Edge-grafting theorems on permanents of the Laplacian matrices of graphs and their applications

TL;DR

This paper investigates the minimal Laplacian permanents of trees and unicyclic graphs with given bipartitions, using edge-grafting transformations to identify those with the smallest values.

Contribution

It introduces edge-grafting techniques to characterize and identify bipartite trees and unicyclic graphs with the smallest Laplacian permanents.

Findings

Identified trees with second and third smallest Laplacian permanent

Characterized bipartite unicyclic graphs with the smallest Laplacian permanents

Determined the top three bipartite unicyclic graphs with minimal Laplacian permanents

Abstract

The trees, respectively unicyclic graphs, on $n$ vertices with the smallest Laplacian permanent are studied. In this paper, by edge-grafting transformations, the $n$-vertex trees of given bipartition having the second and third smallest Laplacian permanent are identified. Similarly, the $n$-vertex bipartite unicyclic graphs of given bipartition having the first, second and third smallest Laplacian permanent are characterized. Consequently, the $n$-vertex bipartite unicyclic graphs with the first, second and third smallest Laplacian permanent are determined.