11institutetext:  Computational Radiology Laboratory of the Department of
Radiology at Boston Children’s Hospital, and Harvard Medical School, Boston,
Massachusetts, USA

# Atlas-powered deep learning (ADL) -
application to diffusion weighted MRI

Davood Karimi and Ali Gholipour

###### Abstract

Deep learning has a great potential for estimating biomarkers in diffusion
weighted magnetic resonance imaging (dMRI). Atlases, on the other hand, are a
unique tool for modeling the spatio-temporal variability of biomarkers. In
this paper, we propose the first framework to exploit both deep learning and
atlases for biomarker estimation in dMRI. Our framework relies on non-linear
diffusion tensor registration to compute biomarker atlases and to estimate
atlas reliability maps. We also use nonlinear tensor registration to align the
atlas to a subject and to estimate the error of this alignment. We use the
biomarker atlas, atlas reliability map, and alignment error map, in addition
to the dMRI signal, as inputs to a deep learning model for biomarker
estimation. We use our framework to estimate fractional anisotropy and neurite
orientation dispersion from down-sampled dMRI data on a test cohort of 70
newborn subjects. Results show that our method significantly outperforms
standard estimation methods as well as recent deep learning techniques. Our
method is also more robust to stronger measurement down-sampling factors. Our
study shows that the advantages of deep learning and atlases can be
synergistically combined to achieve unprecedented accuracy in biomarker
estimation from dMRI data.

###### Keywords:

deep learning atlas estimation diffusion MRI.

##  1 Introduction

Diffusion weighted magnetic resonance imaging (dMRI) is the de-facto imaging
modality for probing the brain micro-structure in vivo. Biomarkers estimated
with dMRI are widely used to study normal and abnormal brain development [2,
16]. Estimation of these biomarkers from the dMRI signal entails solving
inverse problems that can range from non-linear least squares to complex non-
convex optimization problems [26, 14]. Accurate estimation depends on dense
high-quality measurements, which may be difficult or impossible to obtain in
many applications such as neonatal and pediatric studies.

Machine learning, and in particular deep learning (DL), has emerged as a
potent alternative to classical optimization-based estimation methods for dMRI
analysis. Rather than solving the inverse problem directly, DL methods aim to
learn the complex relation between the dMRI signal and the parameter of
interest from a set of training data. Recent studies have used DL models for
estimating diffusion tensor, diffusion kurtosis, multi-compartment models, and
fiber orientation distribution, to name just a few [12, 11, 20, 34, 24]. They
have shown that DL methods can accurately estimate dMRI biomarkers using far
fewer measurements than optimization-based techniques.

Statistical atlases are models of expected anatomy and anatomical variation.
They are routinely used in medical image analysis to characterize normal
anatomy and for identifying anatomical differences between normal and/or
abnormal subjects or populations. In neuroimaging, study-specific atlases
offer higher sensitivity and specificity of analysis than off-the-shelf
atlases. Several prior works have used dMRI atlases to study brain development
and maturation [21, 1], to assess the impact of neurological diseases [28,
15], and for various other purposes [13, 31].

However, no prior work has attempted to leverage DL and atlases for dMRI
biomarker estimation in a unified framework. Such a framework makes much
intuitive sense. While advanced DL methods are very effective in learning the
complex mapping between the dMRI signal and the biomarker of interest, atlases
can supply additional useful information that may be absent from the local
diffusion signal. The information contained in the atlas can be particularly
useful where the local diffusion signal is not adequate for accurate
estimation, such as when the number of measurements is low or the signal is
noisy. Some studies have shown that atlases can improve the performance of DL
methods for segmentation (e.g., [27, 8]). There have also been efforts to use
atlases or other sources of prior information such as distribution of noise
and parameter values with classical estimation methods [37]. For example,
Taquet et al. showed that estimation accuracy for diffusion compartment models
can be improved by using a population-informed prior in a graphical model
[33]. Anderson [3] used probabilistic priors on model parameters in a maximum
likelihood estimation framework for diffusion tensor estimation. Also using a
Bayesian estimation approach, Clayden et al. [6] found that prior information
was very influential for estimating axon radii from dMRI signal. In another
Bayesian work for NODDI estimation, Mozumder et al. [25] incorporated the
priors learned on 35 subjects. However, to the best of our knowledge, no prior
work has used atlases within a DL framework for dMRI biomarker estimation.

On the other hand, designing such a framework is not trivial. An atlas can
only represent the average of a population; it lacks the subtle but important
variations among the individuals in the population. Furthermore, the
correspondence between an atlas and an individual subject’s brain is complex
and spatially-varying. The atlas may match one subject much better than
another subject. Within the brain of a subject, the match between the atlas
and the subject depends on the local anatomy. It is not clear how these
information can be incorporated into a machine learning framework.

In this work, we propose a framework that brings together the power of DL
models and atlases for dMRI biomarker estimation. Our framework nicely
addresses the challenges outlined above by providing methods to compute the
reliability of the atlas and its degree of correspondence with a subject in a
spatially-varying manner. We use our proposed framework to estimate fractional
anisotropy and neurite orientation dispersion from down-sampled dMRI data of
newborn subjects and show that it is significantly more accurate than standard
estimation methods as well as recent DL techniques.

##  2 Materials and methods

###  2.1 Data

We used 300 dMRI scans from the Developing Human Connectome Project (DHCP)
dataset [5]. Each scan is from a different subject. We used 230 of the scans
for model development, including atlas creation and DL model training. We used
the same 230 scans to train the competing DL models (described below). We used
the remaining (completely independent) 70 scans to test our method and the
competing methods. The age of the subjects at the time of scan ranged between
31 and 45 gestational-equivalent weeks. We have focused on this dataset
because dMRI analysis for this age range is especially challenging due to
higher free water content, incomplete myelination, and overall lower data
quality [9].

###  2.2 Atlas development

Because of rapid brain development in the neonatal period, a single atlas
cannot represent the entire age range [30, 36]. Therefore, we built atlases of
the biomarkers of interest separately at one-week intervals. To build atlases
for GA of week 35, for example, we used subjects with GA between 34.5 and
35.5. For each GA, we used 10 scans to build the atlas. Our experience and
prior works [30, 21, 36] have shown that approximately 10 subjects are
sufficient for each GA.

In this study, we focus on two biomarkers: 1) Fractional anisotropy (FA) from
the diffusion tensor model, which is arguably the most widely used dMRI
biomarker, and 2) Orientation dispersion (OD) from the NODDI model [39], which
is a more complex model than diffusion tensor and which has been shown to be a
more specific biomarker in some studies [4, 29].

Given the dMRI signal volumes $\\{s_{i}\\}_{i=1}^{n}$ for $n$ subjects, we
compute the biomarkers of interest separately for each subject. Regardless of
which biomarkers are explored, we also always compute the diffusion tensor for
each subject because we use the tensors for accurate spatial alignment. This
is shown in Figure 1, where we have used $p$ and $q$ to denote the biomarkers
considered in this work and $T$ to denote the diffusion tensor. Given the set
$\\{T_{i}\\}_{i=1}^{n}$ of subject tensors, we compute a set of
transformations $\\{\Phi_{i}\\}_{i=1}^{n}$ that align these tensors into a
common atlas space and compute a mean tensor $\bar{T}$. This is done using an
iterative approach that computes a series of rigid, affine, and diffeomorphic
non-rigid registrations and updates $\bar{T}$ at every iteration, rather
similar to common practice [30, 21]. Specifically, we perform five iterations
of rigid registration, followed by five iterations of affine registration, and
finally ten iterations of non-rigid registration. The final registration
transform $\Phi_{i}$ for subject tensor $T_{i}$ is the composition of the
final affine and non-rigid transforms.

Figure 1: Left: Biomarkers of interest and diffusion tensor are computed for
each subject. Right: tensor-to-tensor registration is used to compute the
atlas and atlas confidence maps for each biomarker.

Using the transformations $\\{\Phi_{i}\\}_{i=1}^{n}$ computed via tensor-to-
tensor registration described above, we transform the biomarker maps into the
common atlas space. We compute the mean of the transformed biomarker maps as
the biomarker atlas. Furthermore, we compute the standard deviation of the
transformed biomarker maps as a measure of atlas confidence. Formally, for
biomarker $p$:

$\bar{p}=\text{mean}\Big{[}\Phi_{i}(p_{i})\Big{]}_{i=1}^{n},\hskip
14.22636ptp^{\sigma}=\text{std}\Big{[}\Phi_{i}(p_{i})\Big{]}_{i=1}^{n}$ (1)

where mean and standard deviation are computed in a voxel-wise manner across
subjects. Larger values of $p^{\sigma}$ indicate higher
variability/disagreement between the biomarker maps of subjects used to create
the atlas and, hence, _lower_ confidence. Figure 1 shows example atlas and
atlas confidence maps for FA and OD.

###  2.3 Biomarker estimation for an individual subject

Given the dMRI signal volume for an individual subject, $s_{k}$, we compute
the desired biomarker(s) for that subject via the following steps, which have
been shown for an example subject in Figure 2.

####  2.3.1 Step 1: Atlas-to-subject alignement.

In order to exploit the information encoded in an atlas, we need to accurately
register it to the subject space. As in our atlas development described above,
we use tensor-to-tensor registration for this alignement. Hence, regardless of
the biomarker(s) being estimated, we compute the diffusion tensor, $T_{k}$,
for the subject. We then compute affine+non-linear registration transforms
that map the atlas tensor $\bar{T}$ to the subject tensor, $T_{k}$. We use
$\Phi_{k}$ to denote the composition of these affine and non-linear
transforms. Hence, $\Phi_{k}$ describes the complete spatial alignment from
the atlas space to the subject. We denote the template tensor transformed to
the subject space with $\bar{T}_{k}=\Phi_{k}(\bar{T})$.

The registration between the template tensor and the subject tensor is never
perfect. The accuracy of this registration is a potentially important piece of
information because it indicates where the prior information is more reliable.
In other words, if at voxel $i$ the registration between $\bar{T}$ and $T_{k}$
is more accurate, then we have a higher incentive to trust the biomarker atlas
at the location of that voxel. The accuracy of this registration is spatially
varying and depends on at least three factors: 1) Accuracy of computation of
$T_{k}$, which in turn depends on the quality of the diffusion signal,
$s_{k}$, 2) Degree of similarity between the subject and the atlas, and 3)
Accuracy of the registration procedure that aligns the atlas to the subject.
We propose the following practical formulation to estimate the error of this
alignment:

$\Phi_{k}^{\text{err}}=\theta\big{(}\bar{T}_{k},T_{k}\big{)}.\exp\Big{(}-\min\big{[}\text{FA}(\bar{T}_{k}),\text{FA}(T_{k})\big{]}/\tau\Big{)}.$
(2)

This formulation has two terms. The first term, $\theta$, measures the angle
between the major eigenvectors of $\bar{T}_{k}$ and $T_{k}$. Clearly, smaller
angles indicate more accurate registration. The second term is introduced to
down-weight the registration accuracy for the location of less anisotropic
tensors such as gray matter and cerebrospinal fluid (CSF). For CSF, for
example, the tensor is spherical and the computed orientation of the major
eigenvector is not reliable but the eigenvectors of $T_{k}$ and $\bar{T}_{k}$
may be very close to each other by chance, hence artificially making $\theta$
very small. By using the minimum of the FAs, if either $\bar{T}_{k}$ or
$T_{k}$ has a low anisotropy, the second term will have a larger value. We set
$\tau=0.2$, which we found empirically to work well.

####  2.3.2 Step 2: Estimation using a DL model.

To compute a specific biomarker for an individual subject, standard techniques
only use the diffusion signal, $s_{k}$. In our framework, we also utilize the
prior information encoded in the biomarker atlas as described above.
Specifically, for estimating a biomarker $p$ for subject $k$, we have three
additional pieces of information; 1) the biomarker atlas registered to the
subject space $\Phi_{k}(\bar{p})$, 2) the biomarker atlas confidence
registered to the subject space $\Phi_{k}(\bar{p})$, and 3) registration error
$\Phi_{k}^{\text{err}}$. Since these three pieces of information are spatially
varying and aligned with the subject space, we simply concatenate them with
the diffusion signal to generate the input to our DL model, as shown in Figure
2. For example, for estimating the biomarker $p$ for subject $k$, the input to
the network is
$\Big{[}s_{k},\Phi_{k}(\mathbf{\bar{p}}),\Phi_{k}(\mathbf{p^{\sigma}}),\Phi_{k}^{\text{err}}\Big{]}$.
The estimation target is the ground-truth parameter map, $p^{\text{g.t.}}$,
computed as described below.

Figure 2: Our proposed method for estimating dMRI biomarkers using atlases and
DL. (a) We compute the diffusion tensor $T_{k}$ from the subject data. (b) We
compute the registration $\phi_{k}$ from the atlas to the subject using
tensor-to-tensor registration, and then the error of this registration
$\Phi_{k}^{\text{err}}$ using Eq. 2. (c) We align the biomarker atlases and
atlas confidence maps to the subject using $\Phi_{k}$. We then feed these
information, along with the dMRI signal $s_{k}$ to the DL model, which aims to
predict the ground truth biomarker values.

We used the UNet++ [41] as our DL architecture. As it has been recently
demonstrated for image segmentation in [17], our experience shows that the
exact network architecture is not critical. We used patches of size $48^{3}$
voxels. The number of network input channels was equal to the number of
diffusion measurements plus 3, as described above. We train a separate network
for each biomarker, hence the network had only one output channel to compute
the scalar biomarker of interest. We set the number of feature maps in the
first stage of the network to be 12, which was the largest possible on our GPU
memory. For training, we sampled blocks from random locations in the training
images. At test time, we used a sliding window with a stride of 16 voxels in
each dimension to estimate the biomarker for an input dMRI volume of arbitrary
size.

####  2.3.3 Compared methods and evaluation strategy.

For FA estimation, we compared our method with: 1) Constrained weighted linear
least-squares (CWLLS) [23], which is the standard method; 2) Deep-DTI [34],
which is a recent DL method based on CNNs. This method exploits the anatomical
T2 image, in addition to the diffusion signal, for estimation. Hence, for
Deep-DTI we also used the T2 image, which we registered to the dMRI volume,
and 3) Super-DTI [24], which is a recent DL method based on CNNs. For OD
estimation, we compared our method with: 1) Dmipy [10], which follows a
standard optimization-based estimation approach, 2) Microstructure Estimation
using a Deep Network (MEDN+) [38], which is a DL method that has been inspired
by AMICO [7] and significantly outperforms AMICO too, and 3) Another recent
CNN-based model [11], which we refer to as CNN-NODDI.

Each DHCP scan includes 20 $b=0$ measurements and 280 diffusion weighted
measurements at $b=400$ (n=64), $b=1000$ (n=88), and $b=2600$ (n=128). For FA
it is known that b values close to 1000 are optimal [18]. Hence, we used all
88 measurements in the $b=1000$ shell (along with all b=0 measurements) to
compute the ground truth using CWLLS. We then selected subsets of 12 and 6
measurements from this shell for each subject, which represent measurement
down-sampling factors of approximately 7 and 15, respectively. To select the 6
measurements, similar to [34, 19], we considered the 6 optimal diffusion
gradient directions proposed in [32] and chose the measurements that were
closest to those directions. To select the 12 measurements, we selected these
measurements to be close to uniformly spread on the sphere, as suggested in
[18, 20].

For OD (and NODDI), we used all 300 measurements to reconstruct the ground
truth using Dmipy [10]. For DL-based reconstruction, prior works have
typically used 20-60 measurements from more than one shell [38, 11]. Here, we
choose either 6 and 15 measurements from each of the $b=1000$ and $b=2600$
shells, for a total of 12 and 30 measurements, which represent downsampling
factors of approximately 24 and 10, respectively. We selected these
measurements to be close to uniformly spread on the sphere, using an approach
similar to [20]. For a fair comparison, for both FA and OD we used the same
down-sampled datasets for our method and for all competing techniques.

####  2.3.4 Implementation and training.

We used the DTI-TK software [40] to compute all registrations, including the
registrations needed in building the atlases. In creating the biomarker
atlases we used MRtrix [35] to compute the diffusion tensors and FA, and we
used Dmipy [10] to compute OD. We implemented all DL techniques in TensorFlow
1.14 under Python 3.7. We ran all algorithms on a Linux machine with 16 CPU
cores and an NVIDIA GeForce GTX 1080 GPU.

We trained our model by minimizing the $\ell_{2}$ norm between the predicted
and ground truth biomarker using Adam [22], a batch size of 10, and an initial
learning rate of $10^{-4}$ that was reduced by half every time the validation
loss did not decrease after a training epoch. For the competing methods, we
followed the training procedures recommended in the original papers.

##  3 Results and Discussion

###  3.1 Comparison with other techniques

Tables 1 and 2 show the reconstruction error for different methods for FA and
OD, respectively, computed on the 70 independent test subjects. For both FA
and OD, the DL methods were substantially more accurate than the standard
optimization-based techniques (i.e., CWLLS and Dmipy). The proposed method
achieved lower errors than the other DL methods for both FA and OD at both
down-sampling factors. We used paired t-tests to compare our method with the
other techniques. For both FA and OD and at both down-sampling factors, the
estimation error for our method was significantly lower than the error for any
of the compared techniques ($p<0.001$). Figure 3 shows example reconstruction
results for different techniques. Our method achieves lower errors across the
brain for both FA and OD.

Table 1: FA estimation errors for the proposed method and compared techniques. no. of measurements |  CWLLS |  Deep-DTI |  Super-DTI |  Proposed
---|---|---|---|---
$n=6$ |  $0.111\pm 0.014$ |  $0.048\pm 0.008$ |  $0.048\pm 0.005$ |  $\bm{0.040\pm 0.005}$
$n=12$ |  $0.053\pm 0.007$ |  $0.044\pm 0.006$ |  $0.043\pm 0.004$ |  $\bm{0.039\pm 0.005}$
Table 2: Comparison of OD estimation errors for the proposed method and competing techniques. no. of measurements |  Dmipy |  MEDN+ |  CNN-NODDI |  Proposed
---|---|---|---|---
$n=16$ |  $0.138\pm 0.032$ |  $0.064\pm 0.028$ |  $0.058\pm 0.011$ |  $\bm{0.047\pm 0.004}$
$n=30$ |  $0.096\pm 0.029$ |  $0.052\pm 0.030$ |  $0.050\pm 0.008$ |  $\bm{0.044\pm 0.005}$
Figure 3: Example FA (top) and OD (bottom) reconstructions by different
methods. In each of the four examples, the bottom row shows maps of absolute
estimation error.

For both FA and OD, our proposed method showed a smaller increase in error as
the down-sampling rate increased. Specifically, the FA estimation error for
our method increased by 2.5% as the number of measurements was reduced from 12
to 6, compared with 9% and 12% for Deep-DTI and Super-DTI, respectively. For
OD, the estimation error for our method increased by 7% as the number of
measurements was decreased from 30 to 16, compared with 23% and 16% for MEDN+
and CNN-NODDI, respectively.

The training time for our model was 10 hours, compared with 10-60 hours for
the other DL methods. To estimate FA or OD for a dMRI test volume, our model
required $73\pm 20$ seconds. Approximately 90% of this time was spent on
computing the atlas-to-subject registration. The average computation time for
the other DL methods ranged from 10 seconds for SuperDTI to 30 minutes for
MEDN+. The average computation times for CWLLS and Dmipy were, respectively,
20 seconds and 3.2 hours. Nonetheless, computation time for dMRI analysis is
not a critical factor since fast estimation is typically not a requirement.

###  3.2 Ablation experiments

Table 3 shows the results of some ablation experiments. We have performed
these experiments to show that the superior accuracy of our method compared to
the other methods is due to the incorporation of the atlas information as we
claim, rather than the differences in network architecture or training.
Results show that the error of our method increases significantly when we
discard the atlas information. When we only use the dMRI signal (last column
in Table 3), the FA accuracy of our method is very close to that of Deep-DTI
and Super-DTI (Table 1) and the OD accuracy of our method is slightly worse
than CNN-NODDI (Table 2). Ablation experiments further show that all three
extra pieces of information contribute to the model accuracy, but the
contribution of atlas and atlas confidence is larger than the contribution of
atlas-to-subject alignment error.

Table 3: Results of some ablation experiments. In the column headings, we use $p$ to denote either FA or OD. input to the model |  $\big{[}s_{k},\Phi_{k}(\mathbf{\bar{p}}),$ $\Phi_{k}(\mathbf{p^{\sigma}}),\Phi_{k}^{\text{err}}\big{]}$ |  $\big{[}s_{k},\Phi_{k}(\mathbf{\bar{p}}),$ $\Phi_{k}(\mathbf{p^{\sigma}})\big{]}$ |  $\big{[}s_{k},\Phi_{k}(\mathbf{\bar{p}})\big{]}$ |  $s_{k}$
---|---|---|---|---
FA, $n=6$ |  $\bm{0.040\pm 0.005}$ |  $0.041\pm 0.005$ |  $0.044\pm 0.006$ |  $0.047\pm 0.006$
OD, $n=16$ |  $\bm{0.047\pm 0.004}$ |  $0.050\pm 0.004$ |  $0.055\pm 0.011$ |  $0.060\pm 0.010$

##  4 Conclusions

We proposed a novel framework to incorporate atlases and DL for dMRI biomarker
estimation. We cannot claim that the design of our framework is “optimal”. For
example, atlas-to-subject alignment may be improved by incorporating
anatomical MRI information in addition to the diffusion tensor information.
Nonetheless, our work has shown, for the first time, that spatio-temporal
atlases can be used within a DL framework to achieve superior biomarker
estimation accuracy from down-sampled data.

## Acknowledgment

This study was supported in part by the National Institute of Biomedical
Imaging and Bioengineering and the National Institute of Neurological
Disorders and Stroke of the National Institutes of Health (NIH) under award
numbers R01EB031849, R01NS106030, and R01EB032366; in part by the Office of
the Director of the NIH under award number S10OD0250111; in part by the
National Science Foundation (NSF) under award 2123061; and in part by a
Technological Innovations in Neuroscience Award from the McKnight Foundation.
The content of this paper is solely the responsibility of the authors and does
not necessarily represent the official views of the NIH, NSF, or the McKnight
Foundation.

The DHCP dataset is provided by the developing Human Connectome Project, KCL-
Imperial-Oxford Consortium funded by the European Research Council under the
European Union Seventh Framework Programme (FP/2007-2013) / ERC Grant
Agreement no. [319456]. We are grateful to the families who generously
supported this trial.

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#  On the distance spectrum and distance-based topological indices of central
vertex-edge join of three graphs

Haritha<EMAIL_ADDRESS>Chithra A<EMAIL_ADDRESS>

1 Department of Mathematics, National Institute of Technology Calicut,
Calicut-673 601, Kerala, India

###### Abstract

Topological indices are molecular descriptors that describe the properties of
chemical compounds. These topological indices correlate specific physico-
chemical properties like boiling point, enthalpy of vaporization, strain
energy, and stability of chemical compounds. This article introduces a new
graph operation based on central graph called central vertex-edge join and
provides its results related to graph invariants like eccentric-connectivity
index, connective eccentricity index, total-eccentricity index, average
eccentricity index, Zagreb eccentricity indices, eccentric geometric-
arithmetic index, eccentric atom-bond connectivity index, and Wiener index.
Also, we discuss the distance spectrum of the central vertex-edge join of
three regular graphs. Furthermore, we obtain new families of $D$-equienergetic
graphs, which are non $D$-cospectral.

Keywords: Distance matrix; Distance eigenvalues; Distance equienergetic
graphs; Topological indices.
2020 Mathematics Subject Classification: 05C50, 05C76, 05C12.

##  1 Introduction

All graphs considered in this paper are simple connected and un-directed. Let
$G_{n}=(V(G_{n}),E(G_{n}))$ be a graph of order $n$ and size $m$, with
vertices $v_{i}$, $1\leq i\leq n$ and edges $e_{j}$, $1\leq j\leq m$. The
adjacency matrix of a graph $G_{n}$ is a square symmetric matrix
$A(G_{n})=(a_{ij})_{n\times n}$, where $a_{ij}=1$ when $v_{i}$ is adjacent to
$v_{j}$ and $a_{ij}=0$ otherwise. Denote the eigenvalues of $A(G_{n})$ by
$\theta_{1}\geq\theta_{2}\geq\cdots\geq\theta_{n}$, then the collection of all
the eigenvalues of $A(G_{n})$ is said to be the adjacency spectrum of $G_{n}$.
The incidence matrix of a graph $G_{n}$ is the $n\times m$ matrix
$Q(G_{n})=(q_{ij})_{n\times m}$, where $q_{ij}=1$ when $v_{i}$ is incident to
$e_{j}$ and $q_{ij}=0$ otherwise. The line graph $L(G_{n})$ of $G_{n}$ is a
graph with $V(L(G_{n}))=E(G_{n})$, in which two vertices in $L(G_{n})$ are
adjacent only if the corresponding edges in $G_{n}$ have an end vertex in
common. For a partitioned matrix we denote its equitable quotient matrix by
$M$ [3] and all one entry matrix by $J_{n\times m}$. We represent by $C_{n}$
the cycle, $K_{p,q}$ $\left(p+q=n\right)$ the complete bipartite graph, and
$K_{n}$ the complete graph each of order $n.$

The distance $d_{G}(v_{i},v_{j})$ between two vertices is the length of the
shortest path connecting them in $G_{n}$ [4]. In particular, for any $v_{i}\in
V(G_{n}),\;d_{G_{n}}(v_{i},v_{i})=0$. The eccentricity of a vertex $v_{i}\in
V(G_{n})$ is defined as $ec_{G_{n}}(v_{i})=max\\{d(v_{i},v_{j}):v_{j}\in
V(G_{n})\\}.$ The square symmetric matrix $D(G_{n})=(d_{ij})_{n\times
n}=d_{G_{n}}(v_{i},v_{j})$ is called the distance matrix of $G_{n}$ and its
eigenvalues are said to be the distance eigenvalues ($D$-eigenvalues). Let
$\mu_{1}\geq\mu_{2}\geq\cdots\geq\mu_{t}$ be the distinct $D$-eigenvalues of
$G_{n}$ with multiplicities $s_{1},s_{2},\ldots,s_{t}$. Then the distance
spectrum($D$-spectrum) of $D(G_{n})$ is given by
$Spec_{D}(G_{n})=\begin{pmatrix}\mu_{1}&\mu_{2}&\dots&\mu_{t}\\\
s_{1}&s_{2}&\dots&s_{t}\end{pmatrix}$. The non-isomorphic graphs of the same
order having identical $D$-spectrum are called $D$-cospectral graphs. The
transmission $Tr(v_{i})$ of a vertex $v_{i}$ is the sum of all distances from
$v_{i}$ to all other vertices in $G.$ See [2] for a survey of the distance
spectra of graphs.

In graph theory, operations on graphs play a significant role. For significant
works on distance spectra of graph operations see [11, 12, 5]. The central
graph $C(G_{n})$ of $G_{n}$ [1], is the graph obtained by adding new vertex to
each edge of $G_{n}$ and joining all the non adjacent vertices in $G_{n}$. In
[10], the authors defined the central vertex(edge) join of two graphs and
their adjacency spectra are investigated when $G_{n_{1}}$ and $G_{n_{2}}$ are
both regular graphs. Inspired by these works, we introduce a new graph
operation based on central graph and join.

######  Definition 1.1.

Let $G_{n_{1}}$, $G_{n_{2}}$ and $G_{n_{3}}$ be any three graphs of orders
$n_{1}$, $n_{2}$, $n_{3}$ and sizes $m_{1}$, $m_{2}$, $m_{3}$ respectively.
The central vertex-edge join (CVE-join) of $G_{n_{1}}$ with $G_{n_{2}}$ and
$G_{n_{3}}$, is the graph $G_{n_{1}}^{C}\triangleright(G_{n_{2}}^{V}\cup
G_{n_{3}}^{E})$, is obtained from $C(G_{n_{1}})$, $G_{n_{2}}$ and $G_{n_{3}}$
by joining each vertex of $G_{n_{1}}$ with every vertex of $G_{n_{2}}$ and
each vertex corresponding to the edges of $G_{n_{1}}$ with every vertex of
$G_{n_{3}}.$ The set of vertices in $C(G_{n_{1}})$ corresponding to the edges
of $G_{n_{1}}$ is denoted by $I(G_{n_{1}}).$

The order and size of $G_{n_{1}}^{C}\triangleright(G_{n_{2}}^{V}\cup
G_{n_{3}}^{E})$ are $n_{1}+m_{1}+n_{2}+n_{3}$ and
$m_{1}+m_{2}+m_{3}+n_{1}n_{2}+m_{1}n_{3}+\frac{n_{1}(n_{1}-1)}{2}$
respectively.

Figure $1.1$. $C_{4}^{C}\triangleright(K_{2}^{V}\cup K_{2}^{E}).$

In general, finding the $D$-spectrum of complicated networks is a difficult
problem. In Section $2$, we obtain the $D$-spectrum of CVE-join of three
regular graphs. Using the results mentioned above, we can find the $D$-spectra
of more complicated networks.

The distance energy ($D$-energy) [7] was introduced by Indulal et al., and it
is defined as $E_{D}(G_{n})=\sum_{i=1}^{n}|\mu_{i}|.$ Two graphs of the same
order having equal $D$-energy are called $D$-equienergetic graphs. In the
literature [9, 8, 12], we can see the constructions of different
$D$-equienergetic graphs. In Section $2$, we provide a new class of
$D$-equienergetic graphs of diameter $3$ using the above operation.

The study of graphs through their topological descriptors is beneficial for
deriving their underlying topologies. This process has many applications in
modeling physico-chemical properties of molecules by QSAR/QSPR analysis
(quantitative structure- activity relationships/quantitative structure-
property relationships). In the studies, QSARs and QSPRs, the topological
descriptors of graphs are used to approximate the biological activities and
properties of chemical compounds. Various topological indices of chemical
structures have emerged as a result of successful applications of QSAR/QSPR
studies. This article gives some distance-based topological indices of the new
graph operation central vertex-edge join of three graphs.
The eccentric connectivity index of $G_{n}$ is introduced in [13] as

$\xi^{c}(G_{n})=\sum_{v_{i}\in V(G_{n})}deg_{G_{n}}(v_{i})ec_{G_{n}}(v_{i}).$
(1)

The connective eccentricity index [15] of $G_{n}$ is

$\xi^{ce}(G_{n})=\sum_{v_{i}\in
V(G_{n})}\frac{deg_{G_{n}}(v_{i})}{ec_{G_{n}}(v_{i})}.$ (2)

By using the eccentric connectivity index, the total-eccentricity index [16]
is given by

$\tau(G_{n})=\sum_{v_{i}\in V(G_{n})}ec_{G_{n}}(v_{i}).$ (3)

The mean value of eccentricity of vertices in $V(G_{n})$ is called the average
eccentricity $aveg(G_{n})$ [17],

$aveg(G_{n})=\frac{1}{n}\sum_{v_{i}\in
V(G_{n})}ec_{G_{n}}(v_{i})=\frac{\tau(G_{n})}{n}.$ (4)

The Zagreb indices [18] of $G_{n}$ in terms of eccentricities is

$\displaystyle M_{1}(G_{n})$ $\displaystyle=\sum_{v_{i}\in
V(G_{n})}ec_{G_{n}}^{2}(v_{i}),$ (5) $\displaystyle M_{2}(G_{n})$
$\displaystyle=\sum_{v_{i}v_{j}\in
E(G_{n})}ec_{G_{n}}(v_{i})ec_{G_{n}}(v_{j}).$

The geometric-arithmetic index [19] in terms of eccentricity is given by

$GA_{4}(G_{n})=\sum_{v_{i}v_{j}\in
E(G_{n})}\frac{2\sqrt{ec_{G_{n}}(v_{i})ec_{G_{n}}(v_{j})}}{ec_{G_{n}}(v_{i})+ec_{G_{n}}(v_{j})}.$
(6)

The atom-bond connectivity index [20] in terms of eccentricity is given by

$ABC_{5}(G_{n})=\sum_{v_{i}v_{j}\in
E(G_{n})}\sqrt{\frac{ec_{G_{n}}(v_{i})+ec_{G_{n}}(v_{j})-2}{ec_{G_{n}}(v_{i})ec_{G_{n}}(v_{j})}}.$
(7)

The Wiener index $W(G)$ [14] of $G_{n}$ is defined as

$W(G_{n})=\frac{1}{2}\sum_{v_{i},v_{j}\in
V(G_{n})}d_{G}(v_{i},v_{j})=\frac{1}{2}\sum_{v_{i}\in V(G_{n})}Tr(v_{i}).$ (8)

Section $3$ gives the eccentric connectivity index, connective eccentricity
index, total-eccentricity index, average eccentricity index, Zagreb indices,
geometric-arithmetic index, atom-bond connectivity index and Wiener index of
central vertex-edge join of three graphs.

##  2 Distance spectrum of CVE-join of three regular graphs

In this section, we discuss the $D$-spectrum of
$G_{n_{1}}^{C}\triangleright(G_{n_{2}}^{V}\cup G_{n_{3}}^{E})$, where
$G_{n_{1}}$ is a triangle-free regular graph and $G_{n_{2}}$, $G_{n_{3}}$ are
two regular graphs.

######  Lemma 2.1.

[6] Let $G_{n}$ be a $k$-regular graph of order $n$ and size $m$ with
$Spec(G_{n})=\\{\theta_{1},\theta_{2},\ldots,\theta_{n}\\}$. Then

$Spec(L(G_{n}))=\begin{pmatrix}2k-2&\theta_{2}+k-2&\dots&\theta_{n}+k-2&-2\\\
1&1&\dots&1&m-n\\\ \end{pmatrix}.$

Moreover, $-2$ is an eigenvalue of $L(G_{n})$ with eigenvector $V$ if and only
if $Q(G_{n})V=0.$

######  Theorem 2.2.

For $i=1,2,3$, let $G_{n_{i}}$ be a $k_{i}$-regular graph of order $n_{i}$ and
size $m_{i}$, where $G_{n_{1}}$ is triangle-free. Then the $D$-eigenvalues of
$G_{n_{1}}^{C}\triangleright(G_{n_{2}}^{V}\cup G_{n_{3}}^{E})$ are the
following

  * (i)

$\frac{-3+\theta_{1j}\pm\sqrt{(\theta_{1j}+1)^{2}+4(\theta_{1j}+k_{1})}}{2}$,
$j=2,3,\ldots,n_{1}.$

  * (ii)

$-2$ with multiplicity $m_{1}-n_{1}.$

  * (iii)

$-\theta_{ij}-2,i=2,3$ and $j=2,3,\ldots,n_{i}.$

  * (iv)

eigenvalues of the matrix

$\begin{bmatrix}n_{1}-1+k_{1}&2m_{1}-k_{1}&n_{2}&2n_{3}\\\
2n_{1}-2&2m_{1}-2&2n_{2}&n_{3}\\\ n_{1}&2m_{1}&2n_{2}-k_{2}-2&3n_{3}\\\
2n_{1}&m_{1}&3n_{2}&2n_{3}-k_{3}-2\end{bmatrix}.$

###### Proof.

By a proper ordering of vertices in
$G_{n_{1}}^{C}\triangleright(G_{n_{2}}^{V}\cup G_{n_{3}}^{E})$, the distance
matrix $D(G_{n_{1}}^{C}\triangleright(G_{n_{2}}^{V}\cup G_{n_{3}}^{E}))$ can
be written as

$\begin{aligned} &D(G_{n_{1}}^{C}\triangleright(G_{n_{2}}^{V}\cup
G_{n_{3}}^{E}))\\\
&=\scriptsize{\begin{bmatrix}J_{n_{1}}-I_{n_{1}}+A(G_{n_{1}})&2J_{n_{1}\times
m_{1}}-Q(G_{n_{1}})&J_{n_{1}\times n_{2}}&2J_{n_{1}\times n_{3}}\\\
2J_{m_{1}\times n_{1}}-Q(G_{n_{1}})^{T}&2(J_{m_{1}}-I_{m_{1}})&2J_{m_{1}\times
n_{2}}&J_{m_{1}\times n_{3}}\\\ J_{n_{2}\times n_{1}}&2J_{n_{2}\times
m_{1}}&2(J_{n_{2}}-I_{n_{2}})-A(G_{n_{2}})&3J_{n_{2}\times n_{3}}\\\
2J_{n_{3}\times n_{1}}&J_{n_{3}\times m_{1}}&3J_{n_{3}\times
n_{2}}&2(J_{n_{3}}-I_{n_{3}})-A(G_{n_{3}})\end{bmatrix}}.\end{aligned}$

Let $k_{i}=\theta_{i1}\geq\theta_{i2}\geq\dots\geq\theta_{in_{i}}$ be the
$A$-eigenvalues of $G_{n_{i}}$, where $i=1,2,3.$ Let $U$ be a vector such that
$A(G_{n_{1}})U=\theta_{1j}U$ (for $j=2,3,\ldots,n_{1}$), then by Perron-
Frobenius theory $J_{1\times n_{1}}U=0$.
Now let, $\phi_{1}=\begin{bmatrix}tU\\\ Q(G_{n_{1}})^{T}U\\\ 0\\\
0\end{bmatrix}$ be an eigenvector of
$D(G_{n_{1}}^{C}\triangleright(G_{n_{2}}^{V}\cup G_{n_{3}}^{E}))$ for the
eigenvalue $\mu$, where $t$ is a real number. Then
$D(G_{n_{1}}^{C}\triangleright(G_{n_{2}}^{V}\cup
G_{n_{3}}^{E}))\phi_{1}=\mu\phi_{1}$.
That is,

$\displaystyle\begin{bmatrix}J_{n_{1}}-I_{n_{1}}+A(G_{n_{1}})&2J_{n_{1}\times
m_{1}}-Q(G_{n_{1}})&J_{n_{1}\times n_{2}}&2J_{n_{1}\times n_{3}}\\\
2J_{m_{1}\times n_{1}}-Q(G_{n_{1}})^{T}&2(J_{m_{1}}-I_{m_{1}})&2J_{m_{1}\times
n_{2}}&J_{m_{1}\times n_{3}}\\\ J_{n_{2}\times n_{1}}&2J_{n_{2}\times
m_{1}}&2(J_{n_{2}}-I_{n_{2}})-A(G_{n_{2}})&3J_{n_{2}\times n_{3}}\\\
2J_{n_{3}\times n_{1}}&J_{n_{3}\times m_{1}}&3J_{n_{3}\times
n_{2}}&2(J_{n_{3}}-I_{n_{3}})-A(G_{n_{3}})\end{bmatrix}\begin{bmatrix}tU\\\
Q(G_{n_{1}})^{T}U\\\ 0\\\ 0\end{bmatrix}$
$\displaystyle=\mu\begin{bmatrix}tU\\\ Q(G_{n_{1}})^{T}U\\\ 0\\\
0\end{bmatrix}$
$\displaystyle\begin{bmatrix}(-t+\theta_{1j}t-\theta_{1j}-k_{1})U\\\
(-t-2)Q(G_{n_{1}})^{T}U\\\ 0\\\ 0\end{bmatrix}=\begin{bmatrix}\mu tU\\\ \mu
Q(G_{n_{1}})^{T}U\\\ 0\\\ 0\end{bmatrix}.$

Therefore,

$\displaystyle\quad-t+\theta_{1j}t-\theta_{1j}-k_{1}$ $\displaystyle=\mu t$
$\displaystyle\quad-t-2$ $\displaystyle=\mu$

from this, we get

$t^{2}+(\theta_{1j}+1)t-(\theta_{1j}+k_{1})=0$

so that $t$ has two values

$\displaystyle
t_{1}=\frac{-(\theta_{1j}+1)+\sqrt{(\theta_{1j}+1)^{2}+4(\theta_{1j}+k_{1})}}{2},$
$\displaystyle
t_{2}=\frac{-(\theta_{1j}+1)-\sqrt{(\theta_{1j}+1)^{2}+4(\theta_{1j}+k_{1})}}{2}.$

That is, $-t_{1}-2$ and $-t_{2}-2$ are eigenvalues of
$D(G_{n_{1}}^{C}\triangleright(G_{n_{2}}^{V}\cup G_{n_{3}}^{E}))$
corresponding to the eigenvalues $\theta_{1j}\neq k_{1}$ of $A(G_{n_{1}})$.
Thus, we get $2(n_{1}-1)$ eigenvalues of
$D(G_{n_{1}}^{C}\triangleright(G_{n_{2}}^{V}\cup G_{n_{3}}^{E}))$.
By Lemma $2.1$, we get $Q(G_{n_{1}})V=0$, where $V$ is an eigenvector of
$L(G_{n_{1}})$ corresponding to the eigenvalue $-2$ (repeated $m_{1}-n_{1}$
times).
If $\phi_{2}=\begin{bmatrix}0\\\ V\\\ 0\\\ 0\end{bmatrix}$, then we have

$D(G_{n_{1}}^{C}\triangleright(G_{n_{2}}^{V}\cup
G_{n_{3}}^{E}))\phi_{2}=-2\phi_{2}.$

Therefore, $-2$ is an eigenvalue of
$D(G_{n_{1}}^{C}\triangleright(G_{n_{2}}^{V}\cup G_{n_{3}}^{E}))$ repeated
$m_{1}-n_{1}$ times.
Consider the eigenvalue $\theta_{2j}$ (for $j=2,3,\ldots,n_{2})$ of
$G_{n_{2}}$ with an eigenvector $W$ such that $J_{1\times n_{2}}W=0$.
Put $\phi_{3}=\begin{bmatrix}0\\\ 0\\\ W\\\ 0\end{bmatrix}$, then we have

$D(G_{n_{1}}^{C}\triangleright(G_{n_{2}}^{V}\cup
G_{n_{3}}^{E}))\phi_{3}=-(\theta_{2j}+2)\phi_{3}.$

This shows that $-(\theta_{2j}+2)$ are eigenvalues of
$D(G_{n_{1}}^{C}\triangleright(G_{n_{2}}^{V}\cup G_{n_{3}}^{E}))$
corresponding to the eigenvalues $\theta_{2j}\neq k_{2}$ of $A(G_{n_{2}})$.
Similarly, for each eigenvalue $\theta_{3j}$ (for $j=2,3,\ldots,n_{3}$) of
$G_{n_{3}}$, we get an eigenvalue $-(\theta_{3j}+2)$ of
$D(G_{n_{1}}^{C}\triangleright(G_{n_{2}}^{V}\cup G_{n_{3}}^{E}))$ with
eigenvector $\begin{bmatrix}0\\\ 0\\\ 0\\\ X\end{bmatrix}.$ Thus, we get
$2n_{1}-2+m_{1}-n_{1}+n_{2}-1+n_{3}-1=n_{1}+m_{1}+n_{2}+n_{3}-4$ eigenvalues
of $D(G_{n_{1}}^{C}\triangleright(G_{n_{2}}^{V}\cup G_{n_{3}}^{E}))$. All the
other eigenvalues of $D(G_{n_{1}}^{C}\triangleright(G_{n_{2}}^{V}\cup
G_{n_{3}}^{E}))$ are the eigenvalues of the equitable quotient matrix

$M=\begin{bmatrix}n_{1}-1+k_{1}&2m_{1}-k_{1}&n_{2}&2n_{3}\\\
2n_{1}-2&2m_{1}-2&2n_{2}&n_{3}\\\ n_{1}&2m_{1}&2n_{2}-k_{2}-2&3n_{3}\\\
2n_{1}&m_{1}&3n_{2}&2n_{3}-k_{3}-2\end{bmatrix}.$

∎

Using the above theorem we present some new families of $D$-equienergetic
graphs.

For a fixed $a\in\mathbb{N}$, let $\mathcal{P}_{a}$ be the family of all
integer partitions of ${}^{\prime}a^{\prime}$, denoted by
$p_{1},p_{2},\ldots,p_{s}$, each of size at least $3$. For
$P=\\{p_{1},p_{2},\ldots,p_{s}\\}\in\mathcal{P}_{a}$, let $C_{P}$ be the union
of cycles with vertices $p_{1},p_{2},\ldots,p_{s}$.

######  Theorem 2.3.

Let $H_{n_{1}}$ be a $k_{1}$-regular triangle free graph of order $n_{1}$ and
size $m_{1}$ and $H_{n_{2}}$ be a $k_{2}$\- regular graph of order $n_{2}$ and
size $m_{2}$ with the least eigenvalue being at least $-2$. Then for every
$a\in\mathbb{N}$, $H_{n_{1}}^{C}\triangleright(C_{P}^{V}\cup H_{n_{2}}^{E})$
forms a family of $D$-equienergetic graphs.

###### Proof.

Let $k_{1}=\theta_{1}\geq\theta_{2}\geq\cdots\geq\theta_{n_{1}}$,
$2=\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{a}$ and
$k_{2}=\gamma_{1}\geq\gamma_{2}\geq\cdots\geq\gamma_{n_{2}}$ be the
$A$-eigenvalues of $H_{n_{1}}$, $C_{P}$ and $H_{n_{2}}$ respectively. From
Theorem $2.2$, the $D$-eigenvalues of
$H_{n_{1}}^{C}\triangleright(C_{P}^{V}\cup H_{n_{2}}^{E})$ are
$\frac{-3+\theta_{j}\pm\sqrt{(\theta_{j}+1)^{2}+4(\theta_{j}+k_{1})}}{2}$,
$j=2,3,\ldots,n_{1}$; $-2$ with multiplicity $m_{1}-n_{1}$;
$-\lambda_{j^{{}^{\prime}}}-2$, $j^{{}^{\prime}}=2,3,\ldots,a$;
$-\gamma_{j^{{}^{\prime\prime}}}-2,j^{{}^{\prime\prime}}=2,3,\ldots,n_{2}$ and
along with the eigenvalues of the equitable quotient matrix $M$ of
$H_{n_{1}}^{C}\triangleright(C_{P}^{V}\cup H_{n_{2}}^{E})$,

$\begin{bmatrix}n_{1}+k_{1}-1&2m_{1}-k_{1}&a&2n_{2}\\\
2n_{1}-2&2m_{1}-2&2a&n_{2}\\\ n_{1}&2m_{1}&2a-4&3n_{2}\\\
2n_{1}&m_{1}&3a&2n_{2}-k_{2}-2\end{bmatrix}$

for every partition $P$ of ${}^{\prime}a^{\prime}$.
Since $\lambda_{j^{{}^{\prime}}}\geq-2$, for $j^{{}^{\prime}}=2,3,\ldots,a$,
we have $-(\lambda_{j^{{}^{\prime}}}+2)\leq 0$. Similarly, we get
$-(\gamma_{j^{{}^{\prime\prime}}}+2)\leq
0,j^{{}^{\prime\prime}}=2,3,\ldots,n_{2}.$
Therefore,

$\displaystyle\sum_{j^{{}^{\prime}}=2}^{a}|{-\lambda_{j^{{}^{\prime}}}-2}|+\sum_{j^{{}^{\prime\prime}}=2}^{n_{2}}|{-\gamma_{j^{{}^{\prime\prime}}}-2}|$
$\displaystyle=\sum_{j^{{}^{\prime}}=2}^{a}(\lambda_{j^{{}^{\prime}}}+2)+\sum_{j^{{}^{\prime\prime}}=2}^{n_{2}}(\gamma_{j^{{}^{\prime\prime}}}+2)$
$\displaystyle=-2+2(a-1)-k_{2}+2(n_{2}-1)$ $\displaystyle=2a-k_{2}+2n_{2}-6.$

Hence for every partition of ${}^{\prime}a^{\prime}$, the energy remains the
same.

∎

##  3 Distance-based topological indices of CVE-join of three graphs

For three graphs $G_{n_{1}},G_{n_{2}},\text{and}\;G_{n_{3}}$, their CVE-join
$\mathcal{G}=G_{n_{1}}^{C}\vee(G_{n_{2}}^{V}\cup G_{n_{2}}^{E})$ has the
following properties,

$deg_{\mathcal{G}}(v)=\begin{cases}n_{1}+n_{2}-1,&\text{if $v\in
V(G_{n_{1}})$}\\\ n_{3}+2,&\text{if $v\in I(G_{n_{1}})$}\\\
deg_{G_{n_{2}}}(v)+n_{1},&\text{if $v\in V(G_{n_{2}})$}\\\
deg_{G_{n_{3}}}(v)+m_{1},&\text{if $v\in V(G_{n_{3}})$}\end{cases}.$ (9)

Now we give the eccentricity of vertices in $\mathcal{G}$ by two different
cases;
Case 1. If $G_{n_{1}}$ is a triangle free graph,

$ec_{\mathcal{G}}(v)=\begin{cases}2,&\text{if $v\in V(G_{n_{1}})$or $v\in
I(G_{n_{1}})$}\\\ 3,&\text{if $v\in V(G_{n_{2}})$ or $v\in
V(G_{n_{3}})$}\end{cases}.$ (10)

Case 2. If $G_{n_{1}}$ is a graph which is not triangle free, then

$ec_{\mathcal{G}}(v)=3,\;\text{for every vertex}\;v\in V(\mathcal{G})$ (11)

This section gives some graph invariants of
$\mathcal{G}=G_{n_{1}}^{C}\vee(G_{n_{2}}^{V}\cup G_{n_{2}}^{E})$ like Wiener
index, Zagreb indices, etc.

######  Theorem 3.1.

For $i=1,2,3$, let $G_{n_{i}}$ be a $k_{i}$\- regular graph of order $n_{i}$
and size $m_{i}$, where $G_{n_{1}}$ is triangle-free. Then the Wiener index of
$\mathcal{G}=G_{n_{1}}^{C}\vee(G_{n_{2}}^{V}\cup G_{n_{2}}^{E})$ is given by,
$W(\mathcal{G})=\frac{1}{2}(n_{1}^{2}-n_{1}+2(n_{2}^{2}+n_{3}^{2}+n_{1}n_{2}-n_{2}-n_{3}+m_{1}n_{3}+m_{1}^{2})+4(n_{1}n_{3}+m_{1}n_{1}+m_{1}n_{2}-m_{1})-(n_{2}k_{2}+n_{3}k_{3})+6n_{2}n_{3}).$

###### Proof.

Let the vertex set of $\mathcal{G}=G_{n_{1}}^{C}\vee(G_{n_{2}}^{V}\cup
G_{n_{2}}^{E})$ be
$V(\mathcal{G})=\\{v_{1},v_{2},\ldots,v_{n_{1}},v_{e_{1}},v_{e_{2}},\ldots,v_{e_{m_{1}}},u_{1},u_{2},\ldots,u_{n_{2}},w_{1},w_{2},\ldots,w_{n_{3}}\\}$,
where $v_{1},v_{2},\ldots,v_{n_{1}}$ are the vertices in $G_{n_{1}}$,
$v_{e_{1}},v_{e_{2}},\ldots,v_{e_{m_{1}}}$ are the vertices corresponding to
the edges of $G_{n_{1}}$ in $C(G_{n_{1}})$, $u_{1},u_{2},\ldots,u_{n_{2}}$ are
the vertices in $G_{n_{2}}$, and $w_{1},w_{2},\ldots,w_{n_{3}}$ are the
vertices in $G_{n_{3}}$. Then

$\displaystyle Tr(v_{i})$
$\displaystyle=n_{1}+n_{2}+2n_{3}+2m_{1}-1,\;i=1,\ldots,n_{1}.$ $\displaystyle
Tr(v_{e_{j}})$
$\displaystyle=2n_{1}+2n_{2}+n_{3}+2m_{1}-4,\;j=1,\ldots,m_{1}.$
$\displaystyle Tr(u_{k})$
$\displaystyle=n_{1}+2n_{2}+3n_{3}+2m_{1}-k_{2}-2,\;k=1,\ldots,n_{2}.$
$\displaystyle Tr(w_{l})$
$\displaystyle=2n_{1}+3n_{2}+2n_{3}+m_{1}-k_{3}-2,\;l=1,\ldots,n_{3}.$

From (8),

$\displaystyle W(\mathcal{G})$ $\displaystyle=\frac{1}{2}\sum_{v_{i}\in
V(\mathcal{G})}Tr(v_{i})$
$\displaystyle=\frac{1}{2}(n_{1}^{2}-n_{1}+2(n_{2}^{2}+n_{3}^{2}+n_{1}n_{2}-n_{2}-n_{3}+m_{1}n_{3}+m_{1}^{2})$
$\displaystyle\;\;\;+4(n_{1}n_{3}+m_{1}n_{1}+m_{1}n_{2}-m_{1})-(n_{2}k_{2}+n_{3}k_{3})+6n_{2}n_{3}).$

∎

######  Theorem 3.2.

For $i=1,2,3$, let $G_{n_{i}}$ be a graph of order $n_{i}$ and size $m_{i}$,
then

  * (i)

$\xi^{c}(\mathcal{G})=\begin{cases}2n_{1}^{2}+5n_{1}n_{2}+5m_{1}n_{3}-2n_{1}+4m_{1}+6m_{2}+6m_{3},&\text{if
$G_{n_{1}}$ is triangle-free}\\\
3n_{1}^{2}+6n_{1}n_{2}+6m_{1}n_{3}-3n_{1}+6(m_{1}+m_{2}+m_{3}),&\text{otherwise}\end{cases}.$

  * (ii)

$\xi^{ce}(\mathcal{G})=\begin{cases}\frac{1}{6}\left(3n_{1}^{2}+5(n_{1}n_{2}+m_{1}n_{3})+6m_{1}+4(m_{2}+m_{3})-3n_{1}\right),&\text{if
$G_{n_{1}}$ is triangle-free}\\\
\frac{1}{3}\left(n_{1}^{2}+2(n_{1}n_{2}+m_{1}n_{3})-n_{1}+2(m_{1}+m_{2}+m_{3})\right),&\text{otherwise}\end{cases}.$

###### Proof.

  * (i)

Let $\mathcal{G}=G_{n_{1}}^{C}\vee(G_{n_{2}}^{V}\cup G_{n_{2}}^{E})$, where
$G_{n_{1}}$ is triangle-free, then from (1) by using (9) and (10), we have

$\displaystyle\xi^{c}(\mathcal{G})$ $\displaystyle=\sum_{v\in
V(G_{n_{1}})}2(n_{1}+n_{2}-1)+\sum_{v\in I(G_{n_{1}})}2(n_{3}+2)$
$\displaystyle\;\;\;+\sum_{v\in
V(G_{n_{2}})}3(deg_{G_{n_{2}}}(v)+n_{1})+\sum_{v\in
V(G_{n_{3}})}3(deg_{G_{n_{3}}}(v)+m_{1})$
$\displaystyle=2n_{1}^{2}+5n_{1}n_{2}+5m_{1}n_{3}-2n_{1}+4m_{1}+6m_{2}+6m_{3}.$

Similarly in the case when $G_{n_{1}}$ is not a triangle-free graph by using
(9) and (10) we get,

$\xi^{c}(\mathcal{G})=3n_{1}^{2}+6n_{1}n_{2}+6m_{1}n_{3}-3n_{1}+6(m_{1}+m_{2}+m_{3}).$

  * (ii)

Let $G_{n_{1}}$ be a triangle-free graph. Then from (2), by using (9) and
(10), we get

$\displaystyle\xi^{ce}(\mathcal{G})$ $\displaystyle=\sum_{v\in
V(G_{n_{1}})}\frac{n_{1}+n_{2}-1}{2}+\sum_{v\in
I(G_{n_{1}})}\frac{n_{3}+2}{2}+\sum_{v\in
V(G_{n_{2}})}\frac{deg_{G_{n_{2}}}+n_{1}}{3}$ $\displaystyle\;\;\;+\sum_{v\in
V(G_{n_{3}})}\frac{deg_{G_{n_{3}}}+m_{1}}{3}$
$\displaystyle=\frac{1}{6}\left(3n_{1}^{2}+5(n_{1}n_{2}+m_{1}n_{3})+6m_{1}+4(m_{2}+m_{3})-3n_{1}\right).$

Similarly in the case when $G_{n_{1}}$ is not a triangle-free graph by using
(9) and (10) we get,

$\xi^{ce}(\mathcal{G})=\frac{1}{3}\left(n_{1}^{2}+2(n_{1}n_{2}+m_{1}n_{3})-n_{1}+2(m_{1}+m_{2}+m_{3})\right).$

∎

######  Theorem 3.3.

For $i=1,2,3$, let $G_{n_{i}}$ be a graph of order $n_{i}$ and size $m_{i}$,
then

  * (i)

$\tau(\mathcal{G})=\begin{cases}2n_{1}+2m_{1}+3n_{2}+3n_{3},&\text{if
$G_{n_{1}}$ is triangle-free}\\\
3(n_{1}+n_{2}+n_{3}+m_{1}),&\text{otherwise}\end{cases}.$

  * (ii)

$aveg(\mathcal{G})=\begin{cases}\frac{2n_{1}+2m_{1}+3n_{2}+3n_{3}}{n_{1}+n_{2}+n_{3}+m_{1}},&\text{if
$G_{n_{1}}$ is triangle-free}\\\ 3,&\text{otherwise}\end{cases}.$

###### Proof.

  * (i)

Consider $\mathcal{G}=G_{n_{1}}^{C}\vee(G_{n_{2}}^{V}\cup G_{n_{2}}^{E})$,
when $G_{n_{1}}$ is triangle-free, then from (3) by using (10), we get

$\displaystyle\tau(\mathcal{G})$ $\displaystyle=2n_{1}+2m_{1}+3n_{2}+3n_{3}.$

Similarly when $G_{n_{1}}$ is not a triangle-free graph we get

$\displaystyle\tau(\mathcal{G})$ $\displaystyle=3(n_{1}+n_{2}+n_{3}+m_{1}).$

  * (ii)

From (4), we get

$\displaystyle aveg(\mathcal{G})$
$\displaystyle=\frac{\tau(\mathcal{G})}{n_{1}+n_{2}+n_{3}+m_{1}}$
$\displaystyle=\begin{cases}\frac{2n_{1}+2m_{1}+3n_{2}+3n_{3}}{n_{1}+n_{2}+n_{3}+m_{1}},&\text{if
$G_{n_{1}}$ is triangle-free}\\\ 3,&\text{otherwise}\end{cases}.$

∎

######  Theorem 3.4.

For $i=1,2,3$, let $G_{n_{i}}$ be a graph of order $n_{i}$ and size $m_{i}$,
then

  * (i)

$M_{1}(\mathcal{G})=\begin{cases}4n_{1}+4m_{1}+9n_{2}+9n_{3},&\text{if
$G_{n_{1}}$ is triangle-free}\\\
9(n_{1}+n_{2}+n_{3}+m_{1}),&\text{otherwise}\end{cases}.$

  * (ii)

$M_{2}(\mathcal{G})=\begin{cases}4m_{1}+9(m_{2}+m_{3})+2(n_{1}^{2}-n_{1})+6(n_{1}n_{2}+m_{1}n_{3}),&\text{if
$G_{n_{1}}$ is triangle-free}\\\
9\left(m_{1}+m_{2}+m_{3}+n_{1}n_{2}+m_{1}n_{3}+\frac{n_{1}^{2}-n_{1}}{2}\right),&\text{otherwise}\end{cases}.$

###### Proof.

  * (i)

Let $\mathcal{G}=G_{n_{1}}^{C}\vee(G_{n_{2}}^{V}\cup G_{n_{2}}^{E})$, where
$G_{n_{1}}$ is triangle-free, then from (5) by using (10) we get

$\displaystyle M_{1}(\mathcal{G})$ $\displaystyle=\sum_{v\in
V(G_{n_{1}})}2^{2}+\sum_{v\in I(G_{n_{1}})}2^{2}+\sum_{v\in
V(G_{n_{2}})}3^{2}+\sum_{v\in V(G_{n_{3}})}3^{2}$
$\displaystyle=4n_{1}+4m_{1}+9n_{2}+9n_{3}.$

Similarly when $G_{n_{1}}$ is not a triangle-free graph we get,

$M_{1}(\mathcal{G})=9(n_{1}+n_{2}+n_{3}+m_{1}).$

  * (ii)

Let $G_{n_{1}}$ be a triangle-free graph. Then from (5) by using (10) we get

$\displaystyle M_{2}(\mathcal{G})$ $\displaystyle=\sum_{uv\in
E(G_{n_{2}})}3\times 3+\sum_{uv\in E(G_{n_{3}})}3\times
3+\sum_{{\begin{subarray}{c}u\in V(G_{n_{1}}),\\\ v\in
V(G_{n_{2}})\end{subarray}}}2\times 3$
$\displaystyle\;\;+\sum_{{\begin{subarray}{c}u\in I(G_{n_{1}}),\\\ v\in
V(G_{n_{3}})\end{subarray}}}2\times 3+\sum_{{\begin{subarray}{c}uv\in
E(\mathcal{G}),\\\ u\in V(G_{n_{1}}),v\in I(G_{n_{1}})\end{subarray}}}2\times
2+\sum_{{\begin{subarray}{c}uv\in E(\mathcal{G}),\\\ u,v\in
V(G_{n_{1}})\end{subarray}}}2\times 2$
$\displaystyle=4m_{1}+9m_{2}+9m_{3}+2n_{1}^{2}-2n_{1}+6n_{1}n_{2}+6m_{1}n_{3}.$

Similarly when $G_{n_{1}}$ is not a triangle-free graph we get,

$M_{2}(\mathcal{G})=9\left(m_{1}+m_{2}+m_{3}+n_{1}n_{2}+m_{1}n_{3}+\frac{n_{1}^{2}-n_{1}}{2}\right).$

∎

######  Theorem 3.5.

For $i=1,2,3$, let $G_{n_{i}}$ be a graph of order $n_{i}$ and size $m_{i}$,
then

  * (i)

$GA_{4}(\mathcal{G})=\begin{cases}m_{1}+m_{2}+m_{3}+\frac{2\sqrt{6}}{5}(n_{1}n_{2}+m_{1}n_{3})+\frac{n_{1}^{2}-n_{1}}{2},&\text{if
$G_{n_{1}}$ is triangle-free}\\\
m_{1}+m_{2}+m_{3}+n_{1}n_{2}+m_{1}n_{3}+\frac{n_{1}^{2}-n_{1}}{2},&\text{otherwise}\end{cases}.$

  * (ii)

$ABC_{5}(\mathcal{G})=\begin{cases}\frac{2}{3}(m_{2}+m_{3})+\frac{1}{\sqrt{2}}\left(n_{1}n_{2}+m_{1}n_{3}+m_{1}+\frac{n_{1}(n_{1}-1)}{2}\right),&\text{if
$G_{n_{1}}$ is triangle-free}\\\
\frac{2}{3}(m_{1}+m_{2}+m_{3}+n_{1}n_{2}+m_{1}n_{3})+\frac{n_{1}(n_{1}-1)}{3},&\text{otherwise}\end{cases}.$

###### Proof.

  * (i)

Let $\mathcal{G}=G_{n_{1}}^{C}\vee(G_{n_{2}}^{V}\cup G_{n_{2}}^{E})$, where
$G_{n_{1}}$ is triangle-free, then from (6) by using (10) we have

$\displaystyle GA_{4}(\mathcal{G})$ $\displaystyle=\sum_{uv\in
E(G_{n_{2}})}2\frac{\sqrt{3\times 3}}{3+3}+\sum_{uv\in
E(G_{n_{3}})}2\frac{\sqrt{3\times 3}}{3+3}+\sum_{{\begin{subarray}{c}u\in
V(G_{n_{1}}),\\\ v\in V(G_{n_{2}})\end{subarray}}}2\frac{\sqrt{2\times
3}}{2+3}$ $\displaystyle+\sum_{{\begin{subarray}{c}u\in I(G_{n_{1}}),\\\ v\in
V(G_{n_{3}})\end{subarray}}}2\frac{\sqrt{2\times
3}}{2+3}+\sum_{{\begin{subarray}{c}u\in V(G_{n_{1}}),\\\ v\in I(G_{n_{1}}),\\\
uv\in E(\mathcal{G})\end{subarray}}}2\frac{\sqrt{2\times
2}}{2+2}+\sum_{{\begin{subarray}{c}uv\in E(\mathcal{G}),\\\ u,v\in
V(G_{n_{1}})\end{subarray}}}2\frac{\sqrt{2\times 2}}{2+2}$
$\displaystyle=m_{1}+m_{2}+m_{3}+\frac{2\sqrt{6}}{5}(n_{1}n_{2}+m_{1}n_{3})+\frac{n_{1}^{2}-n_{1}}{2}.$

Similarly when $G_{n_{1}}$ is not a triangle-free graph we get,

$GA_{4}(\mathcal{G})=m_{1}+m_{2}+m_{3}+n_{1}n_{2}+m_{1}n_{3}+\frac{n_{1}^{2}-n_{1}}{2}.$

  * (ii)

Let $G_{n_{1}}$ be a triangle-free graph. Then from (7) by using (10) we get

$\displaystyle ABC_{5}(\mathcal{G})$ $\displaystyle=\sum_{uv\in
E(G_{n_{2}})}\sqrt{\frac{3+3-2}{3\times 3}}+\sum_{uv\in
E(G_{n_{3}})}\sqrt{\frac{3+3-2}{3\times 3}}+\sum_{{\begin{subarray}{c}u\in
V(G_{n_{1}}),\\\ v\in V(G_{n_{2}})\end{subarray}}}\sqrt{\frac{2+3-2}{2\times
3}}$ $\displaystyle+\sum_{{\begin{subarray}{c}u\in I(G_{n_{1}}),\\\ v\in
V(G_{n_{3}})\end{subarray}}}\sqrt{\frac{2+3-2}{2\times
3}}+\sum_{{\begin{subarray}{c}uv\in E(\mathcal{G}),\\\ u\in V(G_{n_{1}}),\\\
v\in I(G_{n_{1}})\end{subarray}}}\sqrt{\frac{2+2-2}{2\times
2}}+\sum_{{\begin{subarray}{c}uv\in E(\mathcal{G})\\\ u,v\in
V(G_{n_{1}})\end{subarray}}}\sqrt{\frac{2+2-2}{2\times 2}}$
$\displaystyle=\frac{2}{3}(m_{2}+m_{3})+\frac{1}{\sqrt{2}}\left(n_{1}n_{2}+m_{1}n_{3}+m_{1}+\frac{n_{1}(n_{1}-1)}{2}\right).$

Similarly when $G_{n_{1}}$ is not a triangle-free graph we get,

$ABC_{5}(\mathcal{G})=\frac{2}{3}(m_{1}+m_{2}+m_{3}+n_{1}n_{2}+m_{1}n_{3})+\frac{n_{1}(n_{1}-1)}{3}.$

∎

##  4 Conclusion

Topological indices play an important role in mathematical chemistry. In
QSAR/QSPR study, topological indices can be used to estimate the bioactivity
of chemical compounds. In modeling the chemical and other properties of
molecules, we can use topological indices as a tool. This article gives the
results related to topological indices like the eccentric connectivity index,
connective eccentricity index, total- eccentricity index, average eccentricity
index, Zagreb indices, geometric-arithmetic index, atom-bond connectivity
index, and Wiener index of the newly defined graph operation called central
vertex-edge join (CVE-join). Also, the $D$-spectrum of CVE-join of three
regular graphs $G_{n_{1}}$, $G_{n_{2}}$ and $G_{n_{3}}$ when $G_{n_{1}}$ is a
triangle-free are obtained. These results enable us to construct new
$D$-equienergetic graph families.

## Acknowledgements

The authors would like to thank the DST, Government of India, for providing
support to carry out this work under the scheme ’FIST’(No.SR
/FST/MS-I/2019/40).

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