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Critical groups for complete multipartite graphs and Cartesian products of complete graphs. (English) Zbl 1031.05064
Summary: The critical group of a connected graph is a finite abelian group, whose order is the number of spanning trees in the graph, and which is closely related to the graph Laplacian. Its group structure has been determined for relatively few classes of graphs, e.g., complete graphs and complete bipartite graphs. For complete multipartite graphs \(K_{n_1,\dots ,n_k}\), we describe the critical group structure completely. For Cartesian products of complete graphs \(K_{n_1} \times \cdots \times K_{n_k}\), we generalize results of H. Bai [Linear Algebra Appl. 369, 251-261 (2003)] on the \(k\)-dimensional cube, by bounding the number of invariant factors in the critical group, and describing completely its \(p\)-primary structure for all primes \(p\) that divide none of \(n_1,\dots ,n_k\).

MSC:
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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