Inverting non-invertible trees

• Main Authors: Pavlíková, Soňa, Širáň, Jozef
• Format: Journal Article
• Language: English
• Published: 30-12-2017
• Subjects: Mathematics - Combinatorics
• DOI: 10.48550/arxiv.1801.00111

If a graph has a non-singular adjacency matrix, then one may use the inverse
matrix to define a (labeled) graph that may be considered to be the inverse
graph to the original one. It has been known that an adjacency matrix of a tree
is non-singular if and only if the tree has a unique perfect matching; in this
case the determinant of the matrix turns out to be $\pm 1$ and the inverse of
the tree was shown to be `switching-equivalent' to a simple graph [C. Godsil,
Inverses of Trees, Combinatorica 5 (1985), 33--39]. Using generalized inverses
of symmetric matrices (that coincide with Moore-Penrose, Drazin, and group
inverses in the symmetric case) we prove a formula for determining a
`generalized inverse' of a tree.