2022/09/01 by Jendrik Brachter, Brachter, Jendrik, Eda Kaja +1 · 1 citation
Computer Science · Mathematics · #Advanced Graph Theory Research #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.2209.00317
openalex publication_date 2022/09/01 · openalex created_date 2022/09/03 · openalex updated_date 2026/07/28
We prove various properties on the structure of groups whose power graph is chordal. Nilpotent groups with this property have been classified by Manna, Cameron and Mehatari [The Electronic Journal of Combinatorics, 2021]. Here we classify the finite simple groups with chordal power graph, relative to typical number theoretic oracles. We do so by devising several sufficient conditions for the existence and non-existence of long cycles in power graphs of finite groups. We examine other natural group classes, including special linear, symmetric, generalized dihedral and quaternion groups, and we characterize direct products with chordal power graph. The classification problem is thereby reduced to directly indecomposable groups and we further obtain a list of possible socles. Lastly, we give a general bound on the length of an induced path in chordal power graphs, providing another potential road to advance the classification beyond simple groups.