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On the d-dimensional algebraic connectivity of graphs

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Abstract

The d-dimensional algebraic connectivity ad(G) of a graph G = (V,E), introduced by Jordán and Tanigawa, is a quantitative measure of the d-dimensional rigidity of G that is defined in terms of the eigenvalues of stiffness matrices (which are analogues of the graph Laplacian) associated to mappings of the vertex set V into ℝd.

Here, we analyze the d-dimensional algebraic connectivity of complete graphs. In particular, we show that, for d ≥ 3, ad(Kd+1) = 1, and for n ≥ 2d,

$$\left\lceil {{n \over {2d}}} \right\rceil - 2d + 1 \le {a_d}({K_n}) \le {{2n} \over {3(d - 1)}} + {1 \over 3}.$$

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Acknowledgments

Part of this research was done while A.L. was a postdoctoral researcher at the Einstein Institute of Mathematics at the Hebrew University. We thank the anonymous referee for their helpful remarks and suggestions.

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Authors and Affiliations

  1. Deptartment of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Alan Lew

  2. Einstein Institute of Mathematics, The Hebrew University of Jerusalem Givat Ram, Jerusalem, 91904, Israel

    Eran Nevo, Yuval Peled & Orit E. Raz

Authors
  1. Alan Lew
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  2. Eran Nevo
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  3. Yuval Peled
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  4. Orit E. Raz
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Corresponding author

Correspondence to Alan Lew.

Additional information

Dedicated to Nati Linial on the ocassion of his 70th birthday

Alan Lew and Eran Nevo were partially supported by the Israel Science Foundation grant ISF-2480/20.

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Lew, A., Nevo, E., Peled, Y. et al. On the d-dimensional algebraic connectivity of graphs. Isr. J. Math. 256, 479–511 (2023). https://doi.org/10.1007/s11856-023-2519-3

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  • DOI: https://doi.org/10.1007/s11856-023-2519-3

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