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ON THE REGULARITY OF CHARACTER DEGREE GRAPHS

Part of: Representation theory of groups Graph theory

Published online by Cambridge University Press:  08 March 2019

Z. SAYANJALI Open the ORCID record for Z. SAYANJALI [Opens in a new window] ,
Z. AKHLAGHI Open the ORCID record for Z. AKHLAGHI [Opens in a new window]  and
B. KHOSRAVI Open the ORCID record for B. KHOSRAVI [Opens in a new window]
Z. SAYANJALI
Affiliation:
Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), 15914 Tehran, Iran email zohrehsayanjali@aut.ac.ir
Z. AKHLAGHI*
Affiliation:
Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), 15914 Tehran, Iran email z_akhlaghi@aut.ac.ir
B. KHOSRAVI
Affiliation:
Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), 15914 Tehran, Iran email khosravibbb@yahoo.com
*
✉z_akhlaghi@aut.ac.ir
Rights & Permissions [Opens in a new window]

Abstract

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Let $G$ be a finite group and let $\text{Irr}(G)$ be the set of all irreducible complex characters of $G$. Let $\unicode[STIX]{x1D70C}(G)$ be the set of all prime divisors of character degrees of $G$. The character degree graph $\unicode[STIX]{x1D6E5}(G)$ associated to $G$ is a graph whose vertex set is $\unicode[STIX]{x1D70C}(G)$, and there is an edge between two distinct primes $p$ and $q$ if and only if $pq$ divides $\unicode[STIX]{x1D712}(1)$ for some $\unicode[STIX]{x1D712}\in \text{Irr}(G)$. We prove that $\unicode[STIX]{x1D6E5}(G)$ is $k$-regular for some natural number $k$ if and only if $\overline{\unicode[STIX]{x1D6E5}}(G)$ is a regular bipartite graph.

Keywords

finite groupscharacter degree graph

MSC classification

Primary: 20C15: Ordinary representations and characters
Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 100 , Issue 3 , December 2019 , pp. 428 - 433
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

References

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