On regular bipartite divisor graph for the set of irreducible character degrees

Authors : Roghayeh HAFEZİEH

Pages : 1620-1625

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Publication Date : 2019-12-08

Article Type : Research Paper

Abstract :Given a finite group $G$, the \textit{bipartite divisor graph}, denoted by $Binsert ignore into journalissuearticles values(G);$, for its irreducible character degrees is the bipartite graph with bipartition consisting of $cdinsert ignore into journalissuearticles values(G);^{*}$, where $cdinsert ignore into journalissuearticles values(G);^{*}$ denotes the nonidentity irreducible character degrees of $G$ and the $\rhoinsert ignore into journalissuearticles values(G);$ which is the set of prime numbers that divide these degrees, and with $\{p,n\}$ being an edge if $\gcdinsert ignore into journalissuearticles values(p,n);\neq 1$. In [Bipartite divisor graph for the set of irreducible character degress, Int. J. Group Theory, 2017], the author considered the cases where $Binsert ignore into journalissuearticles values(G);$ is a path or a cycle and discussed some properties of $G$. In particular she proved that $Binsert ignore into journalissuearticles values(G);$ is a cycle if and only if $G$ is solvable and $Binsert ignore into journalissuearticles values(G);$ is either a cycle of length four or six. Inspired by $2$-regularity of cycles, in this paper we consider the case where $Binsert ignore into journalissuearticles values(G);$ is an $n$-regular graph for $n\in\{1,2,3\}$. In particular we prove that there is no solvable group whose bipartite divisor graph is $C_{4}+C_{6}$.
Keywords : bipartite divisor graph, irreducible character degrees, regular graph