###### Proof.

###### Proof.

.:

###### Joint Explainability and Sensitivity-Aware Federated Deep Learning for
Transparent 6G RAN Slicing

###### Abstract

In recent years, wireless networks are evolving complex, which upsurges the
use of zero-touch artificial intelligence (AI)-driven network automation
within the telecommunication industry. In particular, network slicing, the
most promising technology beyond 5G, would embrace AI models to manage the
complex communication network. Besides, it is also essential to build the
trustworthiness of the AI black boxes in actual deployment when AI makes
complex resource management and anomaly detection. Inspired by closed-loop
automation and Explainable Artificial intelligence (XAI), we design an
Explainable Federated deep learning (FDL) model to predict per-slice RAN
dropped traffic probability while jointly considering the sensitivity and
explainability-aware metrics as constraints in such non-IID setup. In precise,
we quantitatively validate the faithfulness of the explanations via the so-
called attribution-based _log-odds metric_ that is included as a constraint in
the run-time FL optimization task. Simulation results confirm its superiority
over an unconstrained integrated-gradient (IG) _post-hoc_ FDL baseline.

###### Index Terms:

6G, classification, FL , game theory, proxy-Lagrangian, SLA, stochastic
policy, traffic drop, XAI, ZSM

##  I Introduction

The most promising 6G network slicing technology insists on adopting
autonomous management and orchestration of the end-to-end (E2E) network
resources at the network domains because the isolation of slices may induce a
high cost in terms of efficiency [1, 2].

Figure 1: RAN federated traffic drop classification in NS

So, ETSI standardized zero-touch network and service management (ZSM)
framework has been considered [3].Here, zero-touch refers to the automation
and management of resources without human interference. Besides, developing
cognitive slice management solutions in 6G networks is essential to
automatically orchestrate and manage network slices, particularly network
resources across different technological domains (TDs), along with ensuring
the end-user’s QoE and QoS [4, 5]. Hence, the [6] has proposed an AI-native
network slicing management solution of 6G networks to support emerging AI
services. Also, AI algorithms should be driven by the distributed nature of
datasets to acquire the full potential of network slicing automation, which
will solve the problematic behavior of the cloud-centric traditional ML
schemes. Thus, a decentralized learning approach is required to handle
distributed network slices efficiently. For this, we choose Federated learning
(FL) [7, 8] to handle distributed network slices efficiently like our another
research work [9]. Besides, even if DNN hold the state-of-the-art [10, 11, 12]
in solving resource allocation and orchestration problems of network slicing,
the black-box nature of such ML models impedes understanding of their
decisions, any flaws in the datasets or the model’s performance behavior.
Moreover, the 6G network is going to be "machine-centric" technology which
signifies that all the corresponding "smart things" in the 6G network will
operate intelligently but as a smart black box [13]. Here, the smart black box
is not transparent in its action or decision-making processes and could have
adverse effects on the network’s operations of the 6G technology. In this
concern, XAI provides human interpretable methods to adequately explain the AI
system and its decisions for gaining the human’s trust in the loop. Also, [14]
indicates that it is a prerequisite of any ZSM-based AI models in 6G to enrich
translucency of their models. Viewing this fact, zero-touch XAI-driven FL will
be fetching a particular emphasis for its automation and unique advantages,
which are essential for end-user trust and secured procedure. In contrast, the
conventional XAI focuses only on the interpretability and transparency of any
ML system. Some works of XAI [15, 16, 17] indicate the importance of
explainability and present some research works on handover and resource
allocation, etc., in the beyond 5G networks. In [18], XAI for physical/MAC
layers in 6G networks are focused. In comparison, the authors of [19] present
a trust-aware federated deep reinforcement learning-based device selection
technique in an autonomous driving scenario. And, to evaluate the performance
of XAI models, the paper [20] introduces some essential metrics. So, in this
work, we will present a novel zero-touch Explainable Federated learning (FL)
as the decentralized approach for traffic drop classification in 6G network
slices [7].

###  I-A Contributions

In this paper, we present the following contributions

  * •

We introduce a novel iterative explainable federated learning approach, where
a constrained traffic drop detection classifier and an _explainer_ exchange—in
a closed loop way— attributions of the features as well as predictions to
achieve a transparent zero-touch service management of 6G network slices at
RAN in a non-IID setup.

  * •

We adopt the integrated gradients XAI method to showcase features
attributions.

  * •

The generated attributions are then used to quantitatively validate the
faithfulness of the explanations via the so-called _log-odds_ metric which is
included as a constraint in the FL optimization task.

  * •

We formulate the corresponding joint recall and log-odds-constrained FL
optimization problem under the _proxy-Lagrangian_ framework and solve it via a
non-zero sum two-player game strategy [21], while comparing with the
unconstrained integrated-gradient post-hoc FL baseline.

##  II RAN Architecture and Datasets

A shown in Fig. 1, we consider a radio access network (RAN), which is composed
of a set of $K$ the base station (BSs), wherein a set of $N$ parallel slices
are deployed. Each BS runs a local control closed-loop (CL) which collects
monitoring data and performs traffic drop prediction. Specifically, the
collected data serves to build local datasets for slice $n\,(n=1,\ldots,N)$,
i.e.,
$\mathcal{D}_{k,n}=\\{\mathbf{x}_{k,n}^{(i)},y_{k,n}^{(i)}\\}_{i=1}^{D_{k,n}}$,
where $\mathbf{x}_{k,n}^{(i)}$ stands for the input features vector while
$y_{k,n}^{(i)}$ represents the corresponding output. In this respect, Table I
summarizes the features and the output of the local datasets. These
accumulated datasets are non-IID due to the different traffic profiles induced
by the heterogeneous users’ distribution and channel conditions. Moreover,
since the collected datasets are generally non-exhaustive to train accurate
anomaly detection classifiers, the local CLs take part in a federated learning
task wherein an E2E slice-level federation layer plays the role of a model
aggregator.

Table I: Dataset Features and Output Feature | Description
---|---
Average PRB | Average Physical Resource Block
Latency | Average transmission latency
Channed Quality | SNR value expressing the wireless channel quality
|
Output | Description
Dropped Traffic | Probability of dropped traffic(%)

##  III Explainable FDL for Transparent Traffic Drop Classification

Here, we describe the different stages of the joint explainability and
sensitivity-aware FDL as summarized in Fig. 2.

###  III-A Closed-Loop Description

We propose a federated deep learning architecture where the local learning is
performed iteratively with run-time explanation in a closed loop way as shown
in Fig. 2. We design a deep neural network FL model. For each local epoch, the
Learner module feeds the posterior symbolic model graph to the Tester block
which yields the test features and the corresponding predictions
$\hat{y}_{k,n}^{(i)}$ to the Explainer. The latter first generates the
features attributions using integrated gradients XAI method. The _Log-odds
Mapper_ then uses these attributions to select the top $p$ features that are
then masked. The corresponding soft probability outputs are afterward used to
calculate the the log-odds (LO) metric that is fed back to the Learner to
include it in the local constrained optimization in step 6. Similarly, the
_Recall Mapper_ calculate the recall score $\rho_{k,n}$ based on the
predicated and true positive values at stage 3 and 4 to include it in the
local constrained optimization in step 6. Indeed, for each local CL $(k,n)$,
the predicted traffic drop class $\hat{y}_{k,n}^{(i)},\,(i=1,\ldots,D_{k,n})$,
should minimize the main loss function with respect to the ground truth
$y_{k,n}^{(i)}$, while jointly respecting some long-term statistical
constraints defined over its $D_{k,n}$ samples and jointly corresponding to
recall and explainability log-odds.

As shown in steps 1 and 7 of Fig. 2, the optimized local weights at round $t$,
$\mathbf{W}_{k,n}^{(t)}$, are sent to the server which generates a global FL
model for slice $n$ as,

$\mathbf{W}_{n}^{(t+1)}=\sum_{k=1}^{K}\frac{D_{k,n}}{D_{n}}\mathbf{W}_{k,n}^{(t)},$
(1)

where $D_{n}=\sum_{k=1}^{K}D_{k,n}$ is the total data samples of all datasets
related to slice $n$. The server then broadcasts the global model to all the
$K$ CLs that use it to start the next round of iterative local optimization.
Specifically, it leverages a two-player game strategy to jointly optimize over
the objective and original constraints as well as their smoothed surrogates
and detailed in the sequel.

Figure 2: Explainable FDL building blocks

###  III-B Model Testing and Explanation

As depicted in stage 2 of Fig. 2, upon the reception of the updated model
graph, the Tester uses a batch drawn from the local dataset to reconstruct the
test predictions $\hat{\mathbf{y}}_{k,n}^{(i)}$. All the graph, test dataset
and the predictions are fed to the Explainer at stage 3. After that, at stage
4, Explainer generates the attributions by leveraging the low-complexity
Integrated Gradient (IG) scheme [22], which is based on the gradient variation
when sampling the neighborhood of a feature. Attributions are a quantified
impact of each single feature on the predicted output. Let
$\mathbf{a}_{k,n}^{(i)}\in\mathbb{R}^{Q}$ denote the attribution vector of
sample $i$, which can be generated by any attribution-based XAI method.

###  III-C Log-odds Mapping

To characterize the trustworthiness of the local model, we calculate the log-
odds metric, $\theta_{k,n}$ [23]. It measures the influence of the top-
attributed features on the model’s prediction. Specifically, the log-odds
score is defined as the average difference of the negative logarithmic
probabilities on the predicted class before and after masking the top $p$%
features with zero padding [23]. In this respect, the _log-odds Mapper_ at
stage 5 of Fig. 2 starts by selecting top $p$% features based on their
attributions which is collected from stage 4 and replace them with zero
padding. That is,

$\theta_{k,n}=-\frac{1}{D_{k,n}}\sum_{i=1}^{D_{k,n}}\log\frac{\Pr\left(\hat{y}_{k,n}^{(i)}|\hat{\mathbf{x}}_{k,n}^{(i)}\right)}{\Pr\left(\hat{y}_{k,n}^{(i)}|\mathbf{x}_{k,n}^{(i)}\right)},$
(2)

where, $\hat{y}_{k,n}^{(i)}$ is the predicted class, $\mathbf{x}_{k,n}^{(i)}$
are the features in the original dataset and $\hat{\mathbf{x}}_{k,n}^{(i)}$
denotes the features in the modified dataset with top $p$% features zero-
padded. Finally, the log-odds Mapper reports the log-odds score, which is used
as one of the constraints for the constrained FL optimization task.

###  III-D Joint Recall and Explainability-Aware Traffic Drop Classification

Besides the log-odds score used for explainability,as shown in steps 3 and 4,
we invoke the _recall_ as a measure of the sensitivity of the FL local
classifier, which we denote $\rho_{k,n}$, i.e.,

$\rho_{k,n}=\pi^{+}\left(\mathcal{D}_{k,n}\left[\hat{y}_{k,n}^{(i)}=1\right]\right)$
(3)

Where, $\pi^{+}(\mathcal{D}_{k,n})$ defines the proportion of
$\mathcal{D}_{k,n}$ classified positive, and $\mathcal{D}_{k,n}[*]$ is the
subset of $\mathcal{D}_{k,n}$ satisfying expression *.

In order to trust the traffic drop anomaly detection/classification, a set of
AI SLA is established between the slice tenant and the infrastructure
provider, where a lower bound $\alpha_{n}$ is imposed to the recall score,
while an upper bound $\beta_{n}$ is set for the log-odds score. This
translates into solving a constrained local classification problem in
iterations specified by the epochs as well as in FL rounds
$t\,(t=0,\ldots,T-1)$ i.e.,

$\min_{\mathbf{W}_{k,n}^{(t)}}\,\frac{1}{D_{k,n}}\sum_{i=1}^{D_{k,n}}\ell\left(y_{k,n}^{(i)},\hat{y}_{k,n}^{(i)}\left(\mathbf{W}_{k,n}^{(t)},\mathbf{x}_{k,n}\right)\right),$
(4a)

$\mathrm{s.t.}\hskip 14.22636pt\rho_{k,n}\geq\alpha_{n},$ (4b) $\hskip
19.91692pt\theta_{k,n}\leq\beta_{n},$ (4c)

which is solved by invoking the so-called _proxy Lagrangian_ framework [24],
since the recall is not a smooth constraint. This consists first on
constructing two Lagrangians as follows:

$\begin{split}\mathcal{L}_{\mathbf{W}_{k,n}^{(t)}}=&\frac{1}{D_{k,n}}\sum_{i=1}^{D_{k,n}}\ell\left(y_{k,n}^{(i)},\hat{y}_{k,n}^{(i)}\left(\mathbf{W}_{k,n}^{(t)},\mathbf{x}_{k,n}\right)\right)\\\
&+\lambda_{1}\Psi_{1}\left(\mathbf{W}_{k,n}^{(t)}\right)+\lambda_{2}\Psi_{2}\left(\mathbf{W}_{k,n}^{(t)}\right),\end{split}$
(5a)
$\mathcal{L}_{\lambda}=\lambda_{1}\Phi_{1}\left(\mathbf{W}_{k,n}^{(t)}\right)+\lambda_{2}\Phi_{2}\left(\mathbf{W}_{k,n}^{(t)}\right)$
(5b)

where $\Phi_{1,2}$ and $\Psi_{1,2}$ represent the original constraints and
their smooth surrogates, respectively. In this respect, the recall surrogate
is given by,

$\Psi_{1}=\frac{\sum_{i=1}^{D_{k,n}}{y}_{k,n}^{(i)}\times\min\Bigl{\\{}\hat{y}_{k,n}^{(i)},1\Bigl{\\}}}{\sum_{i=1}^{D_{k,n}}{y}_{k,n}^{(i)}}-\alpha_{n}$
(6)

while $\Psi_{2}=\Phi_{2}=\beta_{n}-\theta_{k,n}$ since the negative logarithm
is already a convex function. It also confirms that the solutions of the
optimization problem are equivalent to those obtained if only the original
constraints were used.

This optimization task turns out to be a non-zero-sum two-player game in which
the $\mathbf{W}_{k,n}^{(t)}$-player aims at minimizing
$\mathcal{L}_{\mathbf{W}_{k,n}^{(t)}}$, while the $\lambda$-player wishes to
maximize $\mathcal{L}_{\lambda}$ [21, Lemma 8]. While optimizing the first
Lagrangian w.r.t. $\mathbf{W}_{k,n}$ requires differentiating the constraint
functions $\Psi_{1}(\mathbf{W}_{k,n}^{(t)})$ and
$\Psi_{2}(\mathbf{W}_{k,n}^{(t)})$, to differentiate the second Lagrangian
w.r.t. $\lambda$ we only need to evaluate
$\Phi_{1}\left(\mathbf{W}_{k,n}^{(t)}\right)$ and
$\Phi_{2}\left(\mathbf{W}_{k,n}^{(t)}\right)$. Hence, a surrogate is only
necessary for the $\mathbf{W}_{k,n}$-player; the $\lambda$-player can continue
using the original constraint functions. The local optimization task can be
written as,

$\begin{split}\min_{\mathbf{W}_{k,n}\in\Delta}\,\,\,\,\max_{\lambda,\,\norm{\lambda}\leq
R_{\lambda}}\,\,\mathcal{L}_{\mathbf{W}_{k,n}^{(t)}}\end{split}$ (7a)
$\max_{\lambda,\,\norm{\lambda}\leq
R_{\lambda}}\,\,\,\,\min_{\mathbf{W}_{k,n}\in\Delta}\mathcal{L}_{\lambda},$
(7b)

where thanks to Lagrange multipliers, the $\lambda$-player chooses how much to
weigh the proxy constraint functions, but does so in such a way as to satisfy
the original constraints, and ends up reaching a nearly-optimal nearly-
feasible solution [25]. These steps are all summarized in Algorithm 1.

Input: $K$, $m$, $\eta_{\lambda}$, $T$, $L$. # See Table II

Server initializes $\mathbf{W}_{n}^{(0)}$ and broadcasts it to the
$\mathrm{CL}$s

for  _$t=0,\ldots,T-1$_ do

parallel for $k=1,\ldots,K$ do

Initialize $M=$ num_constraints and $\mathbf{W}_{k,n,0}=\mathbf{W}_{n}^{(t)}$

Initialize $\mathbf{A}^{(0)}\in\mathbb{R}^{(M+1)\times(M+1)}$ with
$\mathbf{A}_{m^{\prime},m}^{(0)}=1/(M+1)$

for  _$l=0,\ldots,L-1$_ do

Receive the graph $\mathcal{M}_{k,n}$ from the local model

# Test the local model and calculate the attributions

$a_{k,n}^{i,j}=$ Int. Gradient
$\left(\mathcal{M}_{k,n}\left(\mathbf{W}_{k,n,l},\mathbf{x}_{k,n}\right)\right)$

# Mask the top p% dataset based on the attributions with zero padding

# Calculate the log-odds metric

$\theta_{k,n}$ =
$\frac{1}{D_{k,n}}\sum_{i=1}^{D_{k,n}}\log\frac{\Pr\left(\hat{y}_{k,n}^{(i)}|\hat{\mathbf{x}}_{k,n}^{(i)}\right)}{\Pr\left(\hat{y}_{k,n}^{(i)}|\mathbf{x}_{k,n}^{(i)}\right)}$

# Calulate the recall metric

$\rho_{k,n}=\pi^{+}\left(\mathcal{D}_{k,n}\left[\hat{y}_{k,n}^{(i)}=1\right]\right)$

Let $\lambda^{(l)}$ be the top eigenvector of $\mathbf{A}^{(l)}$

# Solve problem (4) via oracle optimization

Let
$\hat{\mathbf{W}}_{k,n,l}=\mathcal{O}_{\delta}\left(\mathcal{L}_{\mathbf{W}_{k,n,l}}(\cdot,\hat{\lambda}^{(l)})\right)$

Let $\Delta_{\lambda}^{(l)}$ be a gradient of
$\mathcal{L}_{\lambda}(\hat{\mathbf{W}}_{k,n,l},\lambda^{(l)})$ w.r.t.
$\lambda$

# Exponentiated gradient ascent

Update
$\tilde{\mathbf{A}}^{(l+1)}=\mathbf{A}^{(l)}\odot\cdot\exp{\eta_{\lambda}\Delta_{\lambda}^{(l)}(\lambda^{(l)})}$

# Colunm-wise normalization

$\mathbf{A}_{m}^{(l+1)}=\tilde{\mathbf{A}}_{m}^{(l+1)}/\norm{\mathbf{A}_{m}^{(l+1)}}_{1},\,m=1,\ldots,M+1$

end for

return
_$\hat{\mathbf{W}}_{k,n}^{(t)}=\frac{1}{L^{\star}}\sum_{l=0}^{L-1}\hat{\mathbf{W}}_{k,n,l}$_

Each local CL $(k,n)$ sends $\hat{\mathbf{W}}_{k,n}^{(t)}$ to the server.

end parallel for

return
_$\mathbf{W}_{n}^{(t+1)}=\sum_{k=1}^{K}\frac{D_{k,n}}{D_{n}}\hat{\mathbf{W}}_{k,n}^{(t)}$_

and broadcasts the value to all local CLs.

end for

Algorithm 1 Explainable Federated Deep Learning

##  IV Results

This section analyzes the proposed Closed loop EFL framework in detail. To
build the explainability-aware constrained traffic drop classification model,
we use feature attributions which is the pillar of this approach. After that,
we present the impact of considering jointly the recall and log-odds metrics
as constraints for optimizing the FL classification problem by showing results
of FL convergence and log-odds score. Finally, we study the correlation
between features attributions, observed predictions, and _true_ predictions
and draw some important conclusions. Specifically, to implement the model
Tester and Explainer, we invoke DeepExplain framework, which includes state-
of-the-art gradient and perturbation-based attribution methods [26]. It
provides an attribution score based on the feature’s contribution to the
model’s output, which we integrate with our proposed constrained traffic drop
classification FL framework in a closed-loop iterative way.

###  IV-A Parameter Settings and Baseline

Three primary slices eMBB, uRLLC and mMTC are considered to analyze the
proposed Explainable FL policy. Here, the datasets are collected from the BSs
and the overall summary of those datasets are presented in Table II. We use
vector $\beta$ for the explainability lower bound threshold and $\alpha$ for
the upper bound of recall score corresponding to the different slices. As a
baseline, we adopt a vanilla FL [27] with post-hoc integrated gradient
explanation,that is, a posterior explanation performed upon the end of the FL
training.

Table II: Settings Parameter | Description | Value
---|---|---
DNN | Deep neural network size |  $2$-hidden layers with $10$ nodes
$N$ | # Slices | $3$
$K$ | # BSs | $50$
$D_{k,n}$ | Local dataset size |  $800$ samples
$T$ | # Max FL rounds | $50$
$U$ | # Total users (All BSs) | $15000$
$L$ | # Local epochs | $100$
$R_{\lambda}$ | Lagrange multiplier radius | Constrained: $10^{-5}$
$\eta_{\lambda}$ | Learning rate | $0.02$

###  IV-B Result Analysis

In this scenario, resources allocated to slices according to their traffic
patterns and radio conditions while ensuring a long term isolation via the
constraints.

Figure 3:  Analysis of FL training loss vs FL rounds of Proposed EFL with
Lower bound of Recall score, $\alpha=[0.9,0.95,0.95]$ and Upper bound of log-
odds score, $\beta=[-0.01,-0.01,-0.01]$ Figure 4:  Analysis of Recall score
with Lower bound of Recall score, $\alpha=[0.9,0.95,0.95]$ and Upper bound of
log-odds score, $\beta=[-0.01,-0.01,-0.01]$

  * •

Convergence: As depicted in Fig. 3, we can conclude that the proposed
constrained EFL resource allocation models of the different slices have
converged faster than the baseline unconstrained IG post-hoc case. Here, the
optimizer of EFL considers the relationships between the objectives and
constraints of the two-player optimization problem, leading to improved
performance compared to the uncon. IG post-hoc one, which accounts for only
the objective function during optimization.

  * •

Sensitivity analysis: To analyze our proposed model’s sensitivity, we choose
the recall metric, which is the rate of actual positive values for measuring
the performance of our binary classification model. From Fig. 4, we can
observe that the recall score of the proposed one for all slices is in close
proximity to the target threshold $\gamma$ (i.e., around $0.88\%$), which is
an acceptable value for operators and slices’ tenants.

  * •

Trustfulness: In Fig. 5-(a), we observe the effect of changing the value top
$p$% on the log-odds, considering proposed model for all slices. Also we
present a comparative analysis of log-odds score in Fig.5-(b) for both cases
which proof the superiority of the proposed constrained EFL model. So, the
statistics of the log-odds score give us an approximate idea of our model’s
reliability and trustworthiness. It shows that the log-odds score is
decreasing with respect to the top $p$% value, which conveys that our model is
explainable and trustworthy in the training phase.

(a) log-odds Score vs. top p $\%$ (b) log-odds Score

Figure 5:  Analysis of log-odds score with Lower bound of Recall score,
$\alpha=[0.9,0.95,0.95]$ and Upper bound of log-odds score,
$\beta=[-0.01,-0.01,-0.01]$

Furthermore, in Fig. 6, the correlation heatmaps of the proposed XAI method of
the eMBB slice has presented for further analysis. It helps us visualize the
strength of relationships between different variables and, in our case,
identify which feature variation impacts the most for SLA variation. To plot
correlation matrix heatmap, we consider one matrix, $\mathbf{R}_{k,n}$ =
[$\mathbf{a}_{k,n},\hat{\mathbf{y}}_{k,n},\mathbf{y}_{k,n}$], where,
$\mathbf{a}_{k,n}$ is the attribution score of features variable with
dimensions $D_{k,n}\times Q$ and $\hat{\mathbf{z}}_{k,n}$ is the predicted
output variable with dimensions $D_{k,n}\times 1$ and $\mathbf{y}_{k,n}$ is
the true predicted value with dimensions $D_{k,n}\times 1$. From the heatmap
we see that the third feature, which is the channel quality, has the most
impact on the recall value. If the third feature increases, the recall value
will increase and vice versa.

##  V Conclusion

This paper has presented a novel closed-loop explainable federated learning
(EFL) approach to achieve transparent zero-touch service management of 6G
network slices at RAN in a non-IID setup. We have jointly considered
explainability and sensitivity metrics as constraints in the traffic drop
prediction task, which we have solved using a proxy-Lagrangian two-player game
strategy. From the results, we conclude that the proposed EFL scheme is
reliable and trustful compared to state-of-the-art unconstrained post-hoc FL.
Finally, the heatmaps of the attributions correlation matrix are presented to
showcase the features whose variation influence more the traffic drop.

##  VI Acknowledgment

This work has been supported in part by the projects SEMANTIC (861165),
6G-BRICKS (101096954) HORIZON-JU-SNS-2022 and ADROIT6G (101095363) HORIZON-JU-
SNS-2022.

Figure 6: Correlation heatmap of eMBB slices based on attribution scores of
features generated by XAI.

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# On the multiplicity of $A_{\alpha}$-eigenvalues and the rank of complex unit
gain graphs

Aniruddha Samanta Department of Mathematics, Indian Institute of Technology
Kharagpur, Kharagpur 721302, India. Email<EMAIL_ADDRESS>M. Rajesh
Kannan Department of Mathematics, Indian Institute of Technology Kharagpur,
Kharagpur 721302, India. Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Let $\Phi=(G,\varphi)$ be a connected complex unit gain graph
($\mathbb{T}$-gain graph) on a simple graph $G$ with $n$ vertices and maximum
vertex degree $\Delta$. The associated adjacency matrix and degree matrix are
denoted by $A(\Phi)$ and $D(\Phi)$, respectively. Let
$m_{\alpha}(\Phi,\lambda)$ be the multiplicity of $\lambda$ as an eigenvalue
of $A_{\alpha}(\Phi):=\alpha D(\Phi)+(1-\alpha)A(\Phi)$, for $\alpha\in[0,1)$.
In this article, we establish that
$m_{\alpha}(\Phi,\lambda)\leq\frac{(\Delta-2)n+2}{\Delta-1}$, and characterize
the classes of graphs for which the equality hold. Furthermore, we establish a
couple of bounds for the rank of $A(\Phi)$ in terms of the maximum vertex
degree and the number of vertices. One of the main results extends a result
known for unweighted graphs and simplifies the proof in [15], and other
results provide better bounds for $r(\Phi)$ than the bounds known in [8].

AMS Subject Classification(2010): 05C50, 05C22, 05C35.

Keywords. Complex unit gain graph, Rank of a graph, Zero forcing set,
$A_{\alpha}$-eigenvalue.

##  1 Introduction

Let $G=(V(G),E(G))$ be a simple graph where
$V(G)=\\{v_{1},v_{2},\dots,v_{n}\\}$ and $E(G)$ are the vertex set and the
edge set of $G$, respectively. If two vertices $v_{s}$ and $v_{t}$ are
connected by an edge, we write $v_{s}\sim v_{t}$. If $v_{s}\sim v_{t}$, then
the edge between them is denoted by $e_{s,t}$. The _degree_ of a vertex
$v_{s}$ is denoted by $d(v_{s})$ and is defined as the number of vertices
adjacent to $v_{s}$. Then the _maximum vertex degree_ of $G$ is denoted by
$\Delta(G)$ (or, simply $\Delta$). The _degree matrix_ of a graph $G$ is a
diagonal matrix, denoted by $D(G)$, is defined by
$D(G):=\operatorname{diag}(d(v_{1}),d(v_{2}),\dots,d(v_{n}))$. The _adjacency
matrix_ $A(G)$ of a graph $G$ is a symmetric matrix whose $(s,t)th$ entry is
$1$ if $v_{s}\sim v_{t}$, and zero otherwise. The nullity of $A(G)$ is the
multiplicity of zero eigenvalue of $A(G)$, and is called the _nullity of $G$_,
denoted by $\eta(G)$. The _rank_ of $G$ is the rank of $A(G)$, and is denoted
by $r(G)$. Thus $\eta(G)=n-r(G)$.

Let $G$ be a simple undirected graph. An oriented edge from the vertex $v_{s}$
to the vertex $v_{t}$ is denoted by $\overrightarrow{e_{s,t}}$. For each
undirected edge $e_{s,t}\in E(G)$, there is a pair of oriented edges
$\overrightarrow{e_{s,t}}$ and $\overrightarrow{e_{t,s}}$. The collection
$\overrightarrow{E(G)}:=\\{\overrightarrow{e_{s,t}},\overrightarrow{e_{t,s}}:e_{s,t}\in
E(G)\\}$ is the _oriented edge set associated with $G$_. Let
$\mathbb{T}=\\{z\in\mathbb{C}:|z|=1\\}$. A _complex unit gain graph (or
$\mathbb{T}$-gain graph)_ on a simple graph $G$ is an ordered pair
$(G,\varphi)$, where the gain function
$\varphi:\overrightarrow{E(G)}\rightarrow\mathbb{T}$ is a mapping such that
$\varphi(\overrightarrow{e_{s,t}})=\varphi(\overrightarrow{e_{t,s}})^{-1}$,
for every $e_{s,t}\in E(G)$. A $\mathbb{T}$-gain graph $(G,\varphi)$ is
denoted by $\Phi$. The _adjacency matrix_ of a $\mathbb{T}$-gain graph
$\Phi=(G,\varphi)$ is a Hermitian matrix, denoted by $A(\Phi)$ and its
$(s,t)th$ entry is defined as follows:

$A(\Phi)_{st}=\begin{cases}\varphi(\overrightarrow{e_{s,t}})&\text{if
}\mbox{$v_{s}\sim v_{t}$},\\\ 0&\text{otherwise.}\end{cases}$

The rank and the nullity of $A(\Phi)$ are the _rank_ and _nullity_ of $\Phi$,
denoted by $r(\Phi)$ and $\eta(\Phi)$, respectively. The degree matrix
$D(\Phi)$ and maximum vertex degree $\Delta(\Phi)$ of $\Phi$ are same as
$D(G)$ and $\Delta(G)$, respectively. The notion of adjacency matrix of
$\mathbb{T}$-gain graphs generalize the notion of adjacency matrix of
undirected graphs, adjacency matrix of signed graphs and the Hermitian
adjacency matrix of a digraph. The notion of gain graph was introduced in [2].
For more information about the properties of gain graphs and $\mathbb{T}$-gain
graphs, we refer to [9, 11, 12, 13, 19, 20].

Let $G$ be a simple graph with adjacency matrix $A(G)$ and degree matrix
$D(G)$. In [10], the author introduced the following new family of matrices,
denoted by $A_{\alpha}(G)$, associated with an undirected graph:

$A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G),~{}~{}~{}\alpha\in[0,1].$

Then $A_{0}(G)=A(G)$, $A_{\frac{1}{2}}(G)=\frac{1}{2}Q(G)$ and
$A_{1}(G)=D(G)$, where $Q(G)$ is the signless Laplacian of $G$. The
eigenvalues of $A_{\alpha}(G)$ are known as _$A_{\alpha}$ -eigenvalues_ of
$G$. For more details about $A_{\alpha}$-eigenvalues, we refer to [10, 15].
Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain graph on underlying graph $G$.
In [6], the authors introduced $A_{\alpha}$-matrix for a complex unit gain
graph $\Phi$, denoted by $A_{\alpha}(\Phi)$, defined as follows:

$A_{\alpha}(\Phi)=\alpha
D(\Phi)+(1-\alpha)A(\Phi),~{}~{}~{}~{}\alpha\in[0,1].$

The eigenvalues of $A_{\alpha}(\Phi)$ are called _$A_{\alpha}$ -eigenvalues_
of $\Phi$. Note that $A_{\alpha}(\Phi)$ is a generalization of $A_{\alpha}(G)$
and hence the matrices $A_{\alpha}(G)$, $A(\Phi)$, $A(G)$ are particular cases
of $A_{\alpha}(\Phi)$. Let $\lambda$ be an eigenvalue of $A_{\alpha}(\Phi)$.
Then the _multiplicity of $\lambda$_ is denoted by $m_{\alpha}(\Phi,\lambda)$.

In [15], authors proved the following upper bound for the multiplicity of
$A_{\alpha}$-eigenvalues of a connected graph $G$.

######  Theorem 1.1.

[15, Theorem 3.1, Theorem 3.3] Let $G$ be a connected graph of $n$ vertices
with maximum vertex degree $\Delta\geq 2$. If $m_{\alpha}(G,\lambda)$ is the
multiplicity of $\lambda$ as an eigenvalue of $A_{\alpha}(G)$, then for
$\alpha\in[0,1)$,

$m_{\alpha}(G,\lambda)\leq\frac{(\Delta-2)n+2}{\Delta-1}$

Equality occur if and only if $G$ and $\lambda$ satisfy one of the following:

  1. (i)

$G=K_{n}$ and $\lambda=\alpha n-1$,

  2. (ii)

$G=C_{n}$ with even order and $\lambda\in\\{2\alpha+2(1-\alpha)\cos(\frac{2\pi
j}{n}):j=1,2,\dots,\frac{n-2}{2}\\}$,

  3. (iii)

$G=C_{n}$ with order order and
$\lambda\in\\{2\alpha+2(1-\alpha)\cos(\frac{2\pi
j}{n}):j=1,2,\dots,\frac{n-1}{2}\\}$,

  4. (iv)

$G=K_{\frac{n}{2},\frac{n}{2}}$ and $\lambda=\frac{\alpha n}{2}$.

One of the main objectives of this article is to extend Theorem 1.1 for
$A_{\alpha}$-matrices of $\mathbb{T}$-gain graphs, and provide an alternate
simple proof (Theorem 3.1).

Establishing bounds for the rank and the nullity of a graph, in terms of
$\Delta$ and $n$, is an interesting problem considered in literature [1, 16,
17, 21]. The following bounds for the rank of $\mathbb{T}$-gain graphs are
known.

######  Theorem 1.2 ([8, Theorem 3.2]).

Let $\Phi=(G,\varphi)$ be any $\mathbb{T}$-gain graph of $n$ vertices with
maximum vertex degree $\Delta$. Then

$r(\Phi)\geq\frac{n}{\Delta}.$

######  Theorem 1.3 ([8, Theorem 3.4]).

Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain graph with $n$ vertices and
maximum vertex degree $\Delta$ such that $2\Delta\nmid n$. Then

$r(\Phi)\geq\frac{n+1}{\Delta}.$

Equality occur if and only if
$\Phi=\frac{n-2\Delta+1}{2\Delta}K^{\varphi}_{\Delta,\Delta}\cup
K^{\varphi}_{(\Delta-1),\Delta}$ and each $C^{\varphi}_{4}$(if any) in
$K^{\varphi}_{\Delta,\Delta}$ and $K^{\varphi}_{(\Delta-1),\Delta}$ is of Type
A.

The second objective of this article is to improve these bounds for
$\mathbb{T}$-gain graphs (Theorem 4.1 and Theorem 4.2). The proofs of the main
results use some known results from the field of zero-forcing sets, and in
[14] the authors used the same techniques to derive bounds for the nullity of
the adjacency matrices of graphs.

##  2 Definitions, notation and preliminary results

The notion of a zero-forcing set of a simple graph $G$ is introduced in [5].

######  Definition 2.1 ([5, Definition 2.1]).

[Color-change rule] Let $G$ be a simple graph such that each vertex of $G$ is
colored either black or white. Suppose vertex $v$ is a black vertex and
exactly one neighbor $w$ of $v$ is white among all other neighbors. Then
change the color of $w$ to black.

The _derived coloring_ of a given coloring of $G$ is the resulting coloring
after applying the color-change rule such that no more changes are possible. A
subset $Z$ of the vertex set of $G$ is called a _zero forcing set_ of $G$, if
initially the vertices of $Z$ are all colored black and the remaining vertices
are colored white, the derived coloring of $G$ are all black. The zero forcing
number of $G$ is defined as $Z(G)=\min\limits_{\text{ Z}}|Z|$, $Z$ is a zero-
forcing set of $G$.

The cycle and the complete graph on $n$ vertices are denoted by $C_{n}$ and
$K_{n}$, respectively. A complete bipartite graph, of partition size $m$ and
$n$, is denoted by $K_{m,n}$. In [3] and [4], the authors established upper
bounds for the zero forcing number of a graph in terms of the maximum vertex
degree.

######  Theorem 2.1 ([3, Theorem 1(ii)]).

If $G$ is a connected graph of $n$ vertices with maximum vertex degree
$\Delta\geq 2$. Then

$Z(G)\leq\frac{(\Delta-2)n+2}{\Delta-1}.$

Equality occurs if and only if $G$ is either $C_{n}$, or $K_{n}$, or
$K_{\frac{n}{2},\frac{n}{2}}$.

Figure 1:  Graph $G_{1}$ and $G_{2}$

######  Theorem 2.2 ([4, Theorem 4]).

Let $G$ be any connected graph of $n$ vertices and the maximum vertex degree
$\Delta\geq 3$. Then

$Z(G)\leq\frac{(\Delta-2)n}{\Delta-1}$

holds if and only if
$G\notin\\{G_{1},G_{2},K_{n},K_{\frac{n}{2},\frac{n}{2}},K_{\frac{n+1}{2},\frac{n-1}{2}}\\}$,
where $G_{1}$ and $G_{2}$ are given in Figure 1.

######  Lemma 2.1 ([5, Proposition 2.2]).

Let $B$ be any square matrix on some field with $\eta(B)>s$. Then there exists
a nonzero vector $y\in\operatorname{Ker}(B)$ vanishing at $s$ specified
positions.

For a graph $G$, any function $\zeta:V(G)\rightarrow\mathbb{T}$ is called a
_switching function_. Let $\Phi_{1}=(G,\varphi_{1})$ and
$\Phi_{2}=(G,\varphi_{2})$ be two $\mathbb{T}$-gain graphs. Then $\Phi_{1}$
and $\Phi_{2}$ are _switching equivalent_ , denoted by $\Phi_{1}\sim\Phi_{2}$,
if there exists a switching function $\zeta$ such that
$\varphi_{1}(\overrightarrow{e_{i,j}})=\zeta(v_{i})^{-1}\varphi_{2}(\overrightarrow{e_{i,j}})\zeta(v_{j})$,
for all $e_{i,j}\in E(G)$. If $\Phi_{1}\sim\Phi_{2}$, then $A(\Phi_{1})$ and
$A(\Phi_{2})$ are diagonally similar and hence have the same spectra. Let
$\overrightarrow{C_{n}}$ be an oriented cycle in a $\mathbb{T}$-gain graph
$\Phi=(G,\varphi)$ with oriented edges
$\overrightarrow{e_{1}},\overrightarrow{e_{2}},\cdots,\overrightarrow{e_{n}}$,
then
$\varphi(\overrightarrow{C_{n}})=\prod\limits_{i=1}^{n}\varphi(\overrightarrow{e_{i}})$.
If $\varphi(\overrightarrow{C_{n}})$ is a real number, then we simply write
$\varphi(C_{n})$. A $\mathbb{T}$-gain graph $\Phi=(G,\varphi)$ is _balanced_
if $\varphi(C)=1$ for all cycle $C$ in $\Phi$. If $\Phi$ is balanced then we
write $\Phi\sim(G,1)$. It is known that
$(C_{n},\varphi_{1})\sim(C_{n},\varphi_{2})$ if and only if
$\varphi_{1}(\overrightarrow{C_{n}})=\varphi_{2}(\overrightarrow{C_{n}})$.
Therefore, if
$\varphi_{1}(\overrightarrow{C_{n}})=\varphi_{2}(\overrightarrow{C_{n}})$,
then $(C_{n},\varphi_{1})$ and $(C_{n},\varphi_{2})$ have the same spectra.

######  Definition 2.2 ([7, Definition 2]).

Let $\Phi=(C_{n},\varphi)$ be any $\mathbb{T}$-gain graph on a cycle $C_{n}$.
Then $\Phi$ is called

$\left\\{\begin{array}[]{cl}\mbox{Type A,}&\mbox{if $n$ is even and
$\varphi(C_{n})=(-1)^{\frac{n}{2}}$}\\\ \mbox{Type B,}&\mbox{if $n$ is even
and $\varphi(C_{n})\neq(-1)^{\frac{n}{2}}$}\\\ \mbox{Type C,}&\mbox{if $n$ is
odd and $Re((-1)^{\frac{(n-1)}{2}}\varphi(C_{n}))>0$}\\\ \mbox{Type
D,}&\mbox{if $n$ is odd and $Re((-1)^{\frac{(n-1)}{2}}\varphi(C_{n}))<0$}\\\
\mbox{Type E,}&\mbox{if $n$ is odd and
$Re((-1)^{\frac{(n-1)}{2}}\varphi(C_{n}))=0.$}\\\ \end{array}\right.$

The rank of the gain adjacency matrices of cycles are known.

######  Theorem 2.3 ([18, Theorem 7]).

Let $\Phi=(C_{n},\varphi)$ be any $\mathbb{T}$-gain graph on $C_{n}$ with $n$
vertices. Then

$r(\Phi)=\left\\{\begin{array}[]{cl}n-2,&\mbox{if $\Phi$ is Type A}\\\
n,&\mbox{if $\Phi$ is Type B}\\\ n,&\mbox{if $\Phi$ is Type C}\\\ n,&\mbox{if
$\Phi$ is Type D}\\\ n-1,&\mbox{if $\Phi$ is Type E.}\end{array}\right.$

The following results will be used in the proofs of the main theorems.

######  Lemma 2.2 ([18, Lemma 3(iii)]).

Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain graph and $\Phi_{1}$ be an
induced subgraph of $\Phi$. Then $r(\Phi_{1})\leq r(\Phi)$.

######  Theorem 2.4 ([9, Theorem 4.1]).

Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain graph on a bipartite graph $G$.
Then the eigenvalues of $A(\Phi)$ are symmetric about origin.

Let us recall a couple of results about the $A_{\alpha}$-eigenvalues of a
$\mathbb{T}$-gain graph $\Phi$.

######  Lemma 2.3 ([6, Lemma 2.9]).

Let $\Phi=(C_{n},\varphi)$ be a $\mathbb{T}$-gain graph such that
$\varphi(\overrightarrow{C_{n}})=e^{i\theta}$. Then the
$A_{\alpha}$-eigenvalues of $\Phi$ are

$\left\\{2\alpha+2(1-\alpha)\cos\left(\frac{\theta+2\pi
j}{n}\right):j=0,1,\dots,(n-1)\right\\}.$

######  Corollary 2.1 ([6, Corollary 2.12]).

Let $\Phi=(C_{n},\varphi)$ be a $\mathbb{T}$-gain graph such that
$\varphi(\overrightarrow{C_{n}})=e^{i\theta}$. If $m_{\alpha}(\Phi,\lambda)$
is the multiplicity of $\lambda$ as an eigenvalue of $A_{\alpha}(\Phi)$, then
$m_{\alpha}(\Phi,\lambda)\leq 2$, for $\lambda\in\mathbb{R}$ and
$\alpha\in[0,1)$. Equality occur if and only if any one of the following
holds:

  1. (i)

$\theta=0$ and $\lambda\in\left\\{2\alpha+2(1-\alpha)\cos\left(\frac{2\pi
j}{n}\right):j=0,1,\dots,\lceil\frac{n}{2}\rceil-1\right\\},$

  2. (ii)

$\theta=\pi$ and
$\lambda\in\left\\{2\alpha+2(1-\alpha)\cos\left(\frac{(2j+1)\pi}{n}\right):j=0,1,\dots,\lfloor\frac{n}{2}\rfloor-1\right\\}.$

##  3 Multiplicity of an $A_{\alpha}$-eigenvalue of $\mathbb{T}$-gain graph

Let $H_{n}$ denote the set of all Hermitian matrices of order $n$. For $B\in
H_{n}$, $\mathcal{G}(B)$ is the matrix with $(i,j)th$ entry defined as
follows:

$\mathcal{G}(B)_{ij}=\begin{cases}\frac{B_{ij}}{|B_{ij}|}&\text{if
}\mbox{$B_{ij}\neq 0$},\\\ 0&\text{otherwise.}\end{cases}$

where $B_{ij}$ denote the $(i,j)$th entry of the matrix $B$. Let
$\Phi=(G,\varphi)$ be any $\mathbb{T}$-gain graph of $n$ vertices. A matrix
$B=(B_{ij})\in H_{n}$ is a matrix of type $\Phi$ if
$\mathcal{G}(B)_{ij}=A(\Phi)_{ij}$ for all $i\neq j$. Define
$\mathcal{H}(\Phi)=\\{B\in H_{n}:B~{}\mbox{is of type}~{}\Phi\\}$. Let
$\eta(B)$ be the nullity of the matrix $B$. Define
$M(\Phi):=\max\\{\eta(B):B\in\mathcal{H}(\Phi)\\}$. Then $\eta(A(\Phi))\leq
M(\Phi)$. For any $\mathbb{T}$-gain graph $\Phi$, the underlying graph is
denoted by $\Gamma(\Phi)$. For $y=(y_{1},y_{2},\dots,y_{n})\in\mathbb{C}^{n}$,
the _support_ of $y$ is the set of indices $j$ such that $y_{j}\neq 0$, and is
denoted by $\operatorname{supp}(y)$. A zero forcing set of a $\mathbb{T}$-gain
graph $\Phi$ is the zero forcing set of its underlying graph $\Gamma(\Phi)$.
The following lemma is an extension of [5, Proposition 2.3] for the complex
matrices. For the sake of completeness we include a proof here.

######  Lemma 3.1.

Let $\Phi$ be any $\mathbb{T}$-gain graph, and $Z$ be a zero forcing set of
$\Phi$. Let $B\in\mathcal{H}(\Phi)$ and $y\in\operatorname{Ker}B$ with
$\operatorname{supp}(y)\cap Z=\phi$. Then $y=0$.

###### Proof.

Let $V(\Phi)$ be the vertex set of $\Phi$. If $Z=V(\Phi)$, then $y=0$. Suppose
$Z\subset V(\Phi)$. Since $Z$ is a zero forcing set, so all the white vertices
in $V(\Phi)\setminus Z$ can be colored black by color change rule. Let
$v_{i}\in Z$ be such that it has exactly one white neighbor vertex $v_{t}$.
Then the $i$-th entry $(By)_{i}=B_{ii}y_{i}+\sum\limits_{v_{i}\sim
v_{j}}B_{ij}y_{j}=B_{it}y_{t}=0$. Thus $y_{t}=0$. As $Z$ is a zero forcing
set, so all the components of $y$ associated with white vertices are zero.
Hence $y=0$. ∎

The following lemma is an extension of [5, Proposition 2.4] for the complex
matrices. For the sake of completeness we include a proof here.

######  Lemma 3.2.

Let $\Phi$ be any $\mathbb{T}$-gain graph. Then $M(\Phi)\leq Z(\Gamma(\Phi))$.

###### Proof.

Let $Z$ be a zero forcing set of $\Phi$. Suppose that $M(\Phi)>|Z|.$ Then
there exists a matrix $B\in\mathcal{H}(\Phi)$ such that $\eta(B)>|Z|$.
Therefore, by Lemma 2.1, there exist a nonzero $y\in Ker(B)$ such that
$\operatorname{supp}(y)\cap Z=\phi$. By Lemma 3.1, we get $y=0$, a
contradiction. Thus $M(\Phi)\leq|Z|$, and hence $M(\Phi)\leq Z(\Gamma(\Phi))$.
∎

######  Lemma 3.3.

Let $\Phi=(K_{n},\varphi)$ be a $\mathbb{T}$-gain graph. If $\mu$ is an
eigenvalue of $\Phi$ with multiplicity $(n-1)$ if and only if
$\left\\{\alpha(n-1)+(1-\alpha)\mu\right\\}$ is an $A_{\alpha}$-eigenvalue of
$\Phi$ with multiplicity $(n-1)$, for $\alpha\in[0,1)$.

Now we are ready to establish one of our main results.

######  Theorem 3.1.

Let $\Phi=(G,\varphi)$ be a connected $\mathbb{T}$-gain graph of $n$ vertices
with maximum vertex degree $\Delta\geq 2$. If $m_{\alpha}(\Phi,\lambda)$ is
the multiplicity of $\lambda$ as an $A_{\alpha}$-eigenvalue of $\Phi$, where
$\alpha\in[0,1)$, then

$m_{\alpha}(\Phi,\lambda)\leq\frac{(\Delta-2)n+2}{(\Delta-1)}.$

Equality occurs if and only if one of the following holds:

  1. (i)

$\Phi=(K_{n},\varphi)$ with $\mu\in\operatorname{spec}(\Phi)$ has multiplicity
$(n-1)$ and $\lambda=\alpha(n-1)+(1-\alpha)\mu$.

  2. (ii)

$\Phi=(C_{n},\varphi)$ with $\varphi(C_{n})=1$ and
$\lambda\in\left\\{2\alpha+2(1-\alpha)\cos\left(\frac{2\pi
j}{n}\right):j=0,1,\dots,\lceil\frac{n}{2}\rceil-1\right\\}$.

  3. (iii)

$\Phi=(C_{n},\varphi)$ with $\varphi(C_{n})=-1$ and
$\lambda\in\left\\{2\alpha+2(1-\alpha)\cos\left(\frac{(2j+1)\pi}{n}\right):j=0,1,\dots,\lfloor\frac{n}{2}\rfloor-1\right\\}$.

  4. (iv)

$\Phi\sim(K_{\frac{n}{2},\frac{n}{2}},1)$ and $\lambda=\frac{\alpha n}{2}$.

###### Proof.

For $\alpha\in[0,1)$, we have $A_{\alpha}(\Phi)=\alpha
D(\Phi)+(1-\alpha)A(\Phi)$, where $D(\Phi)$ is the degree matrix of $\Phi$.
Let $\lambda$ be an eigenvalue of $A_{\alpha}(\Phi)$. Then the matrix
$B:=\left(A_{\alpha}(\Phi)-\lambda I\right)$ is Hermitian, where $I$ is the
identity matrix of order $n$, and hence the multiplicity of the eigenvalue
$\lambda$ is same as the nullity of $B$. That is,
$\eta(B)=m_{\alpha}(\Phi,\lambda)$. Let $B_{ij}$ be the $(i,j)th$-entry of the
matrix $B$. If $B_{ij}\neq 0$, then
$\frac{B_{ij}}{|B_{ij}|}=\frac{(1-\alpha)A(\Phi)_{ij}}{|(1-\alpha)||A(\Phi)_{ij}|}=A(\Phi)_{ij}$,
for $i\neq j$ and $\alpha\in[0,1)$. Thus $B\in\mathcal{H}(\Phi)$. Therefore,
$\eta(B)\leq M(\Phi)$. Now, by combining Lemma 3.2, Theorem 2.1 and the fact
that $\eta(B)\leq M(\Phi)$, we have

$m_{\alpha}(\Phi,\lambda)=\eta(B)\leq M(\Phi)\leq
Z(G)\leq\frac{(\Delta-2)n+2}{\Delta-1}.$

If $m_{\alpha}(\Phi,\lambda)=\frac{(\Delta-2)n+2}{\Delta-1}$, then
$Z(G)=\frac{(\Delta-2)n+2}{\Delta-1}$. Therefore, by Theorem 2.1, $G$ is
either $K_{n}$ or $C_{n}$ or $K_{\frac{n}{2},\frac{n}{2}}$ .

Case 1: Suppose $\Phi=(K_{n},\varphi)$ and
$m_{\alpha}(\Phi,\lambda)=\frac{(\Delta-2)n+2}{\Delta-1}$. Then
$m_{\alpha}(\Phi,\lambda)=n-1.$ Therefore, $\lambda$ is an eigenvalue of
$A_{\alpha}(\Phi)$ with multiplicity $(n-1)$. By Lemma 3.3, statement $(i)$
holds.

Case 2: Suppose $\Phi=(C_{n},\varphi)$ and
$m_{\alpha}(\Phi,\lambda)=\frac{(\Delta-2)n+2}{\Delta-1}.$ Then
$m_{\alpha}(\Phi,\lambda)=2.$ Therefore, by Corollary 2.1, either statement
$(ii)$ or statement $(iii)$ holds.

Case 3: Suppose $\Phi=(K_{\frac{n}{2},\frac{n}{2}},\varphi)$ and
$m_{\alpha}(\Phi,\lambda)=\frac{(\Delta-2)n+2}{\Delta-1}$. Then
$m_{\alpha}(\Phi,\lambda)=n-2$. Then there is an eigenvalue $\mu$ of $A(\Phi)$
with multiplicity $(n-2)$ such that $\lambda=\frac{\alpha
n}{2}+(1-\alpha)\mu$. Since $\Phi$ is bipartite, so by Theorem 2.4, the
eigenvalues are symmetric about origin. Then $\mu=0$. Therefore $r(\Phi)=2$.
Let $(C_{4},\varphi)$ be an induced $4$-cycle in $\Phi$. Using Lemma 2.2,
$2\leq r(C_{4},\varphi)\leq r(\Phi)=2$. Since $r(C_{4},\varphi)=2$, so by
Lemma 2.3, $(C_{4},\varphi)$ is of type A and hence $\varphi(C_{4})=1$.
Therefore, any $4$-cycle in $\Phi$ is neutral. Let us take an arbitrary cycle
$C_{2k}\equiv v_{1}-v_{2}-\cdots-v_{2k}$. Then

$\displaystyle\varphi(\overrightarrow{C_{2k}})$
$\displaystyle=\varphi(\overrightarrow{e_{1,2}})\varphi(\overrightarrow{e_{2,3}})\cdots\varphi(\overrightarrow{e_{(2k-1),2k}})$
$\displaystyle=\\{\varphi(\overrightarrow{e_{1,2}})\varphi(\overrightarrow{e_{2,3}})\varphi(\overrightarrow{e_{3,4}})\varphi(\overrightarrow{e_{4,1}})\\}$
$\displaystyle~{}~{}~{}~{}\\{\varphi(\overrightarrow{e_{1,4}})\varphi(\overrightarrow{e_{4,5}})\varphi(\overrightarrow{e_{5,6}})\varphi(\overrightarrow{e_{6,1}})\\}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\vdots$
$\displaystyle~{}~{}~{}~{}\\{\varphi(\overrightarrow{e_{1,(2k-2)}})\varphi(\overrightarrow{e_{(2k-2),(2k-1)}})\varphi(\overrightarrow{e_{(2k-1),2k}})\varphi(\overrightarrow{e_{2k,1}})\\}$
$\displaystyle=1.$

Therefore $\Phi\sim(K_{\frac{n}{2},\frac{n}{2}},1)$ and $\lambda=\frac{\alpha
n}{2}$. Thus statement $(iv)$ holds.

The converse is easy to verify. ∎

######  Remark 3.1.

It is easy to see that Theorem 3.1 is an extension of Theorem 1.1. Also, the
above proof simplifies the proof of Theorem 1.1.

##  4 Lower bounds of rank for connected $\mathbb{T}$-gain graph

In this section, we establish two lower bounds for the rank of a
$\mathbb{T}$-gain graph $\Phi$ in terms of the number of vertices $n$ and the
maximum vertex degree $\Delta$. The first bound is a consequence of Theorem
3.1.

######  Theorem 4.1.

Let $\Phi=(G,\varphi)$ be a connected $\mathbb{T}$-gain graph of $n$ vertices
with rank $r(\Phi)$ and the maximum vertex degree $\Delta\geq 2$. Then

$r(\Phi)\geq\frac{n-2}{\Delta-1}.$

Equality holds if and only if $\Phi$ is either
$(K_{\frac{n}{2},\frac{n}{2}},1)$ or $(C_{n},\varphi)$, where $n$ is even and
$\varphi(C_{n})=\pm 1$.

###### Proof.

Let $\Phi=(G,\varphi)$ be a connected $\mathbb{T}$-gain graph. Then for
$\alpha\in[0,1)$, we have$A_{\alpha}(\Phi)=\alpha D(\Phi)+(1-\alpha)A(\Phi)$
and $m_{\alpha}(\Phi,\lambda)$ is the multiplicity of $\lambda$ as an
eigenvalue of $A_{\alpha}(\Phi)$. Consider $\alpha=0$ and $\lambda=0$. Then
$A_{0}(\Phi)=A(\Phi)$ and $m_{0}(\Phi,0)$ is the nullity of $\Phi$. That is,
$m_{0}(\Phi,0)=\eta(\Phi)$. Therefore, by Theorem 3.1,
$\eta(\Phi)\leq\frac{(\Delta-2)n+2}{(\Delta-1)}.$ Also $\eta(\Phi)=n-r(\Phi)$,
so

$r(\Phi)\geq\frac{n-2}{\Delta-1}.$ (1)

Since rank of the adjacency matrix of $\Phi=(K_{n},\varphi)$ is at least $2$,
so by Theorem 3.1, equality occurs in (1) if and only if $\Phi$ is either
$(K_{\frac{n}{2},\frac{n}{2}},1)$ or $(C_{n},\varphi)$, where $n$ is even and
$\varphi(C_{n})=\pm 1$. ∎

Let $\Phi=(G,\varphi)$ be a connected $\mathbb{T}$-gain graph with $n$
vertices and the maximum vertex degree $\Delta$. Then either $n<2\Delta$ or
$n\geq 2\Delta$. If $n<2\Delta$, then $r(\Phi)\geq
2>\frac{n}{\Delta}\geq\frac{n-2}{\Delta-1}$. If $n\geq 2\Delta$, then
$r(\Phi)\geq\frac{n-2}{\Delta-1}\geq\frac{n}{\Delta}$. Therefore,
$\frac{n-2}{\Delta-1}$ is better than $\frac{n}{\Delta}$. Thus the bound
derived in the above theorem improves the lower bound of $r(\Phi)$ given in
Theorem 1.2.

Now we establish a bound for $r(\Phi)$ in terms of $n$ and $\Delta$.

######  Theorem 4.2.

Let $\Phi$ be any connected $\mathbb{T}$-gain graph with $n$ vertices and the
maximum vertex degree $\Delta(\Phi)\geq 3$. Then

$r(\Phi)\geq\frac{n}{\Delta-1}$

equality holds if and only if
$\Phi\notin\\{(K_{\frac{n}{2},\frac{n}{2}},1),(K_{\frac{n+1}{2},\frac{n-1}{2}},1)\\}$.

###### Proof.

Let $\Phi$ be any connected $\mathbb{T}$-gain graph. Then, by Lemma 3.2 and
Theorem 2.2,

$\eta(\Phi)\leq M(\Phi)\leq Z(\Gamma(\Phi))\leq\frac{(\Delta-2)n}{\Delta-1}.$
(2)

Now the right most inequality in (2) holds if and only if
$\Gamma(\Phi)\notin\\{G_{1},G_{2},K_{n},K_{\frac{n}{2},\frac{n}{2}},K_{\frac{n+1}{2},\frac{n-1}{2}}\\}$.
Therefore, if
$\Gamma(\Phi)\notin\\{G_{1},G_{2},K_{n},K_{\frac{n}{2},\frac{n}{2}},K_{\frac{n+1}{2},\frac{n-1}{2}}\\}$,
then $r(\Phi)\geq\frac{n}{\Delta-1}$ holds. From (2),
$\eta(\Phi)\leq\frac{(\Delta-2)n}{\Delta-1}$ is possible, even if
$Z(\Gamma(\Phi))>\frac{(\Delta-2)n}{\Delta-1}$. So, for some of the
$\mathbb{T}$-gain graphs whose underlying graphs in
$\\{G_{1},G_{2},K_{n},K_{\frac{n}{2},\frac{n}{2}},K_{\frac{n+1}{2},\frac{n-1}{2}}\\}$
may have rank greater or equal to $\frac{n}{\Delta-1}$. Let
$L=\frac{n}{\Delta-1}.$

Case 1: Let $\Phi$ be a gain graph with $\Gamma(\Phi)=G_{1}.$ Then $L=2.5$.
Suppose $r(\Phi)=2$. Then by Lemma 2.2, all induced subgraphs of $\Phi$ have
rank at most $2$. Since the rank of any non empty graph is at least $2$. So
all non empty induced subgraphs of $\Phi$ have rank $2$. Consider two cyclic
subgraphs of length $4$, namely $\Phi_{1}=(C_{1},\varphi)$ and
$\Phi_{2}=(C_{2},\varphi)$, where $C_{1}\equiv v_{1}-v_{2}-v_{3}-v_{4}-v_{1}$
and $C_{2}\equiv v_{1}-v_{2}-v_{3}-v_{5}-v_{1}$. Now
$r(\Phi_{1})=r(\Phi_{2})=2$. Therefore, by Theorem 2.3, $\Phi_{1}$ and
$\Phi_{2}$ are balanced. Hence both the $3$-cycles in $\Phi$ are balanced. Now
the rank of any balanced $3$-cycles is $3$, a contradiction. Thus
$r(\Phi)\geq\frac{n}{\Delta-1}$ holds for any $\Phi$.

Case 2: Let $\Phi$ be a gain graph with $\Gamma(\Phi)=G_{2}.$ Then
$L=\frac{7}{3}=2.333$. Since $\Phi$ can not have rank $2$, so
$r(\Phi)\geq\frac{n}{\Delta-1}$ holds for any $\Phi$.

Case 3: Let $\Phi$ be a gain graph with $\Gamma(\Phi)=K_{n}.$ Then
$L=1+\frac{2}{n-2}$. Therefore, $r(\Phi)\geq 2\geq\frac{n}{\Delta-1}$ holds
for any $\Phi$.

Case 4: Let $\Phi$ be a gain graph with
$\Gamma(\Phi)=K_{\frac{n}{2},\frac{n}{2}}.$ Then $L>2$. Therefore $r(\Phi)\geq
3$. Using Case 3 of Theorem 3.1, the $\mathbb{T}$-gain graph $\Phi$ is of rank
$2$ if and only if $\Phi$ is balanced. Therefore, inequality holds for any
$\mathbb{T}$-gain graph $\Phi=(K_{\frac{n}{2},\frac{n}{2}},\varphi)$ except
$(K_{\frac{n}{2},\frac{n}{2}},1)$.

Case 5: Let $\Phi$ be a gain graph with
$\Gamma(\Phi)=K_{\frac{n+1}{2},\frac{n-1}{2}}$. Then $L>2$. Therefore, similar
to case 4, $r(\Phi)\geq\frac{n}{\Delta-1}$ holds for any $\Phi$ except
$(K_{\frac{n+1}{2},\frac{n-1}{2}},1)$.

∎

######  Remark 4.1.

It is easy to see that $\frac{n+1}{\Delta}\leq\frac{n}{\Delta-1}$ holds. The
bound for $r(\Phi)$ in Theorem 4.2 is better than that of Theorem 1.3.

## Acknowledgments

Aniruddha Samanta thanks University Grants Commission(UGC) for the financial
support in the form of the Senior Research Fellowship (Ref.No:
19/06/2016(i)EU-V; Roll No. 423206). M. Rajesh Kannan would like to thank the
SERB, Department of Science and Technology, India, for financial support
through the projects MATRICS (MTR/2018/000986) and Early Career Research Award
(ECR/2017/000643).

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