Let p be an odd prime. Let P be a finite p-group of class 2 and exponent p, whose commutator quotient P/[P, P ] is of order p n . We define two parameters for P related to central decompositions. The first parameter, κ(P ), is the smallest integer s for the existence of a subgroup S of P satisfying (1) n−s , and (3) S admits a non-trivial central decomposition. The second parameter, λ(P ), is the smallest integer s for the existence of a central subgroup N of order p s , such that P/N admits a non-trivial central decomposition. While defined in purely group-theoretic terms, these two parameters generalise respectively the vertex and edge connectivities of graphs: For a simple undirected graph G, through the classical procedures of Baer (Trans. Am. Math. Soc., 1938), Tutte (J. Lond. Math. Soc., 1947) and Lovász (B. Braz. Math. Soc., 1989), there is a p-group P G of class 2 and exponent p that is naturally associated with G. Our main results show that the vertex connectivity κ(G) is equal to κ(P G ), and the edge connectivity λ(G) is equal to λ(P G ). We also discuss the relation between κ(P ) and λ(P ) for a general p-group P of class 2 and exponent p, as well as the computational aspects of these parameters.
In REF , the vertex and edge connectivities of a graph G are shown to be equal to certain parameters related to orthogonal decompositions 4 of A G .
Group-theoretic generalisations of vertex and edge connectivities
arXiv: Combinatorics
Mathematics