The stabilizing index and cyclic index of the coalescence and Cartesian product of uniform hypergraphs. (English) Zbl 1476.05145
Summary: Let \(G\) be connected uniform hypergraph and let \(\mathcal{A}(G)\) be the adjacency tensor of \(G\). The stabilizing index of \(G\) is exactly the number of eigenvectors of \(\mathcal{A}(G)\) associated with the spectral radius, and the cyclic index of \(G\) is exactly the number of eigenvalues of \(\mathcal{A}(G)\) with modulus equal to the spectral radius. Let \(G_1\odot G_2\) and \(G_1\square G_2\) be the coalescence and Cartesian product of connected \(m\)-uniform hypergraphs \(G_1\) and \(G_2\) respectively. In this paper, we give explicit formulas for the stabilizing indices and cyclic indices of \(G_1\odot G_2\) and \(G_1\square G_2\) in terms of those of \(G_1\) and \(G_2\) or the invariant divisors of their incidence matrices over \(\mathbb{Z}_m\), respectively.
MSC:
| 05C65 | Hypergraphs |
| 05C76 | Graph operations (line graphs, products, etc.) |
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
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