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

Abstract

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 $r_{ii}=\frac{1}{\lambda }+°v_i$ and $r_{ij}=\mid N_G(V_i)\cap N_G(V_j)\mid$ for $i\ne j$, and we proved that $A_G^{†}=[s_{ij}{]}_{1\le i,j\le n}$ where $s_{ij}=s_{ji}=\underset{\lambda \to +\infty }{\lim }⟨R_{\lambda }^{-1}{e}_j,f_i⟩$. The proof of the main result was based on the Tikhonov regularization.

6, articles