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

#### Roghayeh HAFEZİEH [1]

Given a finite group $G$, the \textit{bipartite divisor graph}, denoted by $B(G)$, for its irreducible character degrees is the bipartite graph with bipartition consisting of $cd(G)^{*}$, where $cd(G)^{*}$ denotes the nonidentity irreducible character degrees of $G$ and the $\rho(G)$ which is the set of prime numbers that divide these degrees, and with $\{p,n\}$ being an edge if $\gcd(p,n)\neq 1$. In [Bipartite divisor graph for the set of irreducible character degress, Int. J. Group Theory, 2017], the author considered the cases where $B(G)$ is a path or a cycle and discussed some properties of $G$. In particular she proved that $B(G)$ is a cycle if and only if $G$ is solvable and $B(G)$ is either a cycle of length four or six. Inspired by $2$-regularity of cycles, in this paper we consider the case where $B(G)$ 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 $C_{4}+C_{6}$.
bipartite divisor graph, irreducible character degrees, regular graph
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Primary Language en Mathematics Mathematics Orcid: 0000-0001-6486-7736Author: Roghayeh HAFEZİEH (Primary Author)Institution: GEBZE TECHNICAL UNIVERSITYCountry: Turkey Publication Date : December 8, 2019
 Bibtex @research article { hujms656649, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2019}, volume = {48}, pages = {1620 - 1625}, doi = {}, title = {On regular bipartite divisor graph for the set of irreducible character degrees}, key = {cite}, author = {HAFEZİEH, Roghayeh} } APA HAFEZİEH, R . (2019). On regular bipartite divisor graph for the set of irreducible character degrees. Hacettepe Journal of Mathematics and Statistics , 48 (6) , 1620-1625 . Retrieved from https://dergipark.org.tr/en/pub/hujms/issue/50516/656649 MLA HAFEZİEH, R . "On regular bipartite divisor graph for the set of irreducible character degrees". Hacettepe Journal of Mathematics and Statistics 48 (2019 ): 1620-1625 Chicago HAFEZİEH, R . "On regular bipartite divisor graph for the set of irreducible character degrees". Hacettepe Journal of Mathematics and Statistics 48 (2019 ): 1620-1625 RIS TY - JOUR T1 - On regular bipartite divisor graph for the set of irreducible character degrees AU - Roghayeh HAFEZİEH Y1 - 2019 PY - 2019 N1 - DO - T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1620 EP - 1625 VL - 48 IS - 6 SN - 2651-477X-2651-477X M3 - UR - Y2 - 2018 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics On regular bipartite divisor graph for the set of irreducible character degrees %A Roghayeh HAFEZİEH %T On regular bipartite divisor graph for the set of irreducible character degrees %D 2019 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 48 %N 6 %R %U ISNAD HAFEZİEH, Roghayeh . "On regular bipartite divisor graph for the set of irreducible character degrees". Hacettepe Journal of Mathematics and Statistics 48 / 6 (December 2019): 1620-1625 . AMA HAFEZİEH R . On regular bipartite divisor graph for the set of irreducible character degrees. Hacettepe Journal of Mathematics and Statistics. 2019; 48(6): 1620-1625. Vancouver HAFEZİEH R . On regular bipartite divisor graph for the set of irreducible character degrees. Hacettepe Journal of Mathematics and Statistics. 2019; 48(6): 1625-1620.