The Inverse of the Incidence Matrix of a Unicyclic Graph

TL;DR

This paper derives a combinatorial formula for the Moore-Penrose inverse of the incidence matrix of even unicyclic graphs, extending known results for odd unicyclic graphs and solving an open problem.

Contribution

It provides the first combinatorial formula for the Moore-Penrose inverse of the incidence matrix of even unicyclic graphs.

Findings

Derived the inverse formula for odd unicyclic graphs.

Presented the Moore-Penrose inverse formula for even unicyclic graphs.

Solved an open problem in graph matrix theory.

Abstract

The vertex-edge incidence matrix of a (connected) unicyclic graph G is a square matrix which is invertible if and only if the cycle of G is an odd cycle. A combinatorial formula of the inverse of the incidence matrix of an odd unicyclic graph was known. A combinatorial formula of the Moore-Penrose inverse of the incidence matrix of an even unicyclic graph is presented solving an open problem.