Nordhaus–Gaddum type results for graph irregularities

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 irr(G)=∑ _{v i v j ∈E(G)} |d _i −d _j |. A similar measure can be defined by irr ₂ (G)=∑ _{v i v j ∈E(G)} (d _i −d _j ) ² . The total irregularity of G is defined by irr _t (G)=[Formula presented]∑ _{v i ,v j ∈V(G)} |d _i −d _j |. The variance of the vertex degrees is defined var(G)=[Formula presented]∑ _i=1 ⁿ d _i ² −([Formula presented]) ² . In this paper, we present some Nordhaus–Gaddum type results for these measures and draw conclusions.