Abstract
A cut vertex of a graph is a vertex whose removal causes the resulting graph to have more connected components. We show that the prime divisor character degree graph of a solvable group has at most one cut vertex. We also prove that a solvable group whose prime divisor character degree graph has a cut vertex has at most two normal nonabelian Sylow subgroups. We determine the structures of those solvable groups whose prime divisor character degree graph has a cut vertex and has two normal nonabelian Sylow subgroups. Finally, we characterize a subgroup determined by the prime associated with the cut vertex in terms of the structure of the prime divisor character degree graph.
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Communicated by J. S. Wilson.
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This work was supported by China Scholarship Council (CSC) (Grant No. 201608410231) and National Natural Science Foundation of China (NSFC) (Grant Nos. 11601121, 11771356) and Science Foundation of Henan University of Technology (Grant No. 31490036).
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Lewis, M.L., Meng, Q. Solvable groups whose prime divisor character degree graphs are 1-connected. Monatsh Math 190, 541–548 (2019). https://doi.org/10.1007/s00605-019-01276-8
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DOI: https://doi.org/10.1007/s00605-019-01276-8