Hafezieh, R. (2017). Bipartite divisor graph for the set of irreducible character degrees. International Journal of Group Theory, 6(4), 41-51. doi: 10.22108/ijgt.2017.21221

Roghayeh Hafezieh. "Bipartite divisor graph for the set of irreducible character degrees". International Journal of Group Theory, 6, 4, 2017, 41-51. doi: 10.22108/ijgt.2017.21221

Hafezieh, R. (2017). 'Bipartite divisor graph for the set of irreducible character degrees', International Journal of Group Theory, 6(4), pp. 41-51. doi: 10.22108/ijgt.2017.21221

Hafezieh, R. Bipartite divisor graph for the set of irreducible character degrees. International Journal of Group Theory, 2017; 6(4): 41-51. doi: 10.22108/ijgt.2017.21221

Bipartite divisor graph for the set of irreducible character degrees

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 $cd(G)=\{\chi(1) : \chi\in Irr(G)\}$. 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 $\rho(G)\cup (cd(G)\setminus\{1\})$, such that an element $p$ of $\rho(G)$ is adjacent to an element $m$ of $cd(G)\setminus\{1\}$ 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.

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