Some Extremal Problems On The Irregularity Of A Graph

The irregularity of a graph is also known as the third Zagreb index,is an important index to measure the irregularity of a graph,it has a natural link to the structural characteristics of a graph,and it also plays an important role in studying the structural properties of organic compounds.This paper mainly studies the properties and extremal problems on the irregularity of some graphs.The effect on the irregularity of a graph under some graph transformations(contraction of non-pendant edges,contraction of non-pendant edges and suspension adding pendant edge,delete one of the vertices of maximum degree or minimum degree of a graph)are explored;the maximum(or minimum)values for the irregularity of two kinds of trees(the trees with given degree sequence and the trees with given branching vertices)are determined,and the corresponding graphs are constructed,respectively;at the same time,the graph with maximum irregularity among the cactus with given number of cycles,cactus with given number of cycles and pendant vertices or cactus with given number of cycles and perfect matchings are determined,respectively.