Algorithm 1: Spectral clustering process with a metric $\rho$.

Input: (data) ${\left\{x_i\right\}}_{i=1}^n$, (kernel function) $f_{\sigma }$, and (scaling parameter) $\sigma >0$ Output: $Y$(labels) with clustered values  1. The weight matrix $W\in {\Re }^{n\times n}$ is computed with $W_{ij}=f_{\sigma }\left(\rho \left(x_i,x_j\right)\right)$.  2. The diagonal degree matrix $D\in {\Re }^{n\times n}$ is computed with $D_{ii}={\displaystyle {\sum }_{j=1}^n{W}_{ij}}$.  3. The symmetric normalized Laplacian $L_{SYM}=I-D^{-\frac{1}{2}}W{D}^{-\frac{1}{2}}$ is formed.  4. The eigendecomposition ${\left\{\left({\varphi }_k,{\lambda }_k\right)\right\}}_{k=1}^n$ is computed and sorted so that $0={\lambda }_1\le {\lambda }_2\le \mathrm{\dots }\le {\lambda }_n$.  5. The number of clusters $K$ is estimated as: $\widehat{K}=\arg {\max }_k{\lambda }_{k+1}-{\lambda }_k$.  6. The row normalized spectral embedding is defined by $v_i={\left({\varphi }_1\left(x_i\right),{\varphi }_2\left(x_i\right),\mathrm{\dots },{\varphi }_{\widehat{k}}\left(x_i\right)\right)/\Vert {\varphi }_1\left(x_i\right),{\varphi }_2\left(x_i\right),\mathrm{\dots },{\varphi }_{\widehat{k}}\left(x_i\right)\Vert }_2$ for $1\le i\le n$.  7. By implementing K-means on the data ${\left\{v_i\right\}}_{i=1}^n$, the labels $Y$ are computed by utilizing $\widehat{K}$ as the total number of clusters, thereby implementing the concept of clustering successfully.