Comparing the irregularity and the total irregularity of graphs

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 ).