Comparing the irregularity and the total irregularity of graphs

Abstract

Albertson has defined the irregularity of a simple undirected graph G as irr(G) = ∑ _{uv ∈ 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) ≤ n²irr(G) / 4.  Moreover, if G is a tree, then irr_t(G) ≤ (n − 2)irr(G).