# Nordhaus–Gaddum type results for graph irregularities

Yuede Ma, Shujuan Cao, Yongtang Shi, Matthias Dehmer, Chengyi Xia

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)

## 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 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.

Original language English 268-272 5 Applied Mathematics and Computation 343 https://doi.org/10.1016/j.amc.2018.09.057 Published - 15 Feb 2019

## Keywords

• Degree
• Graph irregularity