\section{Proof of Lemma \ref{lemma1}}\label{prooflemma1}
The Lagrangian function of \eqref{eqn:C} is

\begin{align}
L(C, \tilde{X}, \boldsymbol{\mu}_1)= & \frac{1}{2}C^{\top}C-C^{\top}A - \boldsymbol{\mu}_1^\top C 
\end{align}
where $\boldsymbol{\mu}_1$ is the dual variable. The KKT conditions of \eqref{eqn:C} is
\begin{align}
C-A-\boldsymbol{\mu}_1=0, \label{eq C 3}\\
\boldsymbol{\mu}^\top C =0,\label{eq C 4}\\
\quad C \geq 0, \label{eq C 5}\\
 \quad  \boldsymbol{\mu}_1 \geq 0 \label{eq C 6}
\end{align}
The optimal solution of $C$ that satisfies all KKT conditions (\ref{eq C 3}-\ref{eq C 6}) is


\begin{align}
C^{t+1}
&=(A)^{+}  \\
&= \left(C^{(t)} - \frac{1}{L}\nabla f\left(C^{(t)}\right)\right)^+
\end{align}
This concludes the proof.


\section{Proof of Theorem \ref{convergence} }\label{prooftheorem}
% NOTE: necessary when ptmx or no mathfont class option is given
The Lagrangian function of \eqref{Main formulation} is
\begin{gather}
L(C, \tilde{X},\boldsymbol{\mu}_1, \boldsymbol{\mu}_2)= -\gamma \text{log det}(C^T\Theta C +J) +\frac{\alpha}{2}||X-C\tilde X||_F^2 +\text{tr}(\tilde{X}^{T}C^T\Theta C\tilde{X}) +\frac \lambda 2 \sum \limits_{i=1}^p \|[C^T]_i\|_1^2 \\ \nonumber + \frac{\beta}{2} \|C^T\Theta C\|_F^2 + \frac{\delta}{2}\|C^TY\|_F^2 - \text{tr} (\boldsymbol{\mu}_1^\top C) + \boldsymbol{\mu}_2^T\Big[ (\|C_1^T\|_2^2-1), \ldots, (\|C_p^T\|_2^2-1) \Big]^T + \frac{\delta}{2}\|C^TY\|_F^2
\end{gather}

where $\boldsymbol{\mu}_{1_(p \times k)}$ and $\boldsymbol{\mu}_{2_(p \times 1)}$ are the dual variables.\\  
 The KKT conditions with respect to $C$ is

\noindent (i)$-2\gamma \Theta C(C^T\Theta C+J)^{-1}+\alpha \left(C\tilde{X} - X\right)\tilde{X}^{T} +2\Theta C\tilde{X}\tilde{X}^T+ \lambda C\pmb{1}_{k \times k} + \delta Y (C^\top Y)^\top- \boldsymbol{\mu}_1 + 2\Big[\mu_{21}C^T_1, \ldots \mu_{2p}C^T_p]^T+2\beta \Theta CC^T\Theta C=0$, \\
(ii) $ \left \{ \mu_{2i}(\|C_i^T\|_2^2-1) =0\right \}_{i=1}^{p}$, \quad $\text{tr}(\boldsymbol{\mu}_1^\top C) =0$\\   (iii) $ C \geq 0$, \ $\|[C^T]_i\|_2^2\leq 1 \ \forall \  i = 1,2, \ldots,p$,\\(iv) $\boldsymbol{\mu}_1 \geq 0$, \quad  $\boldsymbol{\mu}_2 \geq 0$  

where $\textbf{1}_{k\times k}$ is a $k \times k$ matrix whose all entry is one. The variable ${C}$ is derived by using the KKT condition from \eqref{eqn:C}:
\begin{gather}\label{eqns:C}
C^{\infty}-C^{\infty} + \frac{1}{L}\Big(-2\gamma \Theta C^{\infty}\Big((C^{\infty})^T\Theta C^{\infty}+J\Big)^{-1}  +\alpha (C^{\infty}\tilde{X}^{\infty} - X)(\tilde{X}^{\infty})^{T} + 2\beta \Theta C^{\infty}(C^{\infty})^T\Theta C^{\infty}\\ \nonumber  +2\Theta C^{\infty}\tilde{X}^{\infty}(\tilde{X}^{\infty})^{T} + \lambda C^{\infty}\pmb{1}_{k \times k} + \delta Y \big((C^\infty)^\top Y\big)^\top\Big)=0
\end{gather}
For $\boldsymbol{\mu}_1=0$ and $ \mu_{2i}[C^T]^{\infty}_{i}=0 \ \forall \ i=1,2,\ldots p$, we observe that $C^\infty$ satisfies the KKT condition.\\ 
The KKT condition with respect to $\tilde{X}$ is
\[2C^T\Theta C \tilde{X} + \alpha C^T(C\tilde{X} - X) = 0\]
The solution of convex optimization problem \eqref{problemXtilde} is \\
\[\tilde{X}^{\infty} =\left(\frac 2 \alpha (C^{\infty})^T\Theta C^{\infty} + C^TC\right)^{-1} (C^{\infty})^TX^{\infty}\]
which satisfies the KKT condition.
This concludes the proof.


%  \begin{figure*}[h!]
%   \centering
%   \begin{minipage}[b]{0.24\textwidth}
%     \includegraphics[width=\textwidth, height =  3.5 cm]{Original_label_matrixP.png}
%     \caption{Original Graph}
%     \label{tsne original}
%   \end{minipage}
%   \hfill
%   \begin{minipage}[b]{0.22\textwidth}
%     \includegraphics[width=\textwidth, height = 3.5cm]{LAGC_Cora_Coarsening Ratio_0.3.png}
%     \caption{LAGC Coarsened (r=0.3)}
%     \label{fgc 0.3}
%   \end{minipage}
% \hfill
%     \begin{minipage}[b]{0.22\textwidth}
%     \includegraphics[width=\textwidth, height = 3.5 cm ]{LAGC_Cora_Coarsening Ratio_0.1.png}
%     \caption{LAGC Coarsened(r=0.1)}
%     \label{fgc 0.1}
%   \end{minipage}
%   \hfill
%   \begin{minipage}[b]{0.22\textwidth}
%     \includegraphics[width=\textwidth , height = 3.5cm]{LAGC_Cora_Coarsening Ratio_0.05.png}
%     \caption{LAGC Coarsened(r=0.05)}
%     \label{fgc 0.05}
%   \end{minipage}
%     \nextline
    

%   \begin{minipage}[b]{0.22\textwidth}
%     \includegraphics[width=\textwidth, height =  3.5 cm]{Original_label_matrixP.png}
%     \caption{Original Graph}
%     \label{tsne original}
%   \end{minipage}
%   \hfill
%   \begin{minipage}[b]{0.22\textwidth}
%     \includegraphics[width=\textwidth, height = 3.5cm]{FGC_Cora_Coarsening Ratio_0.3.png }
%     \caption{FGC Coarsened (r=0.3)}
%     \label{fgc 0.3}
%   \end{minipage}
% \hfill
%     \begin{minipage}[b]{0.22\textwidth}
%     \includegraphics[width=\textwidth, height = 3.5 cm ]{FGC_Cora_Coarsening Ratio_0.1.png}
%     \caption{FGC Coarsened(r=0.1)}
%     \label{fgc 0.1}
%   \end{minipage}
%   \hfill
%   \begin{minipage}[b]{0.22\textwidth}
%     \includegraphics[width=\textwidth , height = 3.5cm]{FGC_Cora_Coarsening Ratio_0.05.png}
%     \caption{FGC Coarsened(r=0.05)}
%     \label{fgc 0.05}
%   \end{minipage}


 % \begin{figure}[h!]
%   \centering
 
%     \includegraphics[width=2in]{number of misclassified points vs c r.png}
    
%   \hfill
  
%   \caption{In Figure we have plotted number of misclassified points vs Coarsening Ratios of the CORA graph.}
% \end{figure}  




  
%   \caption{Figure (\ref{tsne original}) - (\ref{tsne 0.1}) illustrate the Comparison of Label Assignment by FGC and LAGC Algorithms Across Coarsening Ratios. Top row depicts the label assignment for LAGC, while the bottom row illustrates FGC. Each plot displays the number of classes on the x-axis and the corresponding number of nodes on the y-axis. The figures showcase the labels assigned by the algorithms after coarsening at 0.3, 0.1, and 0.05 coarsening ratios. }
%   \label{visualization}
% \end{figure*}
 