Lower bounds on the irregularity of trees and unicyclic graphs

The imbalance of an edge in a graph is defined as the absolute value of the difference of the degrees of its end-vertices. The irregularity of a simple graph G is defined as the sum of the imbalances of all edges of G. It is used as a measure to quantify the deviation of a graph from being regular. In this paper, a sharp lower bound on the irregularity of trees and unicyclic graphs in terms of the maximum degree is established and the corresponding extremal trees and unicyclic graphs are characterized.