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Given a finite group G, it is an area of research to convey nontrivial information about the structure of G through some sets of invariants associated to G such as the set of conjugacy class sizes or the set of degrees of the irreducible complex characters of G and would be interesting to distinguish the group structure of G influenced by these sets. There is a large literature which is devoted to study the ways in which one can associate a graph with a group, for the purpose of investigating the algebraic structure using properties of the associated graph. In this talk we consider the set of the irreducible complex characters of G, namely Irr(G), and the related degree set cd(G) = {χ(1) : χ ∈ Irr(G)}. Let ρ(G) be the set of all primes which divide some character degrees of G. We consider the notion of bipartite divisor graph, denoted by B(G), with vertex set ρ(G)∪cd(G)\{1} and we determine which finite simple graphs can occur as bipartite divisor graphs of finite groups. This question has attracted many researchers over the years.