Application of Tikhonov Regularization in Generalized Inverse of Adjacency Matrix of Undirected Graph

In this paper, we found the Moore-Penrose generalized inverse of adjacency matrix of an undirected graph, explicitly. We proved that the matrix $R_{\lambda }=[r_{ij}]$ is nonsingular where $r_{ii}=\frac{1}{\lambda }+\mathrm{deg}{v}_i$ and $r_{ij}=\mid N_G(V_i)\cap N_G(V_j)\mid$ for $i\mathrm{\ne }j$, and we proved that $A_G^{†}=[s_{ij}{]}_{1\le i,j\le n}$ where $\displaystyle{s_{ij}=s_{ji}=\lim_{\lambda \rightarrow +\infty} \langle R^{-1}_{\lambda}e_j, f_i \rangle }$. The proof of the main result was based on the Tikhonov regularization.