On The Laplacian Coefficients Of Cacti Graphs

A cactus graph is a connected graph if any two of its cycles have at most one common vertex. Let φ（G,χ）=∑k=1n（-1）kckλn-k be the characteristic polynomial of Laplacian matrix of a n-vertex connected graph G. It is well known that if G is a tree then cn₂and cn₃are equal to the Wiener index and modified hyper-Wiener index of G, respectively. By using some transformations of graphs, we obtain some results of Laplacian coefficients of graphs as follows:（1） Among all cacti graphs with given order, cycle number and matching number, we determine the unique graph that simultaneously minimizes all Laplacian coefficients and characterize the unique graph with the smallest Laplacian-like energy.（2） Among all cacti graphs with given order and cycle number, we determine the unique graph that simultaneously minimizes all Laplacian coefficients and characterize the unique graph with the smallest Laplacian-like energy.（3） We give a transformation of trees and investigate the property of Laplacian coefficients of trees under the transformation.