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

TL;DR

This paper explicitly derives the Moore-Penrose inverse of an undirected graph's adjacency matrix using Tikhonov regularization, providing a new analytical approach for inverse computation.

Contribution

It introduces a novel explicit formula for the generalized inverse of the adjacency matrix of an undirected graph using Tikhonov regularization techniques.

Findings

Explicit formula for the Moore-Penrose inverse of adjacency matrix

Proof of nonsingularity of the regularized matrix R_λ

Limit expression for the inverse as λ approaches infinity

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 $r_{ii}=\frac{1}{\lambda}+ \deg v_i$ and $r_{ij}=\mid N_G(V_i)\cap N_G(V_j)\mid$ for $i\neq j$, and we proved that $A^{\dagger}_G=[s_{ij}]_{1\leq i, j \leq n}$ 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.