Summary: In this paper, we define and compare three new measures of graph irregularity. We use these measures to tighten upper bounds for the chromatic number and the Colin de Verdi\`re parameter. We also strengthen the concise Turán theorem for irregular graphs and investigate to what extent Turán’s theorem can be similarly strengthened for generalized \(r\)-partite graphs. We conclude by relating these new measures to the Randić index and using the measures to devise new normalised indices of network heterogeneity.