Bipartite divisor graph for the set of irreducible character degrees

Let G be a finite groupý. ýWe consider the set of the irreducible complex characters of Gý, ýnamely Irr(G)ý, ýand the related degree set cd(G)={χ(1)ý:ýχ∈Irr(G)}ý. ýLet ρ(G) be the set of all primes which divide some character degree of Gý. ýIn this paper we introduce the bipartite divisor graph for cd(G) as an undirected bipartite graph with vertex set ρ(G)∪(cd(G)∖{1})ý, ýsuch that an element p of ρ(G) is adjacent to an element m of cd(G)∖{1} if and only if p divides mý. ýWe denote this graph simply by B(G)ý. ýThen by means of combinatorial properties of this graphý, ýwe discuss the structure of the group Gý. ýIn particularý, ýwe consider the cases where B(G) is a path or a cycleý.