# Student Probability Seminar

#### Trees in Random Sparse Graphs with Given Degree Sequence

Let $$G^D$$ be the set of graphs $$G(V, E)$$, $$|V| = n$$, with degree sequence equal to $$D = (d_1, d_2, . . . , d_n)$$. What does a graph look like when it is chosen uniformly out of $$G^D$$? This has been studied when $$G$$ is a dense graph ,$$|E| = O(n^2)$$, in the sense of graphons or when $$G$$ is very sparse, $$(d_n)^2 = o(|E|)$$. We investigate this question in the case of sparse graphs with almost given degree sequence, and give the finite tree subgraph structure of $$G$$, under some mild conditions. For graphs with given degree sequence, we re-derive the tree structure in dense and very sparse case to give a continuous picture. Moreover, we are able to show the result for general bipartite graphs with given degree sequence without any further conditions.