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) = Sigma(vivj is an element of E(G)) vertical bar d(i) - d(j)vertical bar. A similar measure can be defined by irr(2)(G) = Sigma(vivj is an element of E(G) )(d(i) - d(j))(2). The total irregularity of G is defined by irr(t) (G) = 1/2 Sigma(vivj is an element of v(G)) vertical bar d(i) - d(j)vertical bar. The variance of the vertex degrees is defined var(G) = 1/n Sigma(n )(i=1)d(i)(2) - (2m/n)(2). In this paper, we present some Nordhaus-Gaddum type results for these measures and draw conclusions. (C) 2018 Elsevier Inc. All rights reserved.