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ON THE REGULARITY OF CHARACTER DEGREE GRAPHS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 100 / Issue 3 / December 2019
- Published online by Cambridge University Press:
- 08 March 2019, pp. 428-433
- Print publication:
- December 2019
-
- Article
- Export citation
-
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.
