Barrière, L.; Comellas, F.; Dalfó, C.; Fiol, M. A. On the hierarchical product of graphs and the generalized binomial tree. (English) Zbl 1225.05206 Linear Multilinear Algebra 57, No. 7, 695-712 (2009). Summary: We follow the study of the hierarchical product of graphs, an operation recently introduced in the context of networks. A well-known example of such a product is the binomial tree which is the (hierarchical) power of the complete graph on two vertices. An appealing property of this structure is that all the eigenvalues are distinct. Here we show how to obtain a graph with this property by applying the hierarchical product. In particular, we propose a generalization of the binomial tree and study some of its main properties. Cited in 2 Documents MSC: 05C76 Graph operations (line graphs, products, etc.) 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C05 Trees Keywords:hierarchical product graphs; binomial tree; adjacency matrix; eigenvalues; eigenvectors PDFBibTeX XMLCite \textit{L. Barrière} et al., Linear Multilinear Algebra 57, No. 7, 695--712 (2009; Zbl 1225.05206) Full Text: DOI References: [1] DOI: 10.1103/RevModPhys.74.47 · Zbl 1205.82086 [2] Barrière L, The hierarchical product of graphs · Zbl 1200.05196 [3] DOI: 10.1016/j.laa.2007.09.039 · Zbl 1134.05051 [4] Biggs N, Algebraic Graph Theory,, 2. ed. (1974) [5] Chartrand, G and Lesniak, L. 1996.Graphs & Digraphs, 3, London: Chapman and Hall. · Zbl 0890.05001 [6] DOI: 10.1090/S0894-0347-1989-0965008-X [7] Cormen TH, Introduction to Algorithms,, 2. ed. (1990) [8] Cvetković DM, Spectra of Graphs. Theory and Applications, 3. ed. (1995) [9] DOI: 10.1016/S0024-3795(98)10238-0 · Zbl 0933.05099 [10] DOI: 10.1016/S0012-365X(01)00255-2 · Zbl 1025.05060 [11] Flajolet P, Analytic Combinatorics (2007) [12] Godsil CD, Algebraic Combinatorics (1993) [13] DOI: 10.1017/S0004972700007760 · Zbl 0376.05049 [14] DOI: 10.1016/0024-3795(95)00199-2 · Zbl 0831.05044 [15] DOI: 10.1002/net.3230180406 · Zbl 0649.90047 [16] Mowshowitz A, Proof Techniques in Graph Theory pp 109– (1969) [17] Schwenk, AJ. Computing the characteristic polynomial of a graph,inGraphs and Combinatorics Proceedings of Capital Conference. pp.153–172. Washington, D.C. Lect. Notes Math. 406(1974) [18] DOI: 10.2307/3620776 · Zbl 02352181 [19] Szegö G, Orthogonal Polynomials, 4th ed., Colloquium Publications (1975) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.