Exact and asymptotic distributions of the degree variance are investigated for Bernoulli graphs and uniform random graphs. In particular the range of values of the degree variance and its maximum value are considered. We show that the degree variance is approximately gamma distributed with parameters obtained from the first two moments of the degree variance. Since centrality of a graph can be interpreted as a measure of its heterogeneity in terms of vertex degrees, we can perform a centrality test with a critical value obtained from the gamma distribution.
Key words: Centrality Testing, Bernoulli Graphs, Degree Variance, Gamma Approximation, Uniform Random Graphs
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