}(G)$ of $G$ is given as

$\operatorname{SDD}_{k}(G)=\sum_{\overset{S\subseteq
V(G)}{|S|=k}}\left(\sum_{v\in S}deg_{G}(v)\right)d_{G}(S)\,.$

The Gutman index of a connected graph $G$ is defined as

$\operatorname{Gut}(G)=\sum_{u,v\in
V(G)}\,deg_{G}(u)\,deg_{G}(v)\,d_{G}(u,v).$

The Gutman index of graphs attracts attention very recently. For its basic
properties and applications, including various lower and upper bounds, see [7,
8, 11] and the references cited therein. Recently, Mao and Das [18] further
extended the concept of Gutman index by incorporating Steiner distance and
consider the weights as a multiplication of degrees. The _Steiner $k$-Gutman
index_ $\operatorname{SGut}_{k}(G)$ of $G$ is defined by

$\operatorname{SGut}_{k}(G)=\sum_{\overset{S\subseteq
V(G)}{|S|=k}}\left(\prod_{v\in S}deg_{G}(v)\right)d_{G}(S)\,.$

Note that this index is a natural generalization of classical Gutman index. In
particular, for $k=2$, $\operatorname{SGut}_{k}(G)=Gut(G)$. This is the reason
the product of the degrees comes to the definition of Steiner $k$-Gutman
index. The weighting of multiplication of degree or expected degree has also
been extensively explored in, for example, the field of random graphs [19, 20]
and proves to be very prolific. For more results on Steiner Wiener index,
Steiner degree distance, and Steiner Gutman index, we refer to the reader to
[13, 16, 18, 21, 22, 23].

For a given a graph parameter $f(G)$ and a positive integer $n$, the well-
known _Nordhaus-Gaddum problem_ is to determine sharp bounds for: $(1)$
$f(G)+f(\overline{G})$ and $(2)$ $f(G)\cdot f(\overline{G})$ over the class of
connected graph $G$, with order $n$, $m$ edges, maximum degree $\Delta$ and
minimum degree $\delta$ characterizing the extremal graphs. Many Nordhaus-
Gaddum type relations have attracted considerable attention in graph theory.
Comprehensive results regarding this topic can be found in e.g. [2, 17, 15,
27, 28].

In Section 2, we obtain sharp upper and lower bounds on ${\rm SGut}_{k}$ of
graph $G$. In Section 3, we obtain sharp upper and lower bounds of ${\rm
SGut}_{k}(G)+{\rm SGut}_{k}(\overline{G})$ and ${\rm SGut}_{k}(G)\cdot{\rm
SGut}_{k}(\overline{G})$ for a connected graph $G$ in terms of $n$, $m$,
maximum degree $\Delta$ and minimum degree $\delta$.

##  2 Sharp bounds for Steiner Gutman index

In [18], the following results have been obtained:

######  Lemma 2.1.

[18] Let $K_{n}$, $S_{n}$ and $P_{n}$ be the complete graph, star graph and
path graph of order $n$, respectively, and let $k$ be an integer such that
$2\leq k\leq n$. Then

$(1)$ ${\rm SGut}_{k}(K_{n})=\binom{n}{k}(n-1)^{n}(k-1)$;

$(2)$ ${\rm SGut}_{k}(S_{n})=(kn-2k+1)\binom{n-1}{k-1}$;

$(3)$ ${\rm SGut}_{k}(P_{n})=2^{k}(k-1)\binom{n}{k+1}$.

For connected graph $G$ of order $n$ with $m$ edges, the authors in [18]
derived the following upper and lower bounds on ${\rm SGut}_{k}(G)$.

######  Lemma 2.2.

[18] Let $G$ be a connected graph of order $n$ with $m$ edges, and let $k$ be
an integer with $2\leq k\leq n$. Then

$(n-1)\left(\frac{2m}{k}\right)^{k}\binom{n-1}{k-1}^{k}\geq{\rm
SGut}_{k}(G)\geq\left\\{\begin{array}[]{ll}2m(k-1)\binom{n-1}{k-1}&\mbox{\rm
if}~{}\delta\geq 2\\\\[5.69054pt] (k-1)\binom{n}{k}&\mbox{\rm
if}~{}\delta=1.\end{array}\right.$

We now give a lower and upper bounds on ${\rm SGut}_{k}(G)$ in terms of $n$,
$m$, maximum degree $\Delta$ and minimum degree $\delta$:

######  Proposition 2.1.

Let $G$ be a connected graph of order $n\geq 3$ with $m$ edges and maximum
degree $\Delta$, minimum degree $\delta$. Also let $k$ be an integer with
$2\leq k\leq n$. Then

$2m(n-1){n-1\choose k-1}\frac{\Delta^{k-1}}{k}\geq{\rm
SGut}_{k}(G)\geq\left\\{\begin{array}[]{ll}2m(k-1){n-1\choose
k-1}\frac{\delta^{k-1}}{k}&\mbox{\rm if}~{}\delta\geq 2\\\\[11.38109pt]
k{p\choose k}+2^{q}(k-1)\left[{n\choose k}-{p\choose k}\right]&\mbox{\rm
if}~{}\delta=1,\end{array}\right.$

where $p$ is the number of pendant vertices in $G$, and $q=\max\\{k-p,1\\}$.
The equality of upper bound holds if and only if $G$ is a regular graph with
$k=n$. The equality of lower bound holds if and only if $G$ is a regular
$(n-k+1)$-connected graph of order $n$ $(\delta\geq 2)$, or $G\cong P_{n}$ and
$k=n>3$ $(\delta=1)$, or $G\cong P_{3}$ and $k=2$ $(\delta=1)$.

Proof. Upper bound: For any $S\subseteq V(G)$ and $|S|=k$, we have $k-1\leq
d_{G}(S)\leq n-1$, and hence

$(k-1)\sum_{\overset{S\subseteq V(G)}{|S|=k}}\left(\prod_{v\in
S}deg_{G}(v)\right)\leq{\rm SGut}_{k}(G)\leq(n-1)\sum_{\overset{S\subseteq
V(G)}{|S|=k}}\left(\prod_{v\in S}deg_{G}(v)\right)\,.$ (1)

Let

$M=\sum_{\overset{S\subseteq V(G)}{|S|=k}}\left(\prod_{v\in
S}deg_{G}(v)\right)=\sum_{\\{v_{1},v_{2},\ldots,v_{k}\\}\subseteq
V(G)}deg_{G}(v_{1})deg_{G}(v_{2})\cdots deg_{G}(v_{k}).$

and

$N=\sum_{\\{v_{1},v_{2},\ldots,v_{k}\\}\subseteq
V(G)}[deg_{G}(v_{1})+deg_{G}(v_{2})+\cdots+deg_{G}(v_{k})].$

We first prove the upper bound. Without loss of generality, we can assume that
$deg_{G}(v_{1})\leq deg_{G}(v_{2})\leq\ldots\leq deg_{G}(v_{k})$. Since

$\displaystyle deg_{G}(v_{1})deg_{G}(v_{2})\ldots
deg_{G}(v_{k})\leq\Delta^{k-1}deg_{G}(v_{1})$ (2) $\displaystyle\leq$
$\displaystyle\frac{\Delta^{k-1}}{k}(deg_{G}(v_{1})+deg_{G}(v_{2})+\cdots+deg_{G}(v_{k})),$
(3)

it follows that

$\displaystyle M$ $\displaystyle=$
$\displaystyle\sum_{\\{v_{1},v_{2},\ldots,v_{k}\\}\subseteq
V(G)}deg_{G}(v_{1})deg_{G}(v_{2})\ldots deg_{G}(v_{k})$ $\displaystyle\leq$
$\displaystyle\frac{\Delta^{k-1}}{k}\sum_{\\{v_{1},v_{2},\ldots,v_{k}\\}\subseteq
V(G)}[deg_{G}(v_{1})+deg_{G}(v_{2})+\cdots+deg_{G}(v_{k})]$
$\displaystyle\leq$ $\displaystyle\frac{\Delta^{k-1}}{k}N.$

For each $v\in V(G)$, there are ${n-1\choose k-1}$ $k$-subsets in $G$ such
that each of them contains $v$. The contribution of vertex $v$ is exactly
${n-1\choose k-1}deg_{G}(v)$. From the arbitrariness of $v$, we have

$N={n-1\choose k-1}\sum_{v\in V(G)}deg_{G}(v)=2m{n-1\choose k-1},$

and hence

${\rm
SGut}_{k}(G)\leq(n-1)M\leq(n-1)\frac{\Delta^{k-1}}{k}N=2m(n-1){n-1\choose
k-1}\frac{\Delta^{k-1}}{k}.$ (4)

Suppose that the left equality holds. Then all the inequalities in the above
must be equalities. From the equality in (3), one can easily see that $G$ is a
regular graph. From the equality in (4), we have $d(S)=n-1$ for any
$S\subseteq V(G)$, $|S|=k$. Since $G$ is connected, then there exists an
$S\subseteq V(G)$ such that $|d_{G}(S)|=k-1$. If $k\leq n-1$, then one can
easily see that the upper bound is strict as $|d_{G}(S)|=k-1\leq n-2$ for some
$S$. Otherwise, $k=n$. Since $G$ is connected, we have $|d_{G}(S)|=n-1$ for
any $S\subseteq V(G)$. Hence $G$ is a regular graph with $k=n$.

Conversely, one can see easily that the left equality holds for regular graph
with $k=n$.

Lower bound: Without loss of generality, we can assume that
$deg_{G}(v_{1})\leq deg_{G}(v_{2})\leq\ldots\leq deg_{G}(v_{k})$. First we
assume that $\delta\geq 2$. Then

$\displaystyle deg_{G}(v_{1})deg_{G}(v_{2})\cdots
deg_{G}(v_{k})\geq\delta^{k-1}deg_{G}(v_{k})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\geq\frac{\delta^{k-1}}{k}(deg_{G}(v_{1})+deg_{G}(v_{2})+\cdots+deg_{G}(v_{k})),~{}~{}~{}$
(5)

since $deg_{G}(v_{1})\leq deg_{G}(v_{2})\leq\cdots\leq deg_{G}(v_{k})$.
Furthermore, we have

$\displaystyle{\rm SGut}_{k}(G)$ $\displaystyle\geq$
$\displaystyle(k-1)\sum_{\\{v_{1},v_{2},\ldots,v_{k}\\}\subseteq
V(G)}deg_{G}(v_{1})deg_{G}(v_{2})\ldots deg_{G}(v_{k})~{}~{}$ (6)
$\displaystyle\geq$
$\displaystyle(k-1)\frac{\delta^{k-1}}{k}\sum_{\\{v_{1},v_{2},\ldots,v_{k}\\}\subseteq
V(G)}[deg_{G}(v_{1})+deg_{G}(v_{2})+\cdots+deg_{G}(v_{k})]~{}~{}$
$\displaystyle=$ $\displaystyle(k-1)\frac{\delta^{k-1}}{k}N$ $\displaystyle=$
$\displaystyle 2m(k-1){n-1\choose k-1}\frac{\delta^{k-1}}{k}.$

Next we assume that $\delta=1$. If
$deg_{G}(v_{1})=deg_{G}(v_{2})=\cdots=deg_{G}(v_{k})=1$, then $d_{G}(S)\geq k$
and $deg_{G}(v_{1})deg_{G}(v_{2})\ldots deg_{G}(v_{k})=1$. If there exists
some $v_{i}$ such that $deg_{G}(v_{i})\geq 2$, then $d_{G}(S)\geq k-1$ and
$deg_{G}(v_{1})deg_{G}(v_{2})\ldots deg_{G}(v_{k})\geq
2^{\max\\{k-p,1\\}}=2^{q}$, where $1\leq i\leq k$. Therefore, we have

$\displaystyle{\rm SGut}_{k}(G)$ $\displaystyle\geq$ $\displaystyle
k\sum_{\overset{\\{v_{1},v_{2},\ldots,v_{k}\\}\subseteq
V(G),}{deg_{G}(v_{1})=deg_{G}(v_{2})=\cdots=deg_{G}(v_{k})=1}}deg_{G}(v_{1})deg_{G}(v_{2})\ldots
deg_{G}(v_{k})$ (8)
$\displaystyle+(k-1)\sum_{\overset{\\{v_{1},v_{2},\ldots,v_{k}\\}\subseteq
V(G),}{some~{}deg_{G}(v_{i})\geq 2}}deg_{G}(v_{1})deg_{G}(v_{2})\ldots
deg_{G}(v_{k})~{}~{}~{}$ $\displaystyle\geq$ $\displaystyle k{p\choose
k}+2^{q}(k-1)\left[{n\choose k}-{p\choose k}\right].~{}~{}~{}$ (9)

Suppose that the right equality holds. Then all the inequalities in the above
must be equalities. Suppose that $\delta\geq 2$. From the equality in (6),
$d_{G}(S)=k-1$ for any $S\subseteq V(G)$ and $|S|=k$, that is, $G[S]$ is
connected for any $S\subseteq V(G)$ and $|S|=k$, and hence $G$ is
$(n-k+1)$-connected. From the equality in (2), we have
$deg_{G}(v_{1})=deg_{G}(v_{2})=\cdots=deg_{G}(v_{k})$ for any
$S=\\{v_{1},v_{2},\ldots,v_{k}\\}\subseteq V(G)$, and hence $G$ is a regular
graph. So $G$ is a regular $(n-k+1)$-connected graph of order $n$.

Next suppose that $\delta=1$. From the equality in (9), we obtain
$deg_{G}(v_{i})=1$ or $deg_{G}(v_{i})=2$ for any vertex $v_{i}\in V(G)$. Since
$G$ is connected, $G\cong P_{n}$ and $p=2$. If $k\geq 3$, then $q=k-p\geq 1$.
In this case $d_{G}(S)=k-1$ for any $S\subseteq V(G)$ and $|S|=k$. One can
easily see that $G\cong P_{n}$ and $k=n>3$ (otherwise, $d_{G}(S)>k-1$ for some
$S\subseteq V(G)$ as $q=k-p$). Otherwise, $k=p=2$ and hence $q=1$. In this
case $G\cong P_{3}$ and $k=2$.

Conversely, one can see easily that the equality holds on lower bound for
regular $(n-k+1)$-connected graph of order $n$ $(\delta\geq 2)$, or $G\cong
P_{n}$ and $k=n>3$ $(\delta=1)$, or $G\cong P_{3}$ and $k=2$ $(\delta=1)$.

######  Example 2.1.

Let $G\cong K_{n}$ with $k=n$. Then

${\rm SGut}_{k}(G)=(n-1)^{n+1}=2m(n-1){n-1\choose k-1}\frac{\Delta^{k-1}}{k}.$

Let $G\cong K_{n}\backslash sK_{2}$ $(n=2s)$ with $k=3$. Then $G$ is a $n-2$
regular graph of order $n$. Then

${\rm SGut}_{k}(G)=2(n-2)^{3}{n\choose 3}=2m(k-1){n-1\choose
k-1}\frac{\delta^{k-1}}{k}.$

Let $G\cong P_{n}$ with $k=n>3$. Then

${\rm SGut}_{k}(G)=2^{n-2}(n-1)=k{p\choose k}+2^{q}(k-1)\left[{n\choose
k}-{p\choose k}\right]\mbox{ as }p=2.$

Let $G\cong P_{n}$ with $k=2$. Then

${\rm SGut}_{k}(G)=6=k{p\choose k}+2^{q}(k-1)\left[{n\choose k}-{p\choose
k}\right]\mbox{ as }p=2.$

##  3 Nordhaus-Gaddum type results on ${\rm SGut}_{k}(G)$

We now in a position to give the Nordhaus-Gaddum type results on ${\rm
SGut}_{k}(G)$.

######  Theorem 3.1.

Let $G$ be a connected graph of order $n$ with $m$ edges, maximum degree
$\Delta$, minimum degree $\delta$ and a connected $\overline{G}$. Also let $k$
be an integer with $2\leq k\leq n$. Then

$(1)$

${\rm SGut}_{k}(G)+{\rm
SGut}_{k}(\overline{G})\leq(n-1)^{2}\binom{n}{k}s_{1}^{k-1}$

and

${\rm SGut}_{k}(G)\cdot{\rm SGut}_{k}(\overline{G})\leq
2m(n^{2}-n-2m)(n-1)^{2}\binom{n-1}{k-1}^{2}\frac{\Delta^{k-1}\,(n-\delta-1)^{k-1}}{k^{2}},$

where $s_{1}=\max\\{\Delta,\,n-\delta-1\\}$. Moreover, the upper bounds are
sharp.

$(2)$

$\displaystyle{\rm SGut}_{k}(G)+{\rm SGut}_{k}(\overline{G})$
$\displaystyle\geq$
$\displaystyle\left\\{\begin{array}[]{ll}(n-1)(k-1)\binom{n}{k}\,t_{1}^{k-1}&\mbox{\rm
if}~{}\delta\geq 2,\ \Delta\leq n-3\\\\[8.53581pt] 2m(k-1){n-1\choose
k-1}\frac{\delta^{k-1}}{k}+k{n\choose k}&\mbox{\rm if}~{}\delta\geq 2,\
\Delta=n-2\\\\[8.53581pt] k{n\choose k}+[n(n-1)-2m](k-1){n-1\choose
k-1}\frac{(n-\Delta-1)^{k-1}}{k}&\mbox{\rm if}~{}\delta=1,\ \Delta\leq
n-3\\\\[8.53581pt] 2k{n\choose k}&\mbox{\rm if}~{}\delta=1,\
\Delta=n-2,\end{array}\right.$

where $t_{1}=\min\\{\delta,\,n-\Delta-1\\}$.

$(3)$

$\displaystyle{\rm SGut}_{k}(G)\cdot{\rm SGut}_{k}(\overline{G})$
$\displaystyle\geq$
$\displaystyle\left\\{\begin{array}[]{ll}2m(n^{2}-n-2m)(k-1)^{2}\binom{n-1}{k-1}^{2}\,\frac{\delta^{k-1}\,(n-\Delta-1)^{k-1}}{k^{2}}&\mbox{\rm
if}~{}\delta\geq 2,\ \Delta\leq n-3\\\\[8.53581pt] 2m(k-1){n\choose
k}{n-1\choose k-1}\delta^{k-1}&\mbox{\rm if}~{}\delta\geq 2,\
\Delta=n-2\\\\[8.53581pt] [n(n-1)-2m](k-1){n\choose k}{n-1\choose
k-1}(n-\Delta-1)^{k-1}&\mbox{\rm if}~{}\delta=1,\ \Delta\leq
n-3\\\\[8.53581pt] k^{2}{n\choose k}^{2}&\mbox{\rm if}~{}\delta=1,\
\Delta=n-2.\end{array}\right.$

Proof. $(1)$ From Proposition 2.1, we have

${\rm SGut}_{k}(G)\leq 2m(n-1){n-1\choose k-1}\frac{\Delta^{k-1}}{k}$

and

${\rm SGut}_{k}(\overline{G})\leq[n(n-1)-2m](n-1){n-1\choose
k-1}\frac{(n-\delta-1)^{k-1}}{k},$

and hence

${\rm SGut}_{k}(G)+{\rm
SGut}_{k}(\overline{G})\leq(n-1)^{2}\binom{n}{k}\,s_{1}^{k-1}$

and

${\rm SGut}_{k}(G)\cdot{\rm SGut}_{k}(\overline{G})\leq
2m(n^{2}-n-2m)(n-1)^{2}\binom{n-1}{k-1}^{2}\frac{\Delta^{k-1}(n-\delta-1)^{k-1}}{k^{2}}.$

$(2)$ From Proposition 2.1, if $\delta\geq 2$ and $\Delta\leq n-3$, then

$\displaystyle{\rm SGut}_{k}(G)+{\rm SGut}_{k}(\overline{G})$
$\displaystyle\geq$ $\displaystyle 2m(k-1){n-1\choose
k-1}\frac{\delta^{k-1}}{k}+[n(n-1)-2m](k-1){n-1\choose
k-1}\frac{(n-\Delta-1)^{k-1}}{k}$ $\displaystyle\geq$
$\displaystyle(n-1)(k-1)\binom{n}{k}\,t^{k-1}_{1}.$

If $\delta(G)\geq 2$ and $\Delta=n-2$, then

$\displaystyle{\rm SGut}_{k}(G)+{\rm SGut}_{k}(\overline{G})$
$\displaystyle\geq$ $\displaystyle 2m(k-1){n-1\choose
k-1}\frac{\delta^{k-1}}{k}+k{p^{\prime}\choose
k}+2^{q^{\prime}}(k-1)\left[{n\choose k}-{p^{\prime}\choose k}\right]$
$\displaystyle\geq$ $\displaystyle 2m(k-1){n-1\choose
k-1}\frac{\delta^{k-1}}{k}+k{p^{\prime}\choose k}+2(k-1)\left[{n\choose
k}-{p^{\prime}\choose k}\right]$ $\displaystyle\geq$ $\displaystyle
2m(k-1){n-1\choose k-1}\frac{\delta^{k-1}}{k}+k{p^{\prime}\choose
k}+k\left[{n\choose k}-{p^{\prime}\choose k}\right]$ $\displaystyle=$
$\displaystyle 2m(k-1){n-1\choose k-1}\frac{\delta^{k-1}}{k}+k{n\choose k},$

where $p^{\prime}$ is the number of pendant vertices in $G$, and
$q^{\prime}=\max\\{k-p^{\prime},1\\}$.

If $\delta=1$ and $\Delta\leq n-3$, then

$\displaystyle{\rm SGut}_{k}(G)+{\rm SGut}_{k}(\overline{G})$
$\displaystyle\geq$ $\displaystyle k{p\choose k}+2^{q}(k-1)\left[{n\choose
k}-{p\choose k}\right]+[n(n-1)-2m](k-1){n-1\choose
k-1}\frac{(n-\Delta-1)^{k-1}}{k}$ $\displaystyle\geq$ $\displaystyle
k{n\choose k}+[n(n-1)-2m](k-1){n-1\choose k-1}\frac{(n-\Delta-1)^{k-1}}{k},$

where $p$ is the number of pendant vertices in $\overline{G}$, and
$q=\max\\{k-p,1\\}$.

If $\delta=1$ and $\Delta=n-2$, then

$\displaystyle{\rm SGut}_{k}(G)+{\rm SGut}_{k}(\overline{G})$
$\displaystyle\geq$ $\displaystyle k{p\choose k}+2^{q}(k-1)\left[{n\choose
k}-{p\choose k}\right]+k{p^{\prime}\choose
k}+2^{q^{\prime}}(k-1)\left[{n\choose k}-{p^{\prime}\choose k}\right]$
$\displaystyle\geq$ $\displaystyle k{n\choose k}+k{n\choose k}\geq 2k{n\choose
k},$

where $p,p^{\prime}$ are the number of pendant vertices in $G,\overline{G}$,
respectively, and $q=\max\\{k-p,1\\}$, $q^{\prime}=\max\\{k-p^{\prime},1\\}$.

From the above argument, we have

$\displaystyle{\rm SGut}_{k}(G)+{\rm SGut}_{k}(\overline{G})$
$\displaystyle\geq$
$\displaystyle\left\\{\begin{array}[]{ll}(n-1)(k-1)\binom{n}{k}\,t^{k-1}_{1}&\mbox{\rm
if}~{}\delta\geq 2,\ \Delta\leq n-3\\\\[8.53581pt] 2m(k-1){n-1\choose
k-1}\frac{\delta^{k-1}}{k}+k{n\choose k}&\mbox{\rm if}~{}\delta\geq 2,\
\Delta=n-2\\\\[8.53581pt] k{n\choose k}+[n(n-1)-2m](k-1){n-1\choose
k-1}\,\frac{(n-\Delta-1)^{k-1}}{k}&\mbox{\rm if}~{}\delta=1,\ \Delta\leq
n-3\\\\[8.53581pt] 2k{n\choose k}&\mbox{\rm if}~{}\delta=1,\
\Delta=n-2.\end{array}\right.$

For $(3)$, from Proposition 2.1, if $\delta\geq 2$ and $\Delta\leq n-3$, then

${\rm SGut}_{k}(G)\cdot{\rm SGut}_{k}(\overline{G})\geq
2m(n^{2}-n-2m)(k-1)^{2}\binom{n-1}{k-1}^{2}\,\frac{\delta^{k-1}\,(n-\Delta-1)^{k-1}}{k^{2}}.$

If $\delta\geq 2$ and $\Delta=n-2$, then

$\displaystyle{\rm SGut}_{k}(G)\cdot{\rm SGut}_{k}(\overline{G})$
$\displaystyle\geq$ $\displaystyle\left[2m(k-1){n-1\choose
k-1}\frac{\delta^{k-1}}{k}\right]\left[k{p^{\prime}\choose
k}+2^{q^{\prime}}(k-1)\left[{n\choose k}-{p^{\prime}\choose k}\right]\right]$
$\displaystyle\geq$ $\displaystyle 2m(k-1){n\choose k}{n-1\choose
k-1}\delta^{k-1},$

where $p^{\prime}$ is the number of pendant vertices in $\overline{G}$, and
$q^{\prime}=\max\\{k-p^{\prime},1\\}$.

If $\delta=1$ and $\Delta\leq n-3$, then

$\displaystyle{\rm SGut}_{k}(G)\cdot{\rm SGut}_{k}(\overline{G})$
$\displaystyle\geq$ $\displaystyle\left[[n(n-1)-2m](k-1){n-1\choose
k-1}\frac{(n-\Delta-1)^{k-1}}{k}\right]\left[k{p\choose
k}+2^{q}(k-1)\left[{n\choose k}-{p\choose k}\right]\right]$
$\displaystyle\geq$ $\displaystyle[n(n-1)-2m](k-1){n\choose k}{n-1\choose
k-1}\,(n-\Delta-1)^{k-1},$

where $p$ is the number of pendant vertices in $G$, and $q=\max\\{k-p,1\\}$.

If $\delta(G)=1$ and $\Delta=n-2$, then

$\displaystyle{\rm SGut}_{k}(G)\cdot{\rm SGut}_{k}(\overline{G})$
$\displaystyle\geq$ $\displaystyle\left[k{p\choose
k}+2^{q}(k-1)\left[{n\choose k}-{p\choose
k}\right]\right]\left[k{p^{\prime}\choose
k}+2^{q^{\prime}}(k-1)\left[{n\choose k}-{p^{\prime}\choose k}\right]\right]$
$\displaystyle\geq$ $\displaystyle k^{2}{n\choose k}^{2},$

where $p,\,p^{\prime}$ are the number of pendant vertices in $G$ and
$\overline{G}$, respectively, and $q=\max\\{k-p,1\\}$,
$q^{\prime}=\max\\{k-p^{\prime},1\\}$.

From the above argument, we have

$\displaystyle{\rm SGut}_{k}(G)\cdot{\rm SGut}_{k}(\overline{G})$
$\displaystyle\geq$
$\displaystyle\left\\{\begin{array}[]{ll}2m(n^{2}-n-2m)(k-1)^{2}\binom{n-1}{k-1}^{2}\,\frac{\delta^{k-1}\,(n-\Delta-1)^{k-1}}{k^{2}}&\mbox{\rm
if}~{}\delta(G)\geq 2,\ \Delta\leq n-3\\\\[8.53581pt] 2m(k-1){n\choose
k}{n-1\choose k-1}\delta^{k-1}&\mbox{\rm if}~{}\delta(G)\geq 2,\
\Delta=n-2\\\\[8.53581pt] [n(n-1)-2m](k-1){n\choose k}{n-1\choose
k-1}(n-\Delta-1)^{k-1}&\mbox{\rm if}~{}\delta(G)=1,\ \Delta\leq
n-3\\\\[8.53581pt] k^{2}{n\choose k}^{2}&\mbox{\rm if}~{}\delta(G)=1,\
\Delta=n-2.\end{array}\right.$

To show the sharpness of the upper bound and the lower bound for
$\delta(G)\geq 2,\,\Delta\leq n-3$, we let $G$ and $\overline{G}$ be two
$\frac{n-1}{2}$-regular graphs of order $n$, where $n$ is odd. If $k=n$, then
${\rm SGut}_{k}(G)=(n-1)(\frac{n-1}{2})^{n}$, ${\rm
SGut}_{k}(\overline{G})=(n-1)(\frac{n-1}{2})^{n}$,
$s_{1}=\max\\{\Delta,\,n-\delta-1\\}=\frac{n-1}{2}$,
$\Delta\,(n-\delta-1)=(\frac{n-1}{2})^{2}$,
$t_{1}=\min\\{\delta,\,n-\Delta-1\\}=\frac{n-1}{2}$, and
$\delta\,(n-\Delta-1)=(\frac{n-1}{2})^{2}$. Furthermore, we have ${\rm
SGut}_{k}(G)+{\rm
SGut}_{k}(\overline{G})=2(n-1)(\frac{n-1}{2})^{n}=(n-1)^{2}\binom{n}{k}s_{1}^{k-1}$,
${\rm SGut}_{k}(G)\cdot{\rm
SGut}_{k}(\overline{G})=(n-1)^{2}(\frac{n-1}{2})^{2n}=2m(n^{2}-n-2m)(n-1)^{2}\binom{n-1}{k-1}^{2}\,\frac{\Delta^{k-1}\,(n-\delta-1)^{k-1}}{k^{2}}$,
${\rm SGut}_{k}(G)+{\rm
SGut}_{k}(\overline{G})=2(n-1)(\frac{n-1}{2})^{n}=(n-1)(k-1)\binom{n}{k}t_{1}^{k-1}$
and ${\rm SGut}_{k}(G)\cdot{\rm
SGut}_{k}(\overline{G})=(n-1)^{2}(\frac{n-1}{2})^{2n}=2m(n^{2}-n-2m)(k-1)^{2}\binom{n-1}{k-1}^{2}\,\frac{\delta^{k-1}\,(n-\Delta-1)^{k-1}}{k^{2}}$.

The following corollary is immediate from above theorem.

######  Corollary 3.1.

Let $G$ be a connected graph of order $n\geq 4$ with maximum degree $\Delta$
and minimum degree $\delta$. Then

$(1)$

$\displaystyle(n-1)^{2}\binom{n}{k}s_{1}^{k-1}\geq{\rm SGut}_{k}(G)+{\rm
SGut}_{k}(\overline{G})$ $\displaystyle\geq$
$\displaystyle\left\\{\begin{array}[]{ll}(n-1)(k-1)\binom{n}{k}\,t_{1}^{k-1}&\mbox{\rm
if}~{}\delta\geq 2,\ \Delta\leq n-3\\\\[8.53581pt] n(k-1){n-1\choose
k-1}\frac{\delta^{k}}{k}+k{n\choose k}&\mbox{\rm if}~{}\delta\geq 2,\
\Delta=n-2\\\\[8.53581pt] k{n\choose k}+n(k-1){n-1\choose
k-1}\frac{(n-\Delta-1)^{k}}{k}&\mbox{\rm if}~{}\delta=1,\ \Delta\leq
n-3\\\\[8.53581pt] 2k{n\choose k}&\mbox{\rm if}~{}\delta=1,\
\Delta=n-2,\end{array}\right.$

where $s_{1}=\min\\{\Delta,\,n-\delta-1\\}$,
$t_{1}=\min\\{\delta,\,n-\Delta-1\\}$;

$(2)$

$\displaystyle
n^{2}\binom{n-1}{k-1}^{2}\,\frac{\Delta^{k-1}\,(n-\delta-1)^{k-1}\,(n-1)^{4}}{4k^{2}}\geq{\rm
SGut}_{k}(G)\cdot{\rm SGut}_{k}(\overline{G})$
$\displaystyle\geq\left\\{\begin{array}[]{ll}n^{2}(k-1)^{2}\binom{n-1}{k-1}^{2}\,\frac{\delta^{k}\,(n-\Delta-1)^{k}}{k^{2}}&\mbox{\rm
if}~{}\delta\geq 2,\ \Delta\leq n-3\\\\[8.53581pt] n(k-1){n\choose
k}{n-1\choose k-1}\delta^{k}&\mbox{\rm if}~{}\delta\geq 2,\
\Delta=n-2\\\\[8.53581pt] n(k-1){n\choose k}{n-1\choose
k-1}\,(n-\Delta-1)^{k}&\mbox{\rm if}~{}\delta=1,\ \Delta\leq
n-3\\\\[8.53581pt] k^{2}{n\choose k}^{2}&\mbox{\rm if}~{}\delta=1,\
\Delta=n-2.\end{array}\right.$

The following is the famous inequality by Pólya and Szegö:

######  Lemma 3.1.

(Pólya-Szegö inequality) [25] Let $(a_{1},\,a_{2},\ldots,\,a_{r})$ and
$(b_{1},\,b_{2},\ldots,\,b_{r})$ be two positive $r$-tuples such that there
exist positive numbers $M_{1}$, $m_{1}$, $M_{2}$, $m_{2}$ satisfying:

$0<m_{1}\leq a_{i}\leq M_{1},~{}0<m_{2}\leq b_{i}\leq M_{2},~{}1\leq i\leq r.$

Then

$\frac{\sum\limits^{r}_{i=1}\,a^{2}_{i}\,\sum\limits^{r}_{i=1}\,b^{2}_{i}}{\left(\sum\limits^{r}_{i=1}\,a_{i}\,b_{i}\right)^{2}}\leq\frac{1}{4}\,\left(\sqrt{\frac{M_{1}\,M_{2}}{m_{1}\,m_{2}}}+\sqrt{\frac{m_{1}\,m_{2}}{M_{1}\,M_{2}}}\right)^{2}.$
(16)

We now give another lower and upper bounds on ${\rm SGut}_{k}(G)\cdot{\rm
SGut}_{k}(\overline{G})$ in terms of $n$, $\Delta$ and $\delta$.

######  Theorem 3.2.

Let $G$ be a connected graph of order $n$ with maximum degree $\Delta$,
minimum degree $\delta$ and a connected $\overline{G}$. Also let $k$ be an
integer with $2\leq k\leq n$. Then

${\rm SGut}_{k}(G)\cdot{\rm
SGut}_{k}(\overline{G})\geq\left\\{\begin{array}[]{ll}(k-1)^{2}\,\delta^{k}\,(n-\delta-1)^{k}\,\binom{n}{k}^{2}&\mbox{\rm
if}~{}\Delta+\delta\leq n-1,\\\\[8.53581pt]
(k-1)^{2}\,\Delta^{k}\,(n-\Delta-1)^{k}\,\binom{n}{k}^{2}&\mbox{\rm
if}~{}\Delta+\delta\geq n-1\end{array}\right.$ (17)

with equality holding if and only if $G$ is a regular graph with
$d_{G}(S)=d_{\overline{G}}(S)=k-1$ for any $S\subseteq V(G)$, $|S|=k$, and

${\rm SGut}_{k}(G)\cdot{\rm
SGut}_{k}(\overline{G})\leq\frac{(n-1)^{2k+2}}{2^{2k+2}}\,\binom{n}{k}^{2}\,\left[\left(\frac{\Delta\,(n-\delta-1)}{\delta\,(n-\Delta-1)}\right)^{k}+\left(\frac{\delta\,(n-\Delta-1)}{\Delta\,(n-\delta-1)}\right)^{k}+2\right],$

Moreover, the equality holds if and only if $G$ is a
$\left(\frac{n-1}{2}\right)$-regular graph with $k=n$, $n$ is odd.

Proof. Lower bound: By Cauchy-Schwarz inequality with (1), we have

$\displaystyle{\rm SGut}_{k}(G)\cdot{\rm SGut}_{k}(\overline{G})$
$\displaystyle\geq$ $\displaystyle(k-1)^{2}\,\sum_{\overset{S\subseteq
V(G)}{|S|=k}}\left(\prod_{v\in S}deg_{G}(v)\right)\,\sum_{\overset{S\subseteq
V(\overline{G})}{|S|=k}}\left(\prod_{v\in S}deg_{\overline{G}}(v)\right)$ (18)
$\displaystyle\geq$ $\displaystyle(k-1)^{2}\,\left(\sum_{\overset{S\subseteq
V(G)}{|S|=k}}\left(\prod_{v\in S}deg_{G}(v)\,\prod_{v\in
S}deg_{\overline{G}}(v)\right)^{1/2}\right)^{2}$ $\displaystyle\geq$
$\displaystyle(k-1)^{2}\,\left(\sum_{\overset{S\subseteq
V(G)}{|S|=k}}\left(\prod_{v\in
S}deg_{G}(v)\,(n-1-deg_{G}(v))\right)^{1/2}\right)^{2}.$

Since $\delta\leq deg_{G}(v)\leq\Delta$, one can easily see that

$\displaystyle deg_{G}(v)\,(n-1-deg_{G}(v))$ $\displaystyle\geq$
$\displaystyle\left\\{\begin{array}[]{ll}\delta\,(n-\delta-1)&\mbox{\rm
if}~{}\Delta+\delta\leq n-1,\\\\[8.53581pt] \Delta\,(n-\Delta-1)&\mbox{\rm
if}~{}\Delta+\delta\geq n-1.\end{array}\right.$ (22)

From the above results, we have

$\displaystyle{\rm SGut}_{k}(G)\cdot{\rm SGut}_{k}(\overline{G})$
$\displaystyle\geq$
$\displaystyle\left\\{\begin{array}[]{ll}(k-1)^{2}\,\delta^{k}\,(n-\delta-1)^{k}\,\binom{n}{k}^{2}&\mbox{\rm
if}~{}\Delta+\delta\leq n-1,\\\\[8.53581pt]
(k-1)^{2}\,\Delta^{k}\,(n-\Delta-1)^{k}\,\binom{n}{k}^{2}&\mbox{\rm
if}~{}\Delta+\delta\geq n-1.\end{array}\right.$

The equality holds in (18) if and only if $d_{G}(S)=d_{\overline{G}}(S)=k-1$
for any $S\subseteq V(G)$ with $|S|=k$. By Cauchy-Schwarz inequality, the
equality holds in (3) if and only if

$\frac{\prod_{v\in S_{1}}deg_{G}(v)}{\prod_{v\in
S_{1}}deg_{\overline{G}}(v)}=\frac{\prod_{v\in S_{2}}deg_{G}(v)}{\prod_{v\in
S_{2}}deg_{\overline{G}}(v)}~{}\mbox{ for any }~{}S_{1},\,S_{2}\in
V(G)~{}\mbox{ with }|S_{1}|=|S_{2}|=k,$

that is, if and only if $deg_{G}(u)=deg_{G}(v)$ for any $u,\,v\in V(G)$, that
is, if and only if $G$ is a regular graph. Hence the equality holds in (17) if
and only if $G$ is a regular graph with $d_{G}(S)=d_{\overline{G}}(S)=k-1$ for
any $S\subseteq V(G)$, $|S|=k$.

Upper bound: Let $\overline{\Delta}$ and $\overline{\delta}$ be the maximum
degree and the minimum degree of graph $\overline{G}$, respectively. Then
$\overline{\Delta}=n-\delta-1$ and $\overline{\delta}=n-\Delta-1$. By (1) and
(16), we have

$\displaystyle{\rm SGut}_{k}(G)\cdot{\rm SGut}_{k}(\overline{G})$
$\displaystyle\leq$ $\displaystyle(n-1)^{2}\,\sum_{\overset{S\subseteq
V(G)}{|S|=k}}\left(\prod_{v\in S}deg_{G}(v)\right)\,\sum_{\overset{S\subseteq
V(\overline{G})}{|S|=k}}\left(\prod_{v\in S}deg_{\overline{G}}(v)\right)$
$\displaystyle\leq$ $\displaystyle(n-1)^{2}\,\left(\sum_{\overset{S\subseteq
V(G)}{|S|=k}}\left(\prod_{v\in S}deg_{G}(v)\,\prod_{v\in
S}deg_{\overline{G}}(v)\right)^{1/2}\right)^{2}\,\frac{1}{4}\,\left(\left(\frac{\Delta\,\overline{\Delta}}{\delta\,\overline{\delta}}\right)^{k/2}+\left(\frac{\delta\,\overline{\delta}}{\Delta\,\overline{\Delta}}\right)^{k/2}\right)^{2}$
$\displaystyle\leq$
$\displaystyle\frac{(n-1)^{2}}{4}\,\left(\sum_{\overset{S\subseteq
V(G)}{|S|=k}}\left(\prod_{v\in
S}deg_{G}(v)\,(n-1-deg_{G}(v))\right)^{1/2}\right)^{2}\,\left(\left(\frac{\Delta\,\overline{\Delta}}{\delta\,\overline{\delta}}\right)^{k/2}+\left(\frac{\delta\,\overline{\delta}}{\Delta\,\overline{\Delta}}\right)^{k/2}\right)^{2}.$

One can easily see that

$deg_{G}(v)\,(n-1-deg_{G}(v))\leq\frac{(n-1)^{2}}{4}~{}\mbox{ for any }v\in
V(G).$

Using this result in the above with $\overline{\Delta}=n-\delta-1$ and
$\overline{\delta}=n-\Delta-1$, we get

${\rm SGut}_{k}(G)\cdot{\rm
SGut}_{k}(\overline{G})\leq\frac{(n-1)^{2k+2}}{2^{2k+2}}\,\binom{n}{k}^{2}\,\left[\left(\frac{\Delta\,(n-\delta-1)}{\delta\,(n-\Delta-1)}\right)^{k}+\left(\frac{\delta\,(n-\Delta-1)}{\Delta\,(n-\delta-1)}\right)^{k}+2\right].$

Moreover, the above equality holds if and only if $G$ is a
$\left(\frac{n-1}{2}\right)$-regular graph with $k=n$, $n$ is odd (very
similar proof of the Proposition 2.1).

######  Example 3.1.

Let $G\cong C_{n}$ with $k=n$. Then $\delta=2$ and hence

${\rm SGut}_{k}(G)\cdot{\rm
SGut}_{k}(\overline{G})=(n-1)^{2}(n-3)^{n}\,2^{n}=(k-1)^{2}\,\delta^{k}\,(n-\delta-1)^{k}\,\binom{n}{k}^{2}.$

Let $G$ be a $\left(\frac{n-1}{2}\right)$-regular graph of order $n$ with
$k=n$ and odd $n$. Then $\Delta=\delta=\frac{n-1}{2}$ and hence

$\displaystyle{\rm SGut}_{k}(G)\cdot{\rm SGut}_{k}(\overline{G})$
$\displaystyle=\frac{(n-1)^{2n+2}}{2^{2n}}$
$\displaystyle=\frac{(n-1)^{2k+2}}{2^{2k+2}}\,\binom{n}{k}^{2}\,\left[\left(\frac{\Delta\,(n-\delta-1)}{\delta\,(n-\Delta-1)}\right)^{k}+\left(\frac{\delta\,(n-\Delta-1)}{\Delta\,(n-\delta-1)}\right)^{k}+2\right].$

We now give another lower and upper bounds on ${\rm SGut}_{k}(G)+{\rm
SGut}_{k}(\overline{G})$ in terms of $n$, $\Delta$ and $\delta$.

######  Theorem 3.3.

Let $G$ be a connected graph of order $n$ with maximum degree $\Delta$,
minimum degree $\delta$ and a connected $\overline{G}$. Also let $k$ be an
integer with $2\leq k\leq n$. Then

${\rm SGut}_{k}(G)+{\rm
SGut}_{k}(\overline{G})\geq\left\\{\begin{array}[]{ll}2\,(k-1)\,\delta^{k/2}\,(n-\delta-1)^{k/2}\,\binom{n}{k}&\mbox{\rm
if}~{}\Delta+\delta\leq n-1,\\\\[14.22636pt]
2\,(k-1)\,\Delta^{k/2}\,(n-\Delta-1)^{k/2}\,\binom{n}{k}&\mbox{\rm
if}~{}\Delta+\delta\geq n-1\end{array}\right.$ (24)

with equality holding if and only if $G$ is a
$\left(\frac{n-1}{2}\right)$-regular graph with odd $n$ and
$d_{G}(S)=d_{\overline{G}}(S)=k-1$ for any $S\subseteq V(G)$, $|S|=k$, and

${\rm SGut}_{k}(G)+{\rm
SGut}_{k}(\overline{G})\leq(n-1)\,\Big{[}\Delta^{k}+(n-\delta-1)^{k}\Big{]}\,\binom{n}{k}$
(25)

with equality holding if and only if $G$ is a regular graph with $k=n$.

Proof. For any two real numbers $a,\,b$, we have $(a-b)^{2}\geq 0$, that is,
$a^{2}+b^{2}\geq 2\,a\,b$ with equality holding if and only if $a=b$.
Therefore we have

$\displaystyle\prod_{v\in S}deg_{G}(v)+\prod_{v\in S}deg_{\overline{G}}(v)$
$\displaystyle\geq$ $\displaystyle 2\,\left(\prod_{v\in
S}deg_{G}(v)\,\prod_{v\in S}deg_{\overline{G}}(v)\right)^{1/2}$
$\displaystyle=$ $\displaystyle 2\,\left(\prod_{v\in
S}deg_{G}(v)\,deg_{\overline{G}}(v)\right)^{1/2}$ $\displaystyle=$
$\displaystyle 2\,\left(\prod_{v\in
S}deg_{G}(v)\,(n-deg_{G}(v)-1)\right)^{1/2}\,.$

From the above result with (22), we get

$\displaystyle\prod_{v\in S}deg_{G}(v)+\prod_{v\in S}deg_{\overline{G}}(v)$
$\displaystyle\geq$
$\displaystyle\left\\{\begin{array}[]{ll}2\,\delta^{k/2}\,(n-\delta-1)^{k/2}&\mbox{\rm
if}~{}\Delta+\delta\leq n-1,\\\\[8.53581pt]
2\,\Delta^{k/2}\,(n-\Delta-1)^{k/2}&\mbox{\rm if}~{}\Delta+\delta\geq
n-1.\end{array}\right.$

Now,

$\displaystyle{\rm SGut}_{k}(G)+{\rm SGut}_{k}(\overline{G})$ $\displaystyle=$
$\displaystyle\sum_{\overset{S\subseteq V(G)}{|S|=k}}\,\left[\left(\prod_{v\in
S}deg_{G}(v)\right)d_{G}(S)+\left(\prod_{v\in
S}deg_{\overline{G}}(v)\right)d_{\overline{G}}(S)\right]$ $\displaystyle\geq$
$\displaystyle(k-1)\,\sum_{\overset{S\subseteq
V(G)}{|S|=k}}\,\left[\prod_{v\in S}deg_{G}(v)+\prod_{v\in
S}deg_{\overline{G}}(v)\right]$ $\displaystyle\geq$
$\displaystyle\left\\{\begin{array}[]{ll}2\,(k-1)\,\delta^{k/2}\,(n-\delta-1)^{k/2}\,\binom{n}{k}&\mbox{\rm
if}~{}\Delta+\delta\leq n-1,\\\\[14.22636pt]
2\,(k-1)\,\Delta^{k/2}\,(n-\Delta-1)^{k/2}\,\binom{n}{k}&\mbox{\rm
if}~{}\Delta+\delta\geq n-1.\end{array}\right.$

From the above, one can easily see that the equality holds in (24) if and only
if $G$ is a $\left(\frac{n-1}{2}\right)$-regular graph with odd $n$ and
$d_{G}(S)=d_{\overline{G}}(S)=k-1$ for any $S\subseteq V(G)$, $|S|=k$.

Upper bound: By arithmetic-geometric mean inequality, we have

$\displaystyle{\rm SGut}_{k}(G)+{\rm SGut}_{k}(\overline{G})$ $\displaystyle=$
$\displaystyle\sum_{\overset{S\subseteq V(G)}{|S|=k}}\,\left[\left(\prod_{v\in
S}deg_{G}(v)\right)d_{G}(S)+\left(\prod_{v\in
S}deg_{\overline{G}}(v)\right)d_{\overline{G}}(S)\right]$ $\displaystyle\leq$
$\displaystyle(n-1)\,\sum_{\overset{S\subseteq
V(G)}{|S|=k}}\,\left[\prod_{v\in S}deg_{G}(v)+\prod_{v\in
S}deg_{\overline{G}}(v)\right]$ $\displaystyle\leq$
$\displaystyle(n-1)\,\sum_{\overset{S\subseteq
V(G)}{|S|=k}}\,\left[\left(\frac{\sum\limits_{v\in
S}\,deg_{G}(v)}{k}\right)^{k}+\left(\frac{\sum\limits_{v\in
S}\,deg_{\overline{G}}(v)}{k}\right)^{k}\right]$ $\displaystyle=$
$\displaystyle\frac{(n-1)}{k^{k}}\,\sum_{\overset{S\subseteq
V(G)}{|S|=k}}\,\left[\left(\sum\limits_{v\in
S}\,deg_{G}(v)\right)^{k}+\left(\sum\limits_{v\in
S}\,(n-deg_{G}(v)-1)\right)^{k}\right]$ $\displaystyle=$
$\displaystyle\frac{(n-1)}{k^{k}}\,\sum_{\overset{S\subseteq
V(G)}{|S|=k}}\,\left[\left(\sum\limits_{v\in
S}\,deg_{G}(v)\right)^{k}+\left(k\,(n-1)-\sum\limits_{v\in
S}\,deg_{G}(v)\right)^{k}\right]$ $\displaystyle\leq$
$\displaystyle\frac{(n-1)}{k^{k}}\,\sum_{\overset{S\subseteq
V(G)}{|S|=k}}\,\left[(k\,\Delta)^{k}+\left(k\,(n-1)-k\,\delta\right)^{k}\right]$
$\displaystyle=$
$\displaystyle(n-1)\,\Big{[}\Delta^{k}+(n-\delta-1)^{k}\Big{]}\,\binom{n}{k}.$

From the above, one can easily see that the equality holds in (25) if and only
if $G$ is a regular graph with $k=n$ (very similar proof of the Proposition
2.1).

######  Example 3.2.

Let $G$ be a $\left(\frac{n-1}{2}\right)$-regular graph with odd $n$ and
$k=n$. Then $\delta=\frac{n-1}{2}$ and hence

${\rm SGut}_{k}(G)+{\rm
SGut}_{k}(\overline{G})=\frac{(n-1)^{n+1}}{2^{n-1}}=2\,(k-1)\,\delta^{k/2}\,(n-\delta-1)^{k/2}\,\binom{n}{k}$

Let $G\cong C_{n}$ with $k=n$. Then $\Delta=\delta=2$,
$\overline{\Delta}=\overline{\delta}=2$ and hence

${\rm SGut}_{k}(G)+{\rm
SGut}_{k}(\overline{G})=(n-1)\,\Big{[}2^{n}+(n-3)^{n}\Big{]}=(n-1)\,\Big{[}\Delta^{k}+(n-\delta-1)^{k}\Big{]}\,\binom{n}{k}.$

Acknowledgment. The authors are much grateful to three anonymous referees for
their valuable comments on our paper, which have considerably improved the
presentation of this paper. Supported by the National Science Foundation of
China (Nos. 12061059, 11601254, 11661068, 11551001, 11161037, and 11461054),
and UoA Flexible Fund from Northumbria University (201920A1001).

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