Expected Maximum Block Size in Critical Random Graphs

Abstract

AbstractLetG(n,M) be a uniform random graph withnvertices andMedges. Let${\wp_{n,m}}$be the maximum block size ofG(n,M), that is, the maximum size of its maximal 2-connected induced subgraphs. We determine the expectation of${\wp_{n,m}}$near the critical pointM=n/2. Whenn− 2M≫n2/3, we find a constantc1such that$$c_1 = \lim_{n \rightarrow \infty} \left({1 - \frac{2M}{n}} \right) \,\E({\wp_{n,m}}).$$Inside the window of transition ofG(n,M) withM= (n/2)(1 + λn−1/3), where λ is any real number, we find an exact analytic expression for$$c_2(\lambda) = \lim_{n \rightarrow \infty} \frac{\E{\left({\wp_{n,{{(n/2)}({1+\lambda n^{-1/3}})}}}\right)}}{n^{1/3}}.$$This study relies on the symbolic method and analytic tools from generating function theory, which enable us to describe the evolution of$n^{-1/3}\,\E{\left({\wp_{n,{{(n/2)}({1+\lambda n^{-1/3}})}}}\right)}$as a function of λ.