The trace and Estrada index of uniform hypergraphs with cut vertices. (English) Zbl 1505.05040
Summary: Let \(\mathcal{H}\) be a uniform hypergraph with cut vertices, which is a coalescence of two nontrivial connected sub-hypergraphs (called branches) at some vertex. The traces of the adjacency tensor \(\mathcal{A}(\mathcal{H})\) of \(\mathcal{H}\), simply called the traces of \(\mathcal{H}\), are important in the expression of the determinant and the characteristic polynomial of \(\mathcal{A}(H)\), and are closely related to the Estrada index of \(\mathcal{H}\). In this paper we give a formula for the traces of \(\mathcal{H}\) in terms of those of its branches, and get some perturbation results on the traces of \(\mathcal{H}\) when a branch of \(\mathcal{H}\) attached at one vertex is relocated to another vertex. We prove that among all hypertrees with fixed number of edges, the hyperpath is the unique one with minimum Estrada index and the hyperstar is the unique one with maximum Estrada index.
MSC:
| 05C09 | Graphical indices (Wiener index, Zagreb index, Randić index, etc.) |
| 05C92 | Chemical graph theory |
| 92E10 | Molecular structure (graph-theoretic methods, methods of differential topology, etc.) |
| 05C12 | Distance in graphs |
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
| 05C65 | Hypergraphs |
| 15A69 | Multilinear algebra, tensor calculus |
| 13P15 | Solving polynomial systems; resultants |
| 14M99 | Special varieties |
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