The signless Laplacian coefficients and the incidence energy of graphs with a given bipartition

Abstract

We consider two classes of the graphs with a given bipartition. One is trees and the other is unicyclic graphs. The signless Laplacian coefficients and the incidence energy are investigated for the sets of trees/unicyclic graphs with $n$ vertices in which each tree/unicyclic graph has an $(n_1,n_2)$-bipartition,  where $n_1$ and $n_2$ are positive integers not less than 2 and $n_1+n_2=n$. Four new graph transformations are proposed for studying the signless Laplacian coefficients.  Among the sets of trees/unicyclic graphs considered, we obtain exactly, for each,  the minimal element with respect to the quasi-ordering according to their signless Laplacian coefficients and the element with the minimal incidence energies.