Comparing the irregularity and the total irregularity of graphs

Abstract

Albertson has defined the irregularity of a simple undirected graph G as irr( G ) = ∑ u v ∈ E ( G ) ∣ d G ( u ) − d G ( v )∣, where d G ( u ) denotes the degree of a vertex u ∈ V ( G ) . Recently, in a new measure of irregularity of a graph, so-called the total irregularity , was defined as irr t ( G ) = 1/2 ∑ u , v ∈ V ( G ) ∣ d G ( u ) − d G ( v )∣. Here, we compare the irregularity and the total irregularity of graphs. For a connected graph G with n vertices, we show that irr t ( G ) ≤ n 2 irr( G ) / 4. Moreover, if G is a tree, then irr t ( G ) ≤ ( n − 2)irr( G ).