Nordhaus–Gaddum type results for graph irregularities

Abstract A graph whose vertices have the same degree is called regular. Otherwise, the graph is irregular. In fact, various measures of irregularity have been proposed and examined. For a given graph G = ( V , E ) with V = { v 1 , v 2 , … , v n } and edge set E ( G ), d i is the vertex degree where 1 ≤  i  ≤  n . The irregularity of G is defined by i r r ( G ) = ∑ v i v j ∈ E ( G ) | d i − d j | . A similar measure can be defined by i r r 2 ( G ) = ∑ v i v j ∈ E ( G ) ( d i − d j ) 2 . The total irregularity of G is defined by i r r t ( G ) = 1 2 ∑ v i , v j ∈ V ( G ) | d i − d j | . The variance of the vertex degrees is defined v a r ( G ) = 1 n ∑ i = 1 n d i 2 − ( 2 m n ) 2 . In this paper, we present some Nordhaus–Gaddum type results for these measures and draw conclusions.