# Log-linear Dynamic Inversion Control with Provable Safety Guarantees in Lie
Groups

Li-Yu Lin , James Goppert , and Inseok Hwang This work is supported in part by
NASA University Leadership Initiative Project on Secure and Safe Assured
Autonomy (S2A2) under Grant 80NSSC20M0161.L. Lin, J. Goppert, and I. Hwang are
with the School of Aeronautics and Astronautics, Purdue University, West
Lafayette, IN 47906, USA (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>ihwang@purdue.edu).

###### Abstract

In this paper, we use the derivative of the exponential map to derive the
exact evolution of the logarithm of the tracking error for mixed-invariant
systems, a class of systems capable of describing rigid body tracking problems
in Lie groups. Additionally, we design a log-linear dynamic inversion-based
control law to remove the nonlinearities due to spatial curvature and enhance
the robustness of the controller. We apply Linear Matrix Inequalities (LMIs)
to bound the tracking error given a bounded disturbance amplified by the
distortion matrix and leverage the tracking error bound to create flow pipes.
To demonstrate the usefulness of our method, we show its application with
Urban Air Mobility (UAM) scenarios using a simplified kinematic aircraft model
and polynomial-based path planning methods.

###### Index Terms:

Lie Group Theory, Mixed-Invariant Systems, Log-Linearization, Dynamic
Inversion Control, Safety Verification

##  I Introduction

Ensuring the safety of an Unmanned Aerial System (UAS) in urban environments
is a challenging task as a UAS operates in a dynamic environment subject to
unknown uncertainties, such as wind, etc. [1]. One of the approaches to ensure
the safety of a dynamical system is to construct an invariant set of system
trajectories in the presence of unknown disturbances. An invariant set
represents the set of trajectories in which the system remains if it starts
within or enters the set [2, 3, 4]. This notion of set invariance provides a
powerful tool to analyze the worst-case impact of the unknown disturbance on
the system.

For constructing invariant sets, the existing methods in the literature either
use a grid-based [5, 6, 7] or a Lyapunov function-based approach [8, 9]. One
of the recent methods in the grid-based approach is FaStrack [7], where the
Hamilton-Jacobi (HJ) reachability of the error dynamics between a tracking
system and a planning system is computed. The authors have successfully
demonstrated that when the control authority is powerful, the tracking error
is bounded, and the HJ partial differential equations (PDEs) converge. Another
popular method for constructing invariant sets of nonlinear systems is using a
Lyapunov function [8]. A Lyapunov function can be found either by hand or
using a Lyapunov-based control design methodology, such as backstepping
control [10, 9, 11]. While the HJ reachability approach can compute the
invariant set for a nonlinear system, it is computationally expensive.
Similarly, Lyapunov-based approaches for non-linear systems can be tedious and
difficult to generalize. In this work, we utilize Linear Matrix Inequalities
(LMIs) [12] to efficiently compute the invariant set of the linearized error
dynamics of the nonlinear tracking system.

For a class of nonlinear systems, group affine systems [13], whose states can
be represented using Lie Groups, a unique property, the log-linear property,
can be employed to convert the nonlinear structure of the Lie Group into a
linear form in the Lie algebra. Modeling the states of the nonlinear system as
a member of a Lie group has been successfully applied in various applications
such as state estimation [13], motion control [14], optimal control [15, 16,
17], etc. In the Lie algebra, a group affine system [13] behaves linearly
despite nonlinearities in the original system on the Lie group, such as those
arising from rigid body rotation. The Lie algebra provides a convenient
coordinate system using the log-linear property for the log-linearization of
error dynamics, where the number of coordinates is equal to the degrees of
freedom of the dynamical system. However, the log-linear property of the error
dynamics governing the deviation of a system from its reference trajectory
does not hold in the presence of feedback control, disturbance, and noise,
which can cause nonlinearities in the Lie algebra dynamical system. These
nonlinearities are often neglected in existing work on log-linearized systems
[13, 18]. As a result, the exact solution is not provided, and thus the
computed invariant set of the nonlinear dynamics is over-approximated.

This paper presents a generalized method for deriving the exact evolution of
the log-linearization of the error dynamics for mixed-invariant systems, as
defined in [19], in Lie groups using the derivative of the exponential map.
While a similar method is mentioned in [20], where they derive the log-
linearization without approximation for a specific Lie group, $SE_{2}(3)$, our
work was developed independently, we provide the exact log-linearization
approach applicable to any Lie group, making our method more generic.
Additionally, to enhance the control robustness, we propose a log-linear
dynamic inversion-based control law. To illustrate the effectiveness of our
method, we demonstrate the application in a UAM scenario example with a
kinematics aircraft model in the $SE(2)$ Lie group. This example involves
calculations of invariant sets and the creation of flow pipes.

In this paper, our contributions are as follows:

  1. 1.

We derive the exact evolution for the log-linearization of the error dynamics
of mixed-invariant systems with disturbance and control input in the vehicle
system for any Lie group.

  2. 2.

We design a log-linear dynamic inversion-based control law to enhance the
robustness of the control system.

  3. 3.

We derive a series-form of the nonlinear input distortion matrix, $U$, for any
Lie group, and compute a closed-form of $U$ for the $SE(2)$ Lie group, which
is the Special Euclidean group of two-dimensional rigid body rotation and
translation.

  4. 4.

We formulate flow-pipe computation for a kinematic aircraft model by using our
log-linear dynamic inversion-based control strategy and applying LMIs to
calculate the invariant sets.

The rest of this paper is organized as follows. The preliminary background and
problem formulation are presented in Section II. In Section III, we use the
derivative of the exponential map in the Lie group to derive the exact log-
linearization for the error dynamics of mixed-invariant systems, and design
the log-linear dynamic inversion controller using the log-linearized system in
the Lie algebra. In Section IV, we embed the two-dimensional kinematic
aircraft model in the $SE(2)$ Lie group, and show an example of log-
linearization in the $SE(2)$ Lie group with disturbances. In Section V, we
illustrate the simulation results for our kinematic aircraft model. Finally,
in Section VI, we present our concluding remarks and immediate future works.

##  II Preliminaries and Problem Statement

###  II-A Background and Preliminaries

######  Definition 1.

(Mixed-Invariant System [19]) The left-invariant system [21, 19, 17] in the
Lie group has the form of $\dot{X}=X[l]^{\wedge}$, and the right-invariant
system in the Lie group is $\dot{X}=[r]^{\wedge}X$, where $X$ is a Lie group
$G$ whose associated Lie algebra is denoted $\mathfrak{g}$ and represents
states of the systems. $l,r\in\mathbb{R}^{n}$ are inputs of the systems and
may be functions of time, and $[l]^{\wedge},[r]^{\wedge}\in\mathfrak{g}$ are
elements in the Lie algebra, which define the left- and right-invariant vector
fields, respectively. $[\cdot]^{\wedge}$ denotes the wedge operator that maps
elements from $\mathbb{R}^{n}$ to the Lie algebra $\mathfrak{g}$. If a system
is the summation of a left-invariant system and a right-invariant system,
$\dot{X}=X{[l]}^{\wedge}+{[r]}^{\wedge}X$, then the system is called a _mixed-
invariant system_.

######  Definition 2.

(Group Affine System [13, 22]) If a system with dynamics $\dot{X}=f(X)$
satisfies $f(AB)=f(A)B+Af(B)-Af(I)B$, then we call the system a _group affine
system_. Here $A$, $B$, $X\in G$, and $I$ is the identity element of the Lie
Group.

######  Lemma 1.

(Mixed-Invariant systems are Group Affine [22, 13]) Let
$f(X)=X[l]^{\wedge}+[r]^{\wedge}X$, then

$\displaystyle f(A)B+Af(B)-Af(I)B$
$\displaystyle=(A[l]^{\wedge}+[r]^{\wedge}A)B+A(B[l]^{\wedge}+[r]^{\wedge}B)-A([l]^{\wedge}+[r]^{\wedge})B$
$\displaystyle=[r]^{\wedge}AB+AB[l]^{\wedge}=f(AB)\hskip
56.9055pt\blacksquare$

######  Lemma 2.

(The Derivative of the Exponential Map [23]) The exponential map of a matrix
Lie group is a map from the matrix Lie algebra to the corresponding matrix Lie
group, $\exp:\mathfrak{g}\rightarrow G$. The _derivative of the exponential
map_ is given by:

$\dfrac{d}{dt}\exp{({[\zeta(t)]}^{\wedge})}=\exp{({[\zeta(t)]}^{\wedge})}\dfrac{I-\exp{(-ad_{[\zeta]^{\wedge}})}}{ad_{[\zeta]^{\wedge}}}\dfrac{d}{dt}[\zeta(t)]^{\wedge}$
(1)

Here, $I$ denotes the identity element of the group, which is the identity
matrix for matrix Lie groups [23, 24], $ad_{[\zeta]^{\wedge}}$ is the adjoint
representation of $[\zeta]^{\wedge}$ in the Lie algebra, and
$\dfrac{I-\exp({-ad_{[\zeta]^{\wedge}}})}{ad_{[\zeta]^{\wedge}}}=\sum\limits_{k=0}^{\infty}\dfrac{(-1)^{k}}{(k+1)!}(ad_{[\zeta]^{\wedge}})^{k}$.

Based on the log-linear property, we can log-linearize the nonlinear system
with feedback control, bounded disturbance, and noise to an exact linear
bounded input system using the derivative of the exponential map. The
invariant set for the linear bounded input system can be found efficiently
employing LMIs.

###  II-B Problem Formulation

Consider the dynamics of two mixed-invariant vector fields, where $X\in G$ is
the state of the vehicle and $\bar{X}\in G$ is the state of the reference
trajectory:

$\displaystyle\dot{X}$ $\displaystyle=X{[l]}^{\wedge}+{[r]}^{\wedge}X$ (2)
$\displaystyle\dot{\bar{X}}$
$\displaystyle=\bar{X}{[\bar{l}]}^{\wedge}+{[\bar{r}]}^{\wedge}\bar{X}$

where $l,r,\bar{l},\bar{r}\in\mathbb{R}^{n}$ are inputs of the systems and may
be functions of time. The relationship between the two inputs is
$l=\bar{l}+u_{l},r=\bar{r}+u_{r}$. $u_{l}$ is the deviation of the system from
the reference left-invariant vector field, $l$, and $u_{r}$ is the deviation
of the system from the reference right-invariant vector field, $r$. The left-
and right-invariant errors, $\eta_{l},\eta_{r}\in G$ are defined as follows
[13]:

$\displaystyle\eta_{l}=X^{-1}\bar{X}\hskip 14.22636pt\eta_{r}=\bar{X}X^{-1}$
(3)

In this paper, we are interested in deriving the exact log-linearization for
the error dynamics, $\dot{\eta_{l}},\dot{\eta_{r}}$, of mixed-invariant
systems, and designing a robust controller with provable safety guarantees.

##  III Main Result

In this section, we first derive the exact log-linearization for the error
dynamics of mixed-invariant systems for both left- and right-invariant error
dynamics. We then design a log-linear dynamic inversion control law to address
the nonlinear input distortion and enhance the robustness of the system.

###  III-A Exact Log-Linearization of Mixed-Invariant System

######  Theorem 1.

(Logarithm of Left-Invariant Error Dynamics) For the systems in 2, denote the
left-invariant error $\eta_{l}$ by $\eta_{l}=\exp([\zeta_{l}]^{\wedge})$, then
the logarithm of the left-invariant error dynamics, $\dot{\zeta}_{l}$, is
governed by the differential equation:

$\displaystyle\dot{\zeta}_{l}$
$\displaystyle=-ad_{[\bar{l}]^{\wedge}}{\zeta_{l}}+U_{l}(u_{l}+Ad_{X^{-1}}u_{r})$
(4) $\displaystyle U_{l}$
$\displaystyle\equiv-\frac{ad_{[\zeta_{l}]^{\wedge}}\exp{(-ad_{[\zeta_{l}]^{\wedge}})}}{I-\exp{(-ad_{[\zeta_{l}]^{\wedge}})}}\hskip
14.22636ptU_{l}^{-1}=-\sum_{k=0}^{\infty}\frac{(ad_{[\zeta_{l}]^{\wedge}})^{k}}{(k+1)!}$

where $\zeta_{l}$ is the state vector of the left-invariant error dynamics in
the Lie algebra, $Ad_{X}$ is the adjoint representation of $X$ in the Lie
group, and $U_{l}$ is the matrix of nonlinear input distortion of inputs in
the Lie algebra for the left-invariant error dynamics. A proof is provided in
Appendix A.

######  Theorem 2.

(Logarithm of Right-Invariant Error Dynamics) For the systems in 2, denote the
right-invariant error $\eta_{r}$ by $\eta_{r}=\exp([\zeta_{r}]^{\wedge})$,
then the logarithm of the right-invariant error dynamics, $\dot{\zeta}_{r}$,
is governed by the differential equation:

$\displaystyle\dot{\zeta}_{r}$
$\displaystyle=ad_{[\bar{r}]^{\wedge}}{\zeta_{r}}+U_{r}(u_{r}+Ad_{X}u_{l})$
(5) $\displaystyle U_{r}$
$\displaystyle\equiv-\frac{ad_{[\zeta_{r}]^{\wedge}}}{I-\exp{(-ad_{[\zeta_{r}]^{\wedge}})}}\hskip
14.22636ptU_{r}^{-1}=-\sum_{k=0}^{\infty}\frac{(-1)^{k}(ad_{[\zeta_{r}]^{\wedge}})^{k}}{(k+1)!}$

where $\zeta_{r}$ is the state vector of right-invariant error dynamics in the
Lie algebra, and $ad_{r}$ is the adjoint representation of input $r$ in the
Lie algebra. $U_{r}$ is the matrix of nonlinear input distortion of inputs in
the Lie algebra for the right-invariant error dynamics. The proof can be
derived with the same process as the proof of 4 in Appendix A.

Based on 4 and 5, we have an exact solution for log-linearization of the error
dynamics of mixed-invariant systems. Since the log-linearization causes
nonlinear input distortion, $U_{l}$ and $U_{r}$, the log-linear dynamics in
the Lie algebra are nearly linear for a small deviation from the reference
trajectory.

######  Remark 1.

It is straightforward to show from 4 and 5 that the dynamics of the
left-/right-invariant error respectively are log-linear and group affine when
$u_{l}=u_{r}=0$. $\dot{\zeta}_{l}=-ad_{[l]^{\wedge}}{\zeta_{l}}$.
$\dot{\zeta}_{r}=ad_{[r]^{\wedge}}{\zeta_{r}}$.

###  III-B Log-Linear Dynamic Inversion Control

For control, 4 and 5 suggest a direct approach for dynamic inversion of the
log-linearized mixed-invariant vector field dynamics. We will focus on 4 but
this can also be extended to 5.

######  Theorem 3.

(Log-Linear Dynamic Inversion Control) Given the dynamics in 4 with
$u_{l}\equiv u+w$ and $u_{r}=0$, where $u$ represents a control input and $w$
represents a disturbance input, if $u$ is designed as:

$u=U_{l}^{-1}BK\zeta_{l}$ (6)

where $B$ is the control input matrix, and $K$ is the corresponding gain
matrix for the state feedback controller.

Then, the closed-loop system becomes a linear system under the impact of
magnitude bounded input disturbance, $||w||_{\infty}$. The induced matrix norm
of $U_{l}$ can be found in [25]. After substitution of 6 into 4, the system is
a linear system with bounded-input bounded-output (BIBO) stability [8, 26]
given by:

$\dot{\zeta}_{l}=(-ad_{[\bar{l}]^{\wedge}}+BK)\zeta_{l}+U_{l}w$ (7)

When using dynamic inversion control, it is important to determine if inputs
are saturated in the controller. Since we use a state feedback control law,
this can be efficiently checked and verified using the invariant set of the
states.

######  Lemma 3.

(Stability of the Log-Linearized System with Log-Linear Dynamic Inversion
Control Law) Since the log-linearized system in the Lie algebra is a linear
system with BIBO stability, a Lyapunov function can be found using LMIs and
used to compute an invariant set to which the system is globally
asymptotically stable (GAS).

Figure 1: Lie Group-Lie Algebra Bijection for Trajectories with Control and
Disturbances in $SE(2)$

Figure 2: Dynamics deviation from nonlinear closed loop system in the $SE(2)$
Lie Group, Left: With Exact Log-Linearization, Right: Without Exact Log-
Linearization

Figure 1 shows a 2D example of the bijection between the vehicle trajectories
in the $SE(2)$ Lie group and the $\mathfrak{se}(2)$ Lie algebra, which are
simulated with sinusoidal disturbances of fixed magnitude at varying
frequencies. In order for a bijection to exist, the exponential map must be
injective and surjective, which holds if the domain of the Lie algebra is
restricted to $[-\pi,\pi]$ [24]. Effectively meaning that our control law will
disregard multiple rotations, which is the intended behaviour. If the
invariant set in the Lie algebra extends beyond the domain of $[-\pi,\pi]$,
this would indicate a possibly instability through a continuous spinning
motion. Thus, the LMI approach is invalidated, as the linear dynamics do not
consider the angle wrapping. The figure on the left in Fig. 2 shows that the
log-linearized system dynamics in the Lie algebra do not deviate from the
closed loop mixed-invariant system dynamics when the exact log-linearization
is applied (the shown deviation can be arbitrarily reduced using numerical
tolerances); however, there are significant deviations from the closed loop
mixed-invariant system if our log-linearization is not applied (figure on the
right). Therefore, we numerically demonstrate 4 and 5 by showing that a
bijection exists between the trajectories in the Lie group and Lie algebra
even in the presence of disturbance and control feedback, given a domain
sufficiently close to the reference trajectory (e.g. avoiding angle wrapping).
This is a useful result, as the Lie algebra is a vector space that is more
convenient for analysis than the Lie group which is a nonlinear manifold [24].

##  IV Example: Kinematic Aircraft Model in the $SE(2)$ Lie Group

In the UAM system, we consider vehicles with 3D rigid body rotation and
translation, the $SE(3)$ Lie group. For constant altitude flight, this can be
simplified to 2D rigid body rotation and translation, the $SE(2)$ Lie group.
Since both VTOL and fixed-wing UAS dynamics can be well approximated as
differentially flat [27, 28], we can calculate a reference trajectory that is
flyable by the aircraft with negligible error in the absence of disturbance
and noise. This is important as our approach focuses on the deviation of the
aircraft from this reference trajectory. To simplify our approach in this
paper, we consider a constant altitude kinematic model of the UAS with bounded
disturbances to account for modelling error, which simplifies the UAS
kinematics to a two-dimensional holonomic model. Although we are limiting our
analysis to the $SE(2)$ Lie group, our algebraic approach can be generalized
to other Lie groups [29] (e.g., the $SE(3)$ Lie Group of rigid body rotation
and translation in three dimensions).

###  IV-A $SE(2)$ Lie Group Embedded Vehicle Kinematics

Consider the kinematic aircraft model evolving on a 2D plane, as follows:

$\frac{d}{dt}\theta=\omega,\;\frac{d}{dt}p_{x}=\cos\theta v_{x}-\sin\theta
v_{y},\;\frac{d}{dt}p_{y}=\sin\theta v_{x}+\cos\theta v_{y}$ (8)

where $\theta\in[-\pi,\pi]$ denotes the heading angle, $p_{x}$ and $p_{y}$
denote the position. $\omega$ represents the angular velocity, $v_{x}$ and
$v_{y}$ represent the forward and side translational velocity respectively. We
assume our kinematic aircraft model as a holonomic model due to the side-slip
angle of the aircraft and mentioned in [27].

The kinematic aircraft model in 8 can be embedded in the $SE(2)$ Lie group in
9, a matrix Lie group. Special in the Special Euclidean group indicates that
all matrices have determinant $1$. A Lie group is a group that is also a
smooth manifold. The associated Lie algebra is a vector space tangent to the
Lie group at the identity [30]. A Lie group and its Lie algebra are closely
related, which allows calculations in one to be mapped into the other. The
exponential map maps the Lie algebra to the Lie group and the logarithm map,
the inverse of the exponential map, maps the Lie group to the Lie algebra
[24].

The corresponding left-invariant system dynamics [14, 15, 19, 17] of the 2D
holonomic aircraft problem, which is a mixed-invariant system with the right-
invariant vector field, $r$, equal to zero, is given by:

$\dot{X}=X{[l]}^{\wedge}$ (9)

where $X$ represents the state of the 2D holonomic aircraft in the $SE(2)$ Lie
group. Note that $l\in\mathbb{R}^{3}$ may be a function of time and represents
the input of the vehicle, including feedback control and disturbances. The
$SE(2)$ Lie group may be represented by matrices of the form:

$X=\begin{bmatrix}R&p\\\ 0&1\end{bmatrix}$ (10)

where $R\in SO(2)$, which is an element of the 2D rotation group, and
$p=\begin{bmatrix}p_{x}&p_{y}\end{bmatrix}^{T}\in\mathbb{R}^{2}$. The $SE(2)$
Lie group is a semi-direct product of the $SO(2)$ and $\mathbb{R}^{2}$ [24].

The corresponding Lie algebra, $\mathfrak{se}(2)$ may be represented by
matrices of the form:

$[x]^{\wedge}=\begin{bmatrix}\omega_{\times}&v\\\ 0&0\end{bmatrix}$ (11)

where $x=\begin{bmatrix}v&\omega\end{bmatrix}^{T}$ is the element in the Lie
algebra. $\omega_{\times}$ is the corresponding skew-symmetric matrix of
$\omega\in\mathbb{R}$, such that $\omega_{\times}=-\omega_{\times}^{T}$,
$v=\begin{bmatrix}v_{x}&v_{y}\end{bmatrix}^{T}\in\mathbb{R}^{2}$. In Lie group
theory, the adjoint representation [24] transports a Lie algebra element from
one tangent space to another. Here, for an element $x$ of the
$\mathfrak{se}(2)$ Lie algebra, we use $ad_{x}$ to represent the adjoint in
the Lie algebra, which can be written as:

$ad_{[x]^{\wedge}}=\begin{bmatrix}\begin{array}[]{c|c}\omega_{\times}&\begin{matrix}v_{y}\\\
-v_{x}\end{matrix}\\\ \hline\cr 0&0\end{array}\end{bmatrix}$ (12)

###  IV-B Log-Linearization in the $SE(2)$ Lie Group

The 2D kinematics aircraft kinematics can be embedded in the $SE(2)$ Lie group
as $\dot{X}=X[l]^{\wedge}$. We also consider the reference kinematics embedded
in the $SE(2)$ Lie group differential equation, as
$\dot{\bar{X}}=\bar{X}{[\bar{l}]}^{\wedge}$, where $\bar{X}$ denotes the state
of the reference trajectory, and $\bar{l}$ denotes the input of the reference
trajectory, and may be functions of time to account for non-trivial reference
trajectories.

As we mention in the previous section, the relationship between $l$ and
$\bar{l}$ is $l=\bar{l}+u_{l}$, here, we consider $u_{l}=u+w$, where $u$ is
the feedback control input, and $w$ is the disturbance. The left-invariant
error between our kinematic aircraft model and the reference kinematics is
$\eta_{l}=X^{-1}\bar{X}$. From 4 and 6, the error dynamics in the Lie algebra
can be written as:

$\displaystyle\dot{\zeta}=(-ad_{[\bar{l}]^{\wedge}}+BK)\zeta+Uw,\hskip
14.22636ptU\equiv U_{l}$ (13)

where $u=U^{-1}BK\zeta$ and
$\zeta\equiv\begin{bmatrix}\zeta_{x}&\zeta_{y}&\zeta_{\theta}\end{bmatrix}^{T}$.

For the $SE(2)$ Lie group, the inverse of the nonlinear input distortion
matrix, $U^{-1}$, can be written in a series-form function and a closed-form
matrix, which is found by reducing the sine and cosine series:

$U^{-1}=-\sum_{k=0}^{\infty}\frac{(ad_{[\zeta]^{\wedge}})^{k}}{(k+1)!}\\\
=\begin{bmatrix}-\frac{\sin(\zeta_{\theta})}{\zeta_{\theta}}&\frac{1-\cos(\zeta_{\theta})}{\zeta_{\theta}}&-\frac{\zeta_{x}a+\zeta_{y}b}{2{\zeta_{\theta}}^{2}(\cos(\zeta_{\theta})-1)}\\\
-\frac{1-\cos(\zeta_{\theta})}{\zeta_{\theta}}&-\frac{\sin(\zeta_{\theta})}{\zeta_{\theta}}&\frac{\zeta_{x}b-\zeta_{y}a}{2{\zeta_{\theta}}^{2}(\cos(\zeta_{\theta})-1)}\\\
0&0&-1\end{bmatrix}$ (14)

where
$a=(-2\zeta_{\theta}\cos(\zeta_{\theta})+2\zeta_{\theta}-2\sin(\zeta_{\theta})+\sin(2\zeta_{\theta}))$
and $b=(-4\cos(\zeta_{\theta})+2\cos(\zeta_{\theta})+3)$. The determinant of
$U^{-1}$ is
$\det(U^{-1})=\frac{2(\cos(\zeta_{\theta})-1)}{\zeta_{\theta}^{2}}=-1+\frac{\zeta_{\theta}^{2}}{12}-\frac{\zeta_{\theta}^{4}}{360}+O(\zeta_{\theta}^{6})$,
since we limit $\zeta_{\theta}\in[-\pi,\pi]$, the determinant will never be
zero, $U^{-1}$ is always invertible. Therefore, the closed-form matrix of $U$
for the $SE(2)$ Lie group is:

$\displaystyle
U=\begin{bmatrix}c&-d&\frac{\zeta_{\theta}\zeta_{x}\sin{\left(\zeta_{\theta}\right)}+\left(1-\cos{\left(\zeta_{\theta}\right)}\right)\left(\zeta_{\theta}\zeta_{y}-2\zeta_{x}\right)}{2\zeta_{\theta}\left(1-\cos{\left(\zeta_{\theta}\right)}\right)}\\\
d&c&\frac{\zeta_{\theta}\zeta_{y}\sin{\left(\zeta_{\theta}\right)}+\left(1-\cos{\left(\zeta_{\theta}\right)}\right)\left(-\zeta_{\theta}\zeta_{x}-2\zeta_{y}\right)}{2\zeta_{\theta}\left(1-\cos{\left(\zeta_{\theta}\right)}\right)}\\\
0&0&-1\end{bmatrix}$ (15)

where
$c=\frac{\zeta_{\theta}\sin{\left(\zeta_{\theta}\right)}}{2\left(\cos{\left(\zeta_{\theta}\right)}-1\right)}$
and $d=\frac{\zeta_{\theta}}{2}$.

The invariant set for a system can be calculated by finding an upper bound for
the singular value of the nonlinear input distortion matrix, $U$, over the
invariant set. The nonlinear input distortion, $U$, depends on the error
state, $\zeta=[\zeta_{x},\zeta_{y},\zeta_{\theta}]^{T}$. If there exists an
invariant set, since the invariant set is a set of the error states, $U$ will
depend on the invariant set itself. And the calculation of the invariant set
also depends on the singular value of $U$, we propose a Log-linear LMI-based
iterative method to compute the invariant set in Algorithm 1. An invariant set
calculated by this algorithm is shown in Figure 3.

Algorithm 1 Calculate Log-Linear LMI-based Invariant Set

1: Guess the maximum singular value, $\sigma_{0}$, of the nonlinear input
distortion matrix, $U$, in the invariant set

2: Compute the invariant set using an LMI by scaling the disturbance by the
maximum singular value of $U$

3: Calculate the maximum singular value, $\sigma_{max}$, of the U matrix over
the invariant set

4: Assign $\sigma_{max}$ to $\sigma_{0}$

5: Repeat stpng 1-4 until converged to desired tolerance
($|\sigma_{0}-\sigma_{max}|<\epsilon$), favor over-approximation for safety
($\sigma_{0}>\sigma_{max}$)

##  V Simulation Studies of Kinematic Aircraft Model

In this section, we consider an illustrative UAM scenario where a fixed-wing
aircraft flies along a reference trajectory at a constant altitude. The
reference trajectory is generated by polynomial trajectory planning [28, 31,
32]. We consider a kinematic model of the fixed-wing aircraft. Our model is
given by the equations in 8, which can be embedded in the $SE(2)$ Lie group,
9. The log-linear error dynamics between the vehicle model and the reference
model in the Lie algebra are given in 13. For the external disturbances in the
$\theta$ direction, we assume $||w_{2}||_{\infty}=0.1$ rad/s. For disturbances
in the $\\{x,y\\}$ direction, we consider two different disturbance
magnitudes. We assume the magnitude of the bounded disturbance inputs,
$||w_{1}||_{\infty}$, as $1$ m/s and $5$ m/s, respectively.

###  V-A Invariant Set

Since the inputs of the reference trajectory may be functions of time, we use
the LMI to calculate the invariant set for a linear polytopic system with
bounded disturbances [12]. The bounds of the inputs to calculate the invariant
set are based on the bounds of the reference inputs, where
$\bar{v}_{x}\in[18,20]$ m/s, $\bar{v}_{y}=0$ m/s, and
$\bar{\omega}\in[-\pi/2,\pi/2]$ rad/s. We use the Linear-Quadratic-Regular
(LQR) method to find the feedback controller for the log-linearized system in
the $\mathfrak{se}(2)$ Lie algebra. After employing our iterative method,
Algorithm 1, to find the desired maximum singular value of $U$, the final
result of the invariant set in the Lie algebra and the Lie group is shown in
Fig. 3. The figures show the invariant set with small initial states in the
$SE(2)$ Lie group, $[x_{0},y_{0},\theta_{0}]=[0.1,0.1,\pi/100]$. The invariant
set in the Lie group is constructed by applying the exponential map from the
set in the Lie algebra.

If the control input does not employ dynamic inversion, the control input $u$
in 6 becomes $BK\zeta$, and the error dynamics in the Lie algebra in 13 can be
rewritten as:

$\dot{\zeta}=(-ad_{[\bar{l}]^{\wedge}}+BK)\zeta+(U-I)BK\zeta+Uw$ (16)

Here, we use the $SE(2)$ example to compare the difference between the two
controllers. For the control input without dynamics inversion, we consider the
nonlinear term of the control input as a bounded input in the LMI calculation.
Blue lines in Fig. 3 show the result of the invariant set for the controller
without dynamic inversion. Since the nonlinear term of the control input is
considered a bounded input, the invariant set has a larger tracking error
bound than the controller with dynamic inversion. Comparing the invariant sets
in Fig. 3, we can see that the invariant set is approximately twice as large
in the case of the control input without dynamic inversion compared to the
case with dynamic inversion. Therefore, the invariant set has a smaller
tracking error bound with our designed controller.

The Lyapunov function in the LMI approach is an ellipsoid, which is the shape
of the invariant set for the log-linearized system in the $\mathfrak{se}(2)$
Lie algebra as shown on the left of Fig. 3 with green lines. The invariant
sets in the $SE(2)$ Lie group, which are mapped from the invariant sets in the
$\mathfrak{se}(2)$ Lie algebra using the exponential map, are shown on the
right of Fig. 3 with green lines. Since the invariant set in the $SE(2)$ Lie
group is not an ellipsoid after mapping, as shown in Fig. 3, and is in the
vehicle body frame; therefore, the invariant set must be rotated along the
trajectory to create flow pipes. The left figure in Fig. 4 shows the invariant
set of our log-linear dynamic inversion control inputs for the small
disturbance case, the saturation bounds for each direction are
$u_{\zeta_{x}}\in[-1.25,1.25]$ m/s, $u_{\zeta_{y}}\in[-0.16,0.16]$ m/s, and
$u_{\zeta_{\theta}}\in[-1.92,1.92]$ rad/s. The right figure in Fig. 4 shows
the time history plot of the control inputs for the simulated trajectories
with sinusoidal and square wave disturbances in the small disturbance case,
and each control input stays within the saturation bound for all time.

Figure 3: Invariant Set Comparison in three dimensions with and without
Dynamic Inversion, Left: Invariant Set in Lie Algebra, Right: Invariant Set in
Lie Group, Top: Small Disturbance, Bottom: Large Disturbance

Figure 4: Left: Saturation Bound of Control Inputs, Right: Time History Plot
of Control Inputs

###  V-B Flow Pipes

To verify the safety properties for the UAM mission, we construct flow pipes
along a reference trajectory. Any Lyapunov function will be sufficient for our
method and we follow a general approach of the flow pipes creation in [4]. To
construct the flow pipe, we first propagate the reference trajectory using
polynomial trajectory planning for a fixed time interval and compute the
interval hull of the reference trajectory. The interval hull is the smallest
box enclosing the set for the propagation duration. Since our invariant set
depends on the direction of the reference trajectory, we sweep the invariant
set from the minimum rotational angle to the maximum rotational angle in that
time interval. Using the swept invariant set and interval hull, we then
compute the convex hull for the flow pipe segment using the Minkowski sum
[33]. Therefore, the flow pipe for the trajectory can be constructed by
repeating the above processes for the next time interval.

Figure 5: Flow Pipes Applications for the UAM Problem, Left: Small
Disturbances, Right: Large Disturbances

Figure 5 shows the application of flow pipes to the UAM problem. The green
squares represent collision zones or obstacles (e.g., buildings) that the
vehicle needs to avoid. Here, we simulate both sinusoidal and square wave wind
disturbances. We simulate two different scenarios for the vehicle. The first
scenario is simulated with small wind disturbances with a magnitude of $1$
m/s, which is shown at the top of Fig. 5. We can see that the green collision
zones, obstacles, and the flow pipes do not intersect; therefore, we can
confirm the safety of the vehicle, since the simulation shows that the vehicle
won’t hit the buildings during the mission. Another scenario is simulated with
large wind disturbances with a magnitude of $5$ m/s, which is shown at the
bottom of Fig. 5. We can see that the buildings and the flow pipes are now
intersecting. Therefore, we are able to confirm that it is not safe for the
vehicle to conduct the mission with wind disturbances of this magnitude. We
can also see that some simulated trajectories collide with obstacles and the
mission is therefore unsafe.

##  VI Conclusion

In this paper, we presented a novel approach to derive the exact log-
linearization of the error dynamics of mixed-invariant systems. This approach
employed the derivative of the exponential map in Lie group theory. Our
analysis of the evolution of the log-linearized system yielded a nonlinear
input distortion matrix. We were able to leverage the singular value of this
bounded input distortion matrix to reformulate our tracking error dynamics to
a linear system with bounded-input bounded-output (BIBO) stability.
Additionally, we designed a new dynamic inversion-based control law to
simplify the log-linearized tracking error dynamics to a linear system with a
bounded disturbance and enhanced the robustness of the system. In the
simulation, we demonstrated the application of our algorithm to safety
verification in an Urban Air Mobility (UAM) scenario. We also compared the
bound of the invariant set with and without the dynamic inversion-based
control law.

As future work, we intend to consider the invariant set as time-varying to
narrow the bound of the flow pipes and extend our approach to more complex
aircraft dynamic models with the $SE_{2}(3)$ and $SE(3)$ Lie groups.

##  Appendix A Proof of 4

Using the product rule, the left-invariant error dynamics, which are the
derivative of $\eta_{l}$, can be written as:

$\displaystyle\dot{\eta}_{l}$
$\displaystyle=-X^{-1}\dot{X}X^{-1}\bar{X}+X^{-1}\dot{\bar{X}}$ (17)
$\displaystyle=\eta_{l}{[\bar{l}]}^{\wedge}-{[\bar{l}+u_{l}]}^{\wedge}\eta_{l}-Ad_{X^{-1}}{[u_{r}]^{\wedge}}\eta_{l}$

Let $\eta_{l}=\exp({[\zeta_{l}]}^{\wedge})$, where $\zeta_{l}$ is the left-
invariant error in the Lie algebra. Based on Lemma 2, the derivative of the
exponential map can be written as:

$[\dot{\zeta}_{l}]^{\wedge}=\dfrac{ad_{[\zeta_{l}]^{\wedge}}}{I-\exp{(-ad_{\zeta_{l})}}}\eta_{l}^{-1}\dot{\eta}_{l}$
(18)

The left-invariant error dynamics in 17 can be rewritten as:

$\eta_{l}^{-1}\dot{\eta}_{l}=[\bar{l}]^{\wedge}-Ad_{{\eta_{l}}^{-1}}[\bar{l}+u_{l}]^{\wedge}-Ad_{{\eta_{l}}^{-1}}Ad_{X^{-1}}[u_{r}]^{\wedge}\\\
=\left(I-\exp{(-ad_{[\zeta_{l}]^{\wedge}})}\right)[\bar{l}]^{\wedge}-\exp{(-ad_{[\zeta_{l}]^{\wedge}})}[u_{l}]^{\wedge}\\\
-\exp{(-ad_{[\zeta_{l}]^{\wedge}})}Ad_{X^{-1}}[u_{r}]^{\wedge}$ (19)

The corresponding dynamics for left-invariant error in the Lie algebra can be
derived as:

∧
$\displaystyle=\frac{ad_{[\zeta_{l}]^{\wedge}}}{I-\exp{(-ad_{[\zeta_{l}]^{\wedge}})}}[\left(I-\exp{(-ad_{[\zeta_{l}]^{\wedge}})}\right)[\bar{l}]^{\wedge}$
(20)
$\displaystyle-\exp{(-ad_{[\zeta_{l}]^{\wedge}})}[u_{l}]^{\wedge}-\exp{(-ad_{[\zeta_{l}]^{\wedge}})}Ad_{X^{-1}}[u_{r}]^{\wedge}]$
$\displaystyle=-ad_{[\bar{l}]^{\wedge}}[{\zeta_{l}}]^{\wedge}$
$\displaystyle-\frac{ad_{[\zeta_{l}]^{\wedge}}\exp{(-ad_{[\zeta_{l}]^{\wedge}})}}{I-\exp{(-ad_{[\zeta_{l}]^{\wedge}})}}([u_{l}]^{\wedge}+Ad_{X^{-1}}[u_{r}]^{\wedge})$

$\frac{ad_{[\zeta_{l}]^{\wedge}}\exp{(-ad_{[\zeta_{l}]^{\wedge}})}}{I-\exp{(-ad_{[\zeta_{l}]^{\wedge}})}}$
can be expressed as a matrix power series of $ad_{[\zeta_{l}]^{\wedge}}$, and
$ad_{[\zeta_{l}]^{\wedge}}$ is a linear operator on the Lie algebra,
therefore, we can use the Lie vee operator, ${[\cdot]}^{\vee}$, which is the
inverse of the wedge operator, to map the error dynamics from a form of
elements of the Lie algebra 20 to elements of the vector space
$\mathbb{R}^{n}$ 4 as shown in 4.

Let
$U_{l}\equiv-\frac{ad_{[\zeta_{l}]^{\wedge}}\exp{(-ad_{[\zeta_{l}]^{\wedge}})}}{I-\exp{(-ad_{[\zeta_{l}]^{\wedge}})}}$,
since each term in this expression is a power of $ad_{[\zeta_{l}]^{\wedge}}$,
multiplication is commutative, thus,
$U_{l}^{-1}=-\frac{I-\exp{(-ad_{[\zeta_{l}]^{\wedge}})}}{ad_{[\zeta_{l}]^{\wedge}}\exp{(-ad_{[\zeta_{l}]^{\wedge}})}}$,
using the taylor series of exponential, we can find a series form for
$U_{l}^{-1}$:

$\displaystyle U_{l}^{-1}$
$\displaystyle=-\frac{I-\exp{\left(-ad_{[\zeta_{l}]^{\wedge}}\right)}}{ad_{[\zeta_{l}]^{\wedge}}\exp{\left(-ad_{[\zeta_{l}]^{\wedge}}\right)}}$
(21)
$\displaystyle=-\frac{1}{ad_{[\zeta_{l}]^{\wedge}}}\left({\exp\left({-ad_{[\zeta_{l}]^{\wedge}}}\right)}^{-1}-I\right)$
$\displaystyle=-\frac{1}{ad_{[\zeta_{l}]^{\wedge}}}\left(\exp{\left({ad_{[\zeta_{l}]^{\wedge}}}\right)}-I\right)$
$\displaystyle=-\frac{1}{ad_{[\zeta_{l}]^{\wedge}}}\left(\sum_{m=0}^{\infty}\frac{\left(ad_{[\zeta_{l}]^{\wedge}}\right)^{m}}{m!}-I\right)$
$\displaystyle=-\sum_{m=1}^{\infty}\frac{\left(ad_{[\zeta_{l}]^{\wedge}}\right)^{m-1}}{m!}=-\sum_{k=0}^{\infty}\frac{\left(ad_{[\zeta_{l}]^{\wedge}}\right)^{k}}{\left(k+1\right)!}$

## References

  * [1] D. P. Thipphavong, R. Apaza, B. Barmore, V. Battiste, B. Burian, Q. Dao, M. Feary, S. Go, K. H. Goodrich, J. Homola _et al._ , “Urban air mobility airspace integration concepts and considerations,” in _2018 AIAA Aviation Technology, Integration, and Operations Conference_ , 2018, p. 3676\.
  * [2] A. Chutinan and B. H. Krogh, “Verification of polyhedral-invariant hybrid automata using polygonal flow pipe approximations,” in _International workshop on hybrid systems: computation and control_. Springer, 1999, pp. 76–90.
  * [3] A. Chutinan and B. Krogh, “Computational techniques for hybrid system verification,” _IEEE Transactions on Automatic Control_ , vol. 48, no. 1, pp. 64–75, 2003.
  * [4] J. M. Goppert, “Security and verification of unmanned vehicles,” Ph.D. dissertation, Purdue University Graduate School, 2019.
  * [5] I. M. Mitchell, A. M. Bayen, and C. J. Tomlin, “A time-dependent hamilton-jacobi formulation of reachable sets for continuous dynamic games,” _IEEE Transactions on automatic control_ , vol. 50, no. 7, pp. 947–957, 2005\.
  * [6] T. Homer and P. Mhaskar, “Constrained control lyapunov function-based control of nonlinear systems,” _Systems & Control Letters_, vol. 110, pp. 55–61, 2017.
  * [7] M. Chen, S. L. Herbert, H. Hu, Y. Pu, J. F. Fisac, S. Bansal, S. Han, and C. J. Tomlin, “Fastrack: a modular framework for real-time motion planning and guaranteed safe tracking,” _IEEE Transactions on Automatic Control_ , vol. 66, no. 12, pp. 5861–5876, 2021.
  * [8] H. Khalil, _Nonlinear Systems_ , ser. Pearson Education. Prentice Hall, 2002.
  * [9] Z. Wang, X. Liu, K. Liu, S. Li, and H. Wang, “Backstepping-based lyapunov function construction using approximate dynamic programming and sum of square techniques,” _IEEE transactions on cybernetics_ , vol. 47, no. 10, pp. 3393–3403, 2016.
  * [10] P. Kokotović and M. Arcak, “Constructive nonlinear control: a historical perspective,” _Automatica_ , vol. 37, no. 5, pp. 637–662, 2001.
  * [11] M. Tayefi and Z. Geng, “Logarithmic control, trajectory tracking, and formation for nonholonomic vehicles on lie group se (2),” _International Journal of Control_ , vol. 92, no. 2, pp. 204–224, 2019.
  * [12] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, _Linear matrix inequalities in system and control theory_. SIAM, 1994.
  * [13] A. Barrau and S. Bonnabel, “The invariant extended kalman filter as a stable observer,” _IEEE Transactions on Automatic Control_ , vol. 62, no. 4, pp. 1797–1812, 2016.
  * [14] N. E. Leonard and P. S. Krishnaprasad, “Motion control of drift-free, left-invariant systems on lie groups,” _IEEE Transactions on Automatic control_ , vol. 40, no. 9, pp. 1539–1554, 1995.
  * [15] A. M. Bloch, P. E. Crouch, J. E. Marsden, and A. K. Sanyal, “Optimal control and geodesics on quadratic matrix lie groups,” _Foundations of Computational Mathematics_ , vol. 8, pp. 469–500, 2008.
  * [16] M. B. Kobilarov and J. E. Marsden, “Discrete geometric optimal control on lie groups,” _IEEE Transactions on Robotics_ , vol. 27, no. 4, pp. 641–655, 2011\.
  * [17] L. J. Colombo and D. V. Dimarogonas, “Symmetry reduction in optimal control of multiagent systems on lie groups,” _IEEE Transactions on Automatic Control_ , vol. 65, no. 11, pp. 4973–4980, 2020.
  * [18] S. Teng, W. Clark, A. Bloch, R. Vasudevan, and M. Ghaffari, “Lie algebraic cost function design for control on lie groups,” in _2022 IEEE 61st Conference on Decision and Control (CDC)_. IEEE, 2022, pp. 1867–1874.
  * [19] A. Khosravian, J. Trumpf, R. Mahony, and T. Hamel, “State estimation for invariant systems on lie groups with delayed output measurements,” _Automatica_ , vol. 68, pp. 254–265, 2016.
  * [20] X. Li, H. Jiang, X. Chen, H. Kong, and J. Wu, “Closed-form error propagation on $se\\_${$n$}$(3)$ group for invariant ekf with applications to vins,” _IEEE Robotics and Automation Letters_ , vol. 7, no. 4, pp. 10 705–10 712, 2022.
  * [21] C. Lageman, J. Trumpf, and R. Mahony, “Gradient-like observers for invariant dynamics on a lie group,” _IEEE Transactions on Automatic Control_ , vol. 55, no. 2, pp. 367–377, 2009.
  * [22] A. Barrau and S. Bonnabel, “Invariant particle filtering with application to localization,” in _53rd IEEE Conference on Decision and Control_. IEEE, 2014, pp. 5599–5605.
  * [23] W. Rossmann, _Lie groups: an introduction through linear groups_. Oxford University Press on Demand, 2006, vol. 5.
  * [24] B. Hall, _Lie groups, Lie algebras, and representations: an elementary introduction_. Springer, 2015, vol. 222\.
  * [25] K. A. Pant, L.-Y. Lin, J. Goppert, and I. Hwang, “Towards robust state estimation in matrix lie groups,” in _Workshop on Geometric Representations, ICRA IEEE,_ , 2023.
  * [26] M. Dahleh, M. A. Dahleh, and G. Verghese, “Lectures on dynamic systems and control,” _A+ A_ , vol. 4, no. 100, pp. 1–100, 2004.
  * [27] J. Levine, _Analysis and control of nonlinear systems: A flatness-based approach_. Springer Science & Business Media, 2009.
  * [28] D. Mellinger and V. Kumar, “Minimum snap trajectory generation and control for quadrotors,” in _2011 IEEE international conference on robotics and automation_. IEEE, 2011, pp. 2520–2525.
  * [29] L.-Y. Lin, J. Goppert, and I. Hwang, “Application of log-linear dynamic inversion control to a multi-rotor,” in _Workshop on Geometric Representations, ICRA IEEE,_ , 2023.
  * [30] A. Baker, _Matrix groups: An introduction to Lie group theory_. Springer Science & Business Media, 2012.
  * [31] C. Richter, A. Bry, and N. Roy, “Polynomial trajectory planning for aggressive quadrotor flight in dense indoor environments,” in _Robotics research_. Springer, 2016, pp. 649–666.
  * [32] K. M. Lynch and F. C. Park, _Modern robotics_. Cambridge University Press, 2017.
  * [33] G. Varadhan and D. Manocha, “Accurate minkowski sum approximation of polyhedral models,” in _12th Pacific Conference on Computer Graphics and Applications, 2004. PG 2004. Proceedings._ IEEE, 2004, pp. 392–401.


# GENERALIZED t-EDGE DISTANCE-BALANCED GRAPHS

Z. Aliannejadi∗, M. Alaeiyan, A. Gilani

###### Abstract.

A connected and nonempty graph $A$ is defined as generalized $t$-edge
distance-balanced, while for each edge $f=\alpha\beta$ the number of edges
nearer to $\alpha$ than $\beta$ are equal to $t$-times of edges nearer to
$\beta$ than to $\alpha$, for $t\in\mathbb{N}$, or vice versa. We determine
some classes of such graphs. Moreover, we investigate edge-szeged index of the
generalized $t$-edge distance-balanced graphs. Also, it is discussed about
their cartesian and lexicographic products.

MSC(2010): Primary: 05C12; Secondary: 05C25,
Keywords: generalized $t$-edge distance-balanced graphs, graph products, edge-
szeged index, complete bipartite graphs
$*$Corresponding author

##  1\. Introduction

The notion of graph is a pivotal tool to make use of the modelling of the
phenomena and it is taken into consideration in many studies in a recent
decades. One of the optimal uses of graphs theory is to classify graphs based
on discriminating quality. This phenomenon can be best observed in distance-
balanced graphs has been determined by [13]. Also, it is investigated in some
papers, we refer the reader to ([1],[2],[3],[4],[5],[6],[10],[11],[14]-[19])
and references therein.

We consider $A$ is a connected, finite and undirected graph throughout of this
paper, in which its vertex set is $V(A)$ and its edge set is $E(A)$. In graph
$A$, the interval among vertices $\alpha,\beta\in V(A)$ is introduced the
number of edges in the least interval joining them and it is indicated by
$d_{A}(\alpha,\beta)$ (see [4, 18]). For every two desired vertices
$\alpha,\beta$ of $V(A)$ we indicate
$n_{\alpha}^{A}(f)=|W_{a,\beta}^{A}|=|\\{a\in
V(A)|d_{A}(a,\alpha)<d_{A}(a,\beta)\\}|$. In the same way, we would define
$n_{\beta}^{A}(f)=|W_{\beta,\alpha}^{A}|$. We name $A$ distance-balanced
($DB$) while for adjacent vertices $\alpha$ and $\beta$ of $A$, we have
$|W_{\alpha,\beta}^{A}|=|W_{\beta,\alpha}^{A}|$.

For each two desired edges $f=\alpha\beta$,
$\acute{f}=\acute{\alpha}\acute{\beta}$, the distance between $f$ and
$\acute{f}$ is introduced via:

$d_{A}(f,\acute{f})=\min\\{d_{A}(\alpha,\acute{f}),d_{A}(\beta,\acute{f})\\}=\min\\{d_{A}(\alpha,\acute{\alpha}),d_{A}(\alpha,\acute{\beta}),d_{A}(\beta,\acute{\alpha}),d_{A}(\beta,\acute{\beta})\\}$.

Set $M_{\alpha}^{A}(f)=\\{\acute{f}\in
E(A)|d_{A}(\alpha,\acute{f})<d_{A}(\beta,\acute{f})\\}$ and
$m_{\alpha}^{A}(f)=|M_{\alpha}^{A}(f)|$,
$M_{\beta}^{A}(f)=\\{\acute{f}\in
E(A)|d_{A}(\beta,\acute{f})<d_{A}(\alpha,\acute{f})\\}$ and
$m_{\beta}^{A}(f)=|M_{\beta}^{A}(f)|$,
and $M_{0}^{A}(f)=\\{\acute{f}\in
E(A)|d_{A}(\alpha,\acute{f})=d_{A}(\beta,\acute{f})\\}$ and
$m_{0}^{A}(f)=|M_{0}^{A}(f)|$.

Persume that $f=\alpha\beta\in E(A)$. For every two integers $i,j$ we
consider:

$\acute{D}_{j}^{i}(f)=\\{\acute{f}\in
E(A)|d_{A}(\acute{f},\alpha)=i,d_{A}(\acute{f},\beta)=j\\}$.

A ”distance partition” of $E(A)$ is concluded by sets $\acute{D}_{j}^{i}(f)$
due to the edge $f=\alpha\beta$. Merely the sets $\acute{D}_{i}^{i-1}(f)$,
$\acute{D}_{i}^{i}(f)$ and, $\acute{D}_{i-1}^{i}(f)$, for each $(1\leqslant
i\leqslant d)$ might be nonempty according to the triangle inequality (The
diameter of the graph $A$ is $d$), as well as $\acute{D}_{0}^{0}(f)=\phi$.

A graph $A$ is said to be generalized $t$-edge distance-balanced for
$t\in\mathbb{N}$ and $t>1$, so that for each edge $f=\alpha\beta$ of $E(A)$,
either $m_{\alpha}^{A}(f)=tm_{\beta}^{A}(f)$ or
$m_{\beta}^{A}(f)=tm_{\alpha}^{A}(f)$. For convenience, we would write $Gt-
EDBG$ instead of generalized $t$-edge distance-balanced graph and also $Gt-
EDB$ instead of generalized $t$-edge distance-balanced. In the same way, we
define generalized $t$-distance-balanced graphs ($Gt-DB$), in which
$n_{\alpha}^{A}(f)=tn_{\beta}^{A}(f)$, or
$n_{\beta}^{A}(f)=tn_{\alpha}^{A}(f)$.

We also remind that the sets $W_{\alpha,\beta}^{A}$ and $M_{\alpha}^{A}(f)$
become visible as the well-known Szeged index and edge-Szeged index of a graph
$A$ in the chemical graph theory, in which $S_{z}(A)=\sum_{\alpha\beta\in
E(A)}|W_{\alpha,\beta}|.|W_{\beta,\alpha}|$ and
$S_{{z}_{e}}(A)=\sum_{\alpha\beta\in E(A)}m_{\alpha}^{A}(f).m_{\beta}^{A}(f)$,
respectively, cf. [7,8,9].

A graph $A$ is called edge distance-balanced (briefly $EDB$), while we have
$m_{\alpha}^{A}(f)=m_{\beta}^{A}(f)$. Also we define $A$ as nicely edge
distance-balanced ($NEDB$), whenever there exists a positive integer
$\acute{\gamma}_{A}$, so that for every edge $f=\alpha\beta$ we have:
$m_{\alpha}^{A}(f)=m_{\beta}^{A}(f)=\acute{\gamma}_{A}$.

It is said $A$ as a graph is strongly edge distance-balanced ($SEDB$),
whenever for each $i\geq 1$ it holds:

$\acute{D}_{i-1}^{i}(f)=\acute{D}_{i}^{i-1}(f)$

According to the stated definition, clearly if graph be $SEDB$, then it is a
$EDB$ graph.
In this segment, we would express some examples and results concerning $Gt-
EDB$ graphs.

##  2\. Main results

At first, we present an example of $Gt-EDB$ graph.

######  Example 2.1.

The complete bipartite graphs $K_{n,tn}$ are a class of $Gt-EDB$ graphs.

###### Proof.

Let $X$ and $Y$ be two separate parts of $K_{n,tn}$, such that for every edge
$f=\alpha\beta$, $\alpha\in X$ and $\beta\in Y$. As we know $K_{n,tn}$ is a
bipartite graph with diameter 2. Then $\alpha$ has $(tn-1)$ adjacent edges and
$\beta$ includes $(n-1)$ adjacent edges. According to the definition of
$m_{\alpha}^{A}(f)$, we obtain $m_{\alpha}^{A}(f)=tn$ and
$m_{\beta}^{A}(f)=n$. Thus, $m_{\alpha}^{A}(f)=tm_{\beta}^{A}(f)$. This
implies that $K_{n,tn}$ is a $Gt-EDB$ graph. ∎

######  Proposition 2.2.

Assume that $A$ is a $Gt-EDB$ bipartite graph, where its diameter is 2. Then
for each pair of adjacent vertices $\alpha\beta$, $deg(\alpha)=tdeg(\beta)$.

###### Proof.

Since $A$ is a $Gt-EDB$ graph, then $m_{\alpha}^{A}(f)=tm_{\beta}^{A}(f)$ and
we have

$|\\{f\\}\bigcup^{d-1}_{i=1}\acute{D}_{i-1}^{i}(f)|=t|\\{f\\}\bigcup^{d-1}_{i=1}\acute{D}_{i}^{i-1}(f)|$.

Therefore,
$\sum^{d-1}_{i=1}|\acute{D}_{i-1}^{i}(f)|=t\sum^{d-1}_{i=1}|\acute{D}_{i}^{i-1}(f)|+(t-1)$.$\hskip
54.06006pt(1)$

If $|\acute{D}^{2}_{1}(f)|=r$, then $|\acute{D}^{1}_{2}(f)|=tr+(t-1)$. Thus,
$deg(\beta)=r+1$ and $deg(\alpha)=tr+t=t(r+1)$. Therefore, always
$deg(\alpha)=tdeg(\beta)$. ∎

######  Lemma 2.3.

Suppose that $A$ is a $Gt-EDB$ bipartite graph with $d=2$. Then $A$ can only
be $K_{n,tn}$.

###### Proof.

Consider that $A$ is a $Gt-EDB$ bipartite graph, in which its diameter is
equal to 2. It is asserted that $A$ is a complete bipartite graph. If not, it
can not have a diameter of 2. The result is obtained from Proposition 2.2
that, $deg(\alpha)=tdeg(\beta)$. If $A$ be a complete bipartite graph, then
$A$ must be $K_{n,tn}$. ∎

######  Proposition 2.4.

A bipartite graph $A$ is $Gt-EDB$ if and only if

$S_{{z}_{e}}(A)=t.\frac{(tn+t+1)^{2}.tn^{2}}{(t+1)^{2}}$.

###### Proof.

Persume that $A$ is $Gt-EDB$. For every edge $f=\alpha\beta$, we have
$m_{\alpha}^{A}(f)=tm_{\beta}^{A}(f)$ and since $A$ is bipartite also
$m_{\alpha}^{A}(f)+m_{\beta}^{A}(f)+m_{0}^{A}(f)=|E(A)|-1$ holds. Therefore,

$m_{\alpha}^{A}(f)+m_{\beta}^{A}(f)=|E(A)|-m_{0}^{A}(f)-1$.

Then,

$t^{2}(m_{\beta}^{A}(f))^{2}+(m_{\beta}^{A}(f))^{2}+2t(m_{\beta}^{A}(f))^{2}=(|E(A)|-m_{0}^{A}(f)-1)^{2}$.

Hence,

$(m_{\beta}^{A}(f))^{2}=\frac{(|E(A)|-m_{0}^{A}(f)-1)^{2}}{(t+1)^{2}}$,

and so,

$\sum_{f\in E(A)}m_{\alpha}^{A}(f).m_{\beta}^{A}(f)=t\sum_{f\in
E(A)}(m_{\beta}^{A}(f))^{2}=$

$t.\frac{(|E(A)|-m_{0}^{A}(f)-1)^{2}.|E(A)|}{(t+1)^{2}}=t.\frac{(tn+t+1)^{2}.tn^{2}}{(t+1)^{2}}$.

For converse, let $S_{{z}_{e}}(A)=t.\frac{(tn+t+1)^{2}.tn^{2}}{(t+1)^{2}}$.
Since $A$ is a graph, we have
$m_{\alpha}^{A}(f)+m_{\beta}^{A}(f)+m_{0}^{A}(f)=|E(A)|-1$ and also

$m_{0}^{A}(f)+(t+1)m_{\beta}^{A}(f)=|E(A)|-1$.

Hence,

$m_{\beta}^{A}(f)=\frac{(|E(A)|-m_{0}^{A}(f)-1)}{(t+1)}$.

As well as

$m_{\alpha}^{A}(f).m_{\beta}^{A}(f)=t.\frac{(|E(A)|-m_{0}^{A}(f)-1)^{2}}{(t+1)^{2}}$,

and hence

$m_{\alpha}^{A}(f)=tm_{\beta}^{A}(f)=t.\frac{(|E(A)|-m_{0}^{A}(f)-1)}{(t+1)}$.

This implies that $A$ is $Gt-EDB$. ∎

##  3\. $Gt-EDB$ graphs and graph products

We would now investigate situations in which the Cartesian product leads to a
$Gt-EDBG$. We mention that such product graphs, formed by graphs $A$ and $B$,
its vertex set is $V(A\square B)=V(A)\times V(B)$. Consider that
$(a_{1},b_{1})$ and $(a_{2},b_{2})$ are detached vertices in $V(A\square B)$.
In the Cartesian product $A\square B$, if vertices $(a_{1},b_{1})$ and
$(a_{2},b_{2})$ are coincident in one coordinate and adjacent in the another
coordinate, then they are adjacent, that is, $a_{1}=a_{2}$ and $b_{1}b_{2}\in
E(B)$, or $b_{1}=b_{2}$ and $a_{1}a_{2}\in E(A)$. Obviously, for vertices we
have:

$d_{A\square
B}((a_{1},b_{1}),(a_{2},b_{2}))=d_{A}(a_{1},a_{2})+d_{B}(b_{1},b_{2})$.

For edges we have:

$d_{A\square
B}((a,b)(a_{1},b_{1}),(\acute{a},\acute{b})(\acute{a}_{1},\acute{b}_{1}))=$

$\min\\{d_{A\square B}((a,b),(\acute{a},\acute{b})),d_{A\square
B}((a,b),(\acute{a}_{1},\acute{b}_{1})),d_{A\square
B}((a_{1},b_{1}),(\acute{a},\acute{b}))$,

$d_{A\square B}((a_{1},b_{1}),(\acute{a}_{1},\acute{b}_{1}))\\}=$

$\min\\{d_{A}(a,\acute{a})+d_{B}(b,\acute{b}),d_{A}(a,\acute{a}_{1})+d_{B}(b,\acute{b}_{1}),d_{A}(a_{1},\acute{a})+d_{B}(b_{1},\acute{b}),d_{A}(a_{1},\acute{a}_{1})+d_{B}(b_{1},\acute{b}_{1})\\}$.

######  Proposition 3.1.

Suppose that $A$ and $B$ are connected graphs. Then $A\square B$ is a $Gt-EDB$
graph if and only if $A$ and $B$ are both $Gt-EDB$ and $Gt-DB$ graphs.

###### Proof.

Consider that $a_{1},a_{2}$ are adjacent vertices in $V(A)$, and $b_{1},b_{2}$
are two adjacent vertices in $V(B)$. We assume that
$(a_{1},b_{1}),(a_{2},b_{1}),(a_{1},b_{2})\in V(A\square B)$. We observe that

$(a,b)(\acute{a},\acute{b})\in
M_{(a_{1},b_{1})}((a_{1},b_{1})(a_{2},b_{1}))\Leftrightarrow$

$\min\\{d_{A\square B}((a,b),(a_{1},b_{1})),d_{A\square
B}((\acute{a},\acute{b}),(a_{1},b_{1}))\\}<\min\\{d_{A\square
B}((a,b),(a_{2},b_{1})),d_{A\square
B}((\acute{a},\acute{b}),(a_{2},b_{1}))\\}\Leftrightarrow$

$\min\\{d_{A}(a,a_{1})+d_{B}(b,b_{1}),d_{A}(\acute{a},a_{1})+d_{B}(\acute{b},b_{1})\\}<\min\\{d_{A}(a,a_{2})+d_{B}(b,b_{1}),d_{A}(\acute{a},a_{2})+d_{B}(\acute{b},b_{1})\\}\Leftrightarrow$

$\min\\{d_{A}(a,a_{1}),d_{A}(\acute{a},a_{1})\\}<\min\\{d_{A}(a,a_{2}),d_{A}(\acute{a},a_{2})\\}$.

We deduce that

$M_{(a_{1},b_{1})}((a_{1},b_{1})(a_{2},b_{2}))=\\{(a,b)(\acute{a},\acute{b})\in
E(A\square B)|a\acute{a}\in E(A),b=\acute{b}$ or

$b\acute{b}\in E(B)$ , $a=\acute{a}$ ,

$\min\\{d_{A}(a,a_{1}),d_{A}(\acute{a},a_{1})\\}<\min\\{d_{A}(a,a_{2}),d_{A}(\acute{a},a_{2})\\}\\}$,
(2)

and hence applying (2) and by [19. Theorem 2.1] we conclude that:

$m^{A\square
B}_{(a_{1},b_{1})}((a_{1},b_{1})(a_{2},b_{2}))=m^{A}_{a_{1}}(a_{1}a_{2}).|V(B)|+n^{A}_{a_{1}}(a_{1}a_{2}).|E(B)|$.
(3)

Similar to this process we obtain

$m^{A\square
B}_{(a_{2},b_{2})}((a_{2},b_{2})(a_{1},b_{1}))=m^{A}_{a_{2}}(a_{1}a_{2}).|V(B)|+n^{A}_{a_{2}}(a_{1}a_{2}).|E(B)|$,
(4)

$m^{A\square
B}_{(a_{1},b_{1})}((a_{1},b_{1})(a_{1},b_{2}))=m^{B}_{b_{1}}(b_{1}b_{2}).|V(A)|+n^{B}_{b_{1}}(b_{1}b_{2}).|E(A)|$,
(5)

$m^{A\square
B}_{(a_{1},b_{2})}((a_{1},b_{2})(a_{1},b_{1}))=m^{B}_{b_{2}}(b_{1}b_{2}).|V(A)|+n^{B}_{b_{2}}(b_{1}b_{2}).|E(A)|$.
(6)

Persume that both $A$ and $B$ are both $Gt-EDBG$ and $Gt-DBG$. Since $A$ is
$Gt-EDB$ and $Gt-DB$, then by Equalities (3) and (4) we have:

$m^{A}_{a_{1}}(a_{1}a_{2}).|V(B)|+n^{A}_{a_{1}}(a_{1}a_{2}).|E(B)|=tm^{A}_{a_{2}}(a_{1}a_{2}).|V(B)|+tn^{A}_{a_{2}}(a_{1}a_{2}).|E(B)|$,

Therefore,

$m^{A\square B}_{(a_{1},b_{1})}((a_{1},b_{1})(a_{2},b_{2}))=tm^{A\square
B}_{(a_{2},b_{2})}((a_{1},b_{1})(a_{2},b_{2}))$.

By analogy, using (5) and (6), we conclude that

$m^{A\square B}_{(a_{1},b_{1})}((a_{1},b_{1})(a_{1},b_{2}))=tm^{A\square
B}_{(a_{1},b_{2})}((a_{1},b_{1})(a_{1},b_{2}))$,

and hence $A\square B$ is $Gt-EDB$.
For converse, let $A\square B$ is $Gt-EDB$, then applying (3) and (4) we
observe that

$m^{A\square B}_{(a_{1},b_{1})}((a_{1},b_{1})(a_{2},b_{2}))=tm^{A\square
B}_{(a_{2},b_{2})}((a_{1},b_{1})(a_{2},b_{2}))\Rightarrow$

$m^{A}_{a_{1}}(a_{1}a_{2}).|V(B)|+n^{A}_{a_{1}}(a_{1}a_{2}).|E(B)|=tm^{A}_{a_{2}}(a_{1}a_{2}).|V(B)|+tn^{A}_{a_{2}}(a_{1}a_{2}).|E(B)|$,

Hence $A$ is $Gt-EDB$ and $Gt-DB$. In the same way, (5) and (6) yield that $B$
is $Gt-EDB$ and $Gt-DB$. This completes the result. ∎

We would define the lexicographic product graphs. Product graph $A[B]$ of the
graphs $A$ and $B$, where its vertex set is $V(A[B])=V(A)\times V(B)$, and two
its adjacent vertices are $(a_{1},b_{1}),(a_{2},b_{2})$ is defined the
_lexicographic product_ if $a_{1}a_{2}\in E(A)$ or if $a_{1}=a_{2}$ and also
$b_{1}b_{2}\in E(B)$ (for more information see [12,p.22]). Since $A$ is a
graph, thus it is easily seen for vertices that

$\
d_{A[B]}((a_{1},b_{1}),(a_{2},b_{2}))=\left\\{\begin{array}[]{cc}d_{A}(a_{1},a_{2})&ifa_{1}\neq
a_{2}\\\ \min\\{2,d_{B}(b_{1},b_{2})\\}&ifa_{1}=a_{2}.\end{array}\right.$

And for edges we have:

$d_{A[B]}((a,b)(a_{1},b_{1}),(\acute{a},\acute{b})(\acute{a}_{1},\acute{b}_{1}))=$

$\min\left\\{\begin{array}[]{cccc}d_{A}(a,\acute{a})&ifa\neq\acute{a},&\min\\{2,d_{B}(b,\acute{b})\\}&ifa=\acute{a}\\\
d_{A}(a,\acute{a_{1}})&ifa\neq\acute{a_{1}},&\min\\{2,d_{B}(b,\acute{b_{1}})\\}&ifa=\acute{a_{1}}\\\
d_{A}(a_{1},\acute{a})&ifa_{1}\neq\acute{a},&\min\\{2,d_{B}(b_{1},\acute{b})\\}&ifa_{1}=\acute{a}\\\
d_{A}(a_{1},\acute{a}_{1})&ifa_{1}\neq\acute{a}_{1},&\min\\{2,d_{B}(b_{1},\acute{b}_{1})\\}&ifa_{1}=\acute{a}_{1}\end{array}\right\\}.$

######  Proposition 3.2.

Persume that $A$ and $B$ are graphs. Then there is the lexicographic product
$A[B]$, and $A[B]$ is $Gt-EDB$ if and only if $A$ is $Gt-EDB$ and also $B$ is
an empty graph.

###### Proof.

Let us choose $A$ as a connected graph, and $A[B]$ as a $Gt-EDBG$. Therefore,
it yields $A[B]$ is a bipartite graph based on Lemma 2.3. Since $B$ has at
least an edge, then it is understandable that $A[B]$ is not a bipartite graph.
Therefore, $B$ is an empty graph. Assume that $a_{1}$ and $a_{2}$ be adjacent
vertices in $A$, and $b_{1},b_{2}\in V(B)$ ($b_{1}$ and $b_{2}$ are not
necessarily distinct). We inspire of the definition of the lexicographic
product that $(a_{1},b_{1})$ and $(a_{2},b_{2})$ are adjacent, and using (2)
we observe that,

$M^{A[B]}_{(a_{1},b_{1})}=\\{(a_{1},b_{1})(a,b)\\}\cup\\{(a_{2},\acute{b})(a,b)|\acute{b}\in
V(B)\backslash\\{b_{2}\\}\\}$

$\cup\\{(a,b)(a_{1},\acute{b})|aa_{1}\in
M^{A}_{a_{1}}(a_{1}a_{2}),a\acute{a}\neq a_{1}a_{2},\acute{b}\in V(B)$},

where $a\acute{a}\neq a_{1}a_{2}$, that is $\\{a\neq a_{1},\acute{a}\neq
a_{2}\\}$ or $\\{\acute{a}\neq a_{2},\acute{a}\neq a_{1}\\}$.
Therefore,
$m^{A[B]}_{(a_{1},b_{1})}((a_{1},b_{1})(a_{2},b_{2}))=|V(B)|.m^{A}_{a_{1}}(a_{1}a_{2})$.
Similarly we attain
$m^{A[B]}_{(a_{2},b_{2})}((a_{1},b_{1})(a_{2},b_{2}))=|V(B)|.m^{A}_{a_{2}}(a_{1}a_{2})$.
For an empty graph $B$ it is simple to be realized, the lexicographic product
$A[B]$ is $Gt-EDB$ if and only if $A$ is $Gt-EDB$. The proof is completed. ∎

##  4\. $Gt-NEDB$ and $Gt-SEDB$ graphs

For every edge $f=\alpha\beta\in E(A)$ in a $Gt-EDB$ graph,
$m_{\alpha}^{A}(f)=tm_{\beta}^{A}(f)$. Consider that for every arbitrary edge
$f\in E(A)$, the constant of $m_{\beta}^{A}(f)$ is $\acute{\gamma}_{A}$. Then
$A$ is introduced as Generalized $t$-nicely edge distance-balanced ($Gt-
NEDB$). Each $Gt-NEDB$ graph is $Gt-EDB$ and every bipartite $Gt-EDB$ graph is
$Gt-NEDB$. At first, we start with the below lemma.

######  Lemma 4.1.

If $A$ be a connected $Gt-NEDB$ graph with diameter of $d$, then for each
arbitrary edge $f=\alpha\beta\in E(A)$, there will precisely be
$|E(A)|-(t+1)\acute{\gamma}_{A}$ edges of $A$, such that will be at the equal
distance from $\alpha$ and $\beta$. On the other hand,
$m_{0}^{A}(f)=|E(A)|-(t+1)\acute{\gamma}_{A}-1$.

###### Proof.

It is directly deduced from
$m_{\alpha}^{A}(f)+m_{\beta}^{A}(f)+m_{0}^{A}(f)=|E(A)|-1$. ∎

######  Lemma 4.2.

Suppose that $A$ is a connected $Gt-NEDB$ graph with diameter $d$. Then
$d-1\leq t\acute{\gamma}_{A}$.

###### Proof.

Let a path $a_{0},a_{1},a_{2},...,a_{d-1},a_{d}$. Also, we have
$h_{1},h_{2},h_{3},...,h_{d}$ as a series of edges, in which
$d_{A}(h_{1},h_{d})=d$. Now, let $h_{1}=a_{0}a_{1}$ be an arbitrary edge of
$A$. without loss of generality, we asuume that
$m_{a_{1}}^{A}(h_{1})=tm_{a_{0}}^{A}(h_{1})$. Hence,
$|\\{h_{2},...,h_{d}\\}|=d-1\leq m_{a_{1}}^{A}(h_{1})=tm_{a_{0}}^{A}(h_{1})$.
This shows that $d-1\leq t\acute{\gamma}_{A}$. ∎

######  Lemma 4.3.

Persume that $A$ and $B$ are graphs. Then, $A\square B$ is $Gt-NEDB$ if and
only if both $A$ and $B$ are $Gt-NEDB$ and

$|V(B)|.\acute{\gamma}_{A}+|E(B)|.\gamma_{A}=|V(A)|.\acute{\gamma}_{B}+|E(A)|.\gamma_{B}$.

###### Proof.

Consider that vertices $(a_{1},b_{1})$ and $(a_{2},b_{2})$ are in $V(A\square
B)$. According to the definition, either $b_{1}=b_{2}$ and $a_{1}$ and $a_{2}$
are adjacent in $A$, or $a_{1}=a_{2}$ and $b_{1}$ and $b_{2}$ are adjacent in
$B$. Let now $b_{1}=b_{2}$ and $a_{1}$ and $a_{2}$ be adjacent in $A$. Assume
now that $A\square B$ is $Gt-NEDB$. Using equations (3),(4),(5) and (6), we
obtain

$m_{a_{1}}^{A}(a_{1}a_{2}).|V(B)|+n_{a_{1}}^{A}(a_{1}a_{2}).|E(B)|=t\\{m_{a_{2}}^{A}(a_{1}a_{2}).|V(B)|+n_{a_{2}}^{A}(a_{1}a_{2}).|E(B)|\\}$=
$m_{b_{1}}^{B}(b_{1}b_{2}).|V(A)|+n_{b_{1}}^{B}(b_{1}b_{2}).|E(A)|=t\\{m_{b_{2}}^{B}(b_{1}b_{2}).|V(A)|+n_{b_{2}}^{B}(b_{1}b_{2}).|E(A)|\\}$=

$t(\acute{\gamma}_{A\square B}+\gamma_{A\square B})$,$\hskip 130.88284pt(7)$

in which

$\gamma_{A\square
B}=n_{a_{1}}^{A}(a_{1}a_{2}).|E(B)|=n_{a_{2}}^{A}(a_{1}a_{2}).|E(B)|$,

and

$\acute{\gamma}_{A\square
B}=m_{a_{1}}^{A}(a_{1}a_{2}).|V(B)|=m_{a_{2}}^{A}(a_{1}a_{2}).|V(B)|$.

We conclude that from (7) that $A$ and $B$ are both $Gt-NEDB$ and $Gt-NDB$ and
$|V(B)|.\acute{\gamma}_{A}+|E(B)|.\gamma_{A}=|V(A)|.\acute{\gamma}_{B}+|E(A)|.\gamma_{B}$
holds. Conversely, if $A$ and $B$ are both $Gt-NEDB$ and $Gt-NDB$ with
$|V(B)|.\acute{\gamma}_{A}+|E(B)|.\gamma_{A}=|V(A)|.\acute{\gamma}_{B}+|E(A)|.\gamma_{B}$.
By equations (3),(4),(5) and (6), $A\square B$ is a $Gt-NEDB$ graph. ∎

While for each arbitrary edge $f=\alpha\beta\in E(A)$ and every $i\in[1,d-1]$
in a graph $A$ with diameter $d$, we have
$|\acute{D}_{i-1}^{i}(f)|=t|\acute{D}_{i}^{i-1}(f)|+(t-1)$, then such graphs
are called $Gt-SEDB$. Let $t=1$. Then graph $A$ is a $SEDB$ graph.

######  Example 4.4.

A class of $Gt-SEDB$ graphs are complete bipartite graphs $K_{n,tn}$.

###### Proof.

Let $f=\alpha\beta\in E(K_{n,tn})$. Since $K_{n,tn}$ is bipartite with
diameter 2, we attain $\acute{D}_{1}^{2}(f)=(n-1)$ and
$\acute{D}_{2}^{1}(f)=t(n-1)$. Therefore,
$\acute{D}_{2}^{1}(f)=t\acute{D}_{1}^{2}(f)+(t-1)$ and it completes the
result. ∎

######  Lemma 4.5.

Consider that $A$ is a $Gt-SEDB$ graph with diameter 2. Then $A$ is a $Gt-EDB$
graph for $t\in\mathbb{N}$ and $t>1$.

###### Proof.

According to (1), we know that the graph $A$ is $Gt-EDB$ if and only if for
each arbitrary edge $f=\alpha\beta\in E(A)$ and each $i\in[1,d-1]$, we have

$\sum_{i=1}^{d-1}|\acute{D}_{i}^{i-1}(f)|=t\sum_{i=1}^{d-1}|\acute{D}_{i-1}^{i}(f)|+(t-1)$.

Let now $A$ be a $Gt-SEDB$ graph. Then for each $f=\alpha\beta\in E(A)$ and
each $i\in[1,d-1]$, it is obtained

$|\acute{D}_{i}^{i-1}(f)|=t|\acute{D}_{i-1}^{i}(f)|+(t-1)$.

Thus,

$\sum_{i=1}^{d-1}|\acute{D}_{i}^{i-1}(f)|=t\sum_{i=1}^{d-1}|\acute{D}_{i-1}^{i}(f)|+(t-1)(d-1)$.

If $d=2$, then we obtain

$|\acute{D}_{2}^{1}(f)|=t|\acute{D}_{1}^{2}(f)|+(t-1)$.

The proof is completed. ∎

Pay attention that If $t=1$, then every $SEDB$ graph is $EDB$.

## References

  * [1] A.Abedi, M. Alaeiyan, A. Hujdurović, K. Kutnar, Quasi-$\lambda$-distance-balanced graphs, Discrete Appl. Math. 227 (2017), 21-28.
  * [2] Z. Aliannejadi, A. Gilani, M. Alaeiyan, J. Asadpour, On some properties of edge quasi-distance-balanced graphs, Journal of Mathematical Extension. 16 (2022), 1-13.
  * [3] Z. Aliannejadi, M. Alaeiyan, A. Gilani, Strongly edge distance-balanced graph products, 7th International Conference on Combinatorics, Cryptography, Computer Science and Computing (2022).
  * [4] K. Balakrishman, M. Changat, I. Peterin, S.P. Špacapan, P. Šparal, A. R. Subhamathi, Strongly distance-balanced graph and graph product, European. J. Combin. 30 (2009), 1048-1053.
  * [5] D.M. Cvetkocić, M. Doob, H. Sachs, Spectra of Graphs-Theory and Application, Academic Press, New York. (1980).
  * [6] M. Faghani, A.R. Ashrafi, Revised and edge revised Szeged indices of graphs, Ars Math. Contemp. 7 (2014), 153-160.
  * [7] A. Graovac, M. Juvan, M. Petkovsek, A. Vesel, J. Zerovnik, The Szeged index of fasciagraphs, Match Common. Chem. 49 (2003), 47-66.
  * [8] I. Gutman, L. Popovic, P.V. Khadikar, S. Karmarkar, S. Joshi, M. Mandloi, Relations between Wiener and Szeged indices of monocyclic molecules, Match Common. Math. Comput. Chem. 35 (1997), 91-103.
  * [9] I. Gutman, A.R. Ashrafi, The edge version of the Szeged index, Croat. Chem. Acta. 81 (2008), 263-266.
  * [10] A. Hujdurović, On some properties of quasi-distance-balanced graphs, Bull. Aust. Math. Soc. 97 (2018), 177-184.
  * [11] A. Ilič, S. Klavžar, M. Milanović, On distance-balanced graphs, European. J. Combin. 31 (2010), 733-737.
  * [12] W. Imrich, S. Klavžar, Product Graphs: Structure and Recognition, Wiley, New York, USA, (2000).
  * [13] J. Jerebic, S. Klavžar, D.F. Rall, Distance-balanced graphs, Ann. Comb. 12 (2008), 71-79.
  * [14] M.H. Khalifeh, H. Yousefi-Azari, A.R. Ashrafi, S.G. Wagner, Some new results on distance-based graph invariants, European J. Combin. 30 (2009), 1149-1163.
  * [15] K. Kutnar, A. Malnič, D. Marušič, Š. Miklavič, Distance-balanced graphs:symmetry conditions,Discrete Math. 306 (2006), 1881-1894.
  * [16] K. Kutnar, A. Malnič, D. Marušič, Š. Miklavič, The strongley distance-balanced property of generalized petersen graphs, Ars Math. Contemp. 2 (2009), 41-47.
  * [17] K. Kutnar, Š. Miklavič, Nicely distance-balanced graphs,European. j. Combin. 39 (2014), 57-67.
  * [18] Š. Miklavič, P. Šparl, On the connectivity of bipartite distance-balanced graphs, European. J. Combin. 33 (2012), 237-247.
  * [19] M. Tavakoli, H. Yousefi-azari, A.R. Ashrafi, Note on Edge Distance-Balanced Graphs, Transaction on Combinatorics, University of Isfehan 1 no. 1 (2012), 1-6.

Zohreh Aliannejadi
Department of mathematics, Islamic Azad university, South-Branch, Tehran,
Iran.
<EMAIL_ADDRESS>

Mehdi Alaeiyan
Department of Mathematics, Department of mathematics, Iran university of
science and technology, Narmak, Tehran 16844. Iran.
Email<EMAIL_ADDRESS>

Alireza Gilani
Department of mathematics, Islamic Azad university, South-Branch, Tehran,
Iran.
Email: a_<EMAIL_ADDRESS>