BIPARTITE DIVISOR GRAPH FOR THE SET OF IRREDUCIBLE CHARACTER DEGREES

Abstract

Let G be a finite group. We consider the set of the irreducible complex characters of G, namely Irr(G), and the related degree set cd(G) = {chi(1) : chi is an element of Irr(G)}. Let rho(G) be the set of all primes which divide some character degree of G. In this paper we introduce the bipartite divisor graph for cd(G) as an undirected bipartite graph with vertex set rho(G) (cd(G) \ {1}), such that an element p of p(G) is adjacent to an element m of cd(G) \ {1} if and only if p divides m. We denote this graph simply by B(G). Then by means of combinatorial properties of this graph, we discuss the structure of the group G. In particular, we consider the cases where B(G) is a path or a cycle.