A Graph Associated with $|\cd(G)|-1$ Degrees of a Solvable Group
Received:June 11, 2009  Revised:September 15, 2009
Key Word: solvable groups   irreducible character degrees.
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.10871032), Innovation Project for the Development of Science and Technology (IHLB) (Grant No.201098) and the Specific Research Fund of the Doctoral Program of Higher Education of China (Grant No.20060285002).
 Author Name Affiliation Deng Feng LIANG College of Computer and Information Engineering, Beijing Technology and Business University, Beijing 100048, P. R. China Wu Jie SHI Department of Mathematics, Suzhou University, Jiangsu 215006, P. R. China
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Let $G$ be a group. We consider the set $\cd(G)\backslash\{m\}$, where $m\in \cd(G)$. We define the graph $\Delta(G-m)$ whose vertex set is $\rho(G-m)$, the set of primes dividing degrees in $\cd(G)\backslash\{m\}$. There is an edge between $p$ and $q$ in $\rho(G-m)$ if $pq$ divides a degree $a\in \cd(G)\backslash\{m\}$. We show that if $G$ is solvable, then $\Delta(G-m)$ has at most two connected components.