On the Moore-Penrose pseudo-inversion of block symmetric matrices and its application in the graph theory

TL;DR

This paper analyzes the Moore-Penrose pseudo-inversion of block symmetric matrices, introduces new concepts of positive and negative pseudo-inverses in graph theory, and provides explicit formulas for constructing pseudo-inverse graphs.

Contribution

It introduces novel concepts of positive and negative pseudo-inverses for matrices and graphs, and derives explicit conditions and formulas for pseudo-inversion of block symmetric matrices.

Findings

Explicit form of Moore-Penrose pseudo-inverse for certain block symmetric matrices

Construction method for pseudo-inverse graphs with pendent vertices or paths

Introduction of positively and negatively pseudo-inverse matrices and graphs

Abstract

The purpose of this paper is to analyze the Moore-Penrose pseudo-inversion of symmetric real matrices with application in the graph theory. We introduce a novel concept of positively and negatively pseudo-inverse matrices and graphs. We also give sufficient conditions on the elements of a block symmetric matrix yielding an explicit form of its Moore-Penrose pseudo-inversion. Using the explicit form of the pseudo-inverse matrix we can construct pseudo-inverse graphs for a class of graphs which are constructed from the original graph by adding pendent vertices or pendant paths.