character degree graph
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2020 ◽  
Vol 23 (2) ◽  
pp. 217-234
Author(s):  
Mark L. Lewis ◽  
Qingyun Meng

AbstractLet G be a solvable group, and let {\Delta(G)} be the character degree graph of G. In this paper, we generalize the definition of a square graph to graphs that are block squares. We show that if G is a solvable group so that {\Delta(G)} is a block square, then G has at most two normal nonabelian Sylow subgroups. Furthermore, we show that when G is a solvable group that has two normal nonabelian Sylow subgroups and {\Delta(G)} is block square, then G is a direct product of subgroups having disconnected character degree graphs.



2019 ◽  
Vol 100 (3) ◽  
pp. 428-433 ◽  
Author(s):  
Z. SAYANJALI ◽  
Z. AKHLAGHI ◽  
B. KHOSRAVI

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.



2019 ◽  
Vol 198 (5) ◽  
pp. 1595-1614 ◽  
Author(s):  
Zeinab Akhlaghi ◽  
Carlo Casolo ◽  
Silvio Dolfi ◽  
Emanuele Pacifici ◽  
Lucia Sanus


2018 ◽  
Vol 68 (4) ◽  
pp. 1149-1157
Author(s):  
Maryam Khademi ◽  
Behrooz Khosravi


2017 ◽  
Vol 146 (4) ◽  
pp. 1505-1513 ◽  
Author(s):  
Zeinab Akhlaghi ◽  
Carlo Casolo ◽  
Silvio Dolfi ◽  
Khatoon Khedri ◽  
Emanuele Pacifici


2016 ◽  
Vol 215 (2) ◽  
pp. 523-558 ◽  
Author(s):  
Carlo Casolo ◽  
Silvio Dolfi ◽  
Emanuele Pacifici ◽  
Lucia Sanus


SpringerPlus ◽  
2016 ◽  
Vol 5 (1) ◽  
Author(s):  
Shitian Liu ◽  
Yunxia Xie


2015 ◽  
Vol 44 (6) ◽  
pp. 2635-2644
Author(s):  
Yongcai Ren


2014 ◽  
Vol 178 (2) ◽  
pp. 251-257 ◽  
Author(s):  
Behrooz Khosravi ◽  
Behnam Khosravi ◽  
Bahman Khosravi ◽  
Zahra Momen


2014 ◽  
Vol 21 (02) ◽  
pp. 355-360
Author(s):  
Xianxiu Zhang ◽  
Guangxiang Zhang

In this article, we prove that a finite solvable group with character degree graph containing at least four vertices has Fitting height at most 4 if each derived subgraph of four vertices has total degree not more than 8. We also prove that if the vertex set ρ(G) of the character degree graph Δ(G) of a solvable group G is a disjoint union ρ(G) = π1 ∪ π2, where |πi| ≥ 2 and pi, qi∈ πi for i = 1,2, and no vertex in π1 is adjacent in Δ(G) to any vertex in π2 except for p1p2 and q1q2, then the Fitting height of G is at most 4.



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