Abstract
Martin and Wagner determined the integral eigenvalue spectrum of the simplicial rook graphs on the triangular lattice by explicitly constructing their eigenvectors. In this work we deduce the same result by instead constructing the characteristic polynomials for this class of graphs. The resulting analysis provides a neat explanation for the observed spectral structure.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Notes
Least eigenvalues are of interest in their own right, but also because they are connected to the independence number \(\alpha \) of a graph via the Hoffman bound, which states that \(\alpha \le |V|/(1-k/\tau )\), where |V| is the number of vertices, k is the degree, and \(\tau \) is the least eigenvalue. But the Hoffman bound is not always exact. For instance, for S(n, 3) with \(n>3\), the Hoffman bound gives \(\alpha <3(n+2)(n+1)/(4n+6)\), which is weaker than the actual bound given above.
Remarks. (1) We refer to \(L_{ij}^\alpha \) as a ‘line’, although technically it is just a finite set of points lying along a lattice line. (2) The lines \(L_{ij}^\alpha \) are cliques of S(n, d), but we prefer the more geometric language in this context.
We may also obtain the eigenvectors of M by letting u and v vary over the eigenvectors of K and \(B(\mu )\), respectively. It would be interesting to relate these to the hexagon and line eigenvectors used in [5].
We need not worry about zero divisors as long as we treat ‘x’ as an indeterminate. In any case, the denominators will all clear in the end.
References
Ahadi, A., Mollahajiaghaei, M., Dehghan, A.: On the maximum number of non-attacking rooks on a high-dimensional simplicial chessboard. Graphs Combin. 38, 52 (2022)
Blackburn, S.R., Paterson, M.B., Stinson, D.R.: Putting dots in triangles. J. Combin. Math. Combin. Comput. 78, 23–32 (2011)
Brouwer, A.E., Cioabă, S.M., Haemers, W.H., Vermette, J.R.: Notes on simplicial rook graphs. J. Algebraic Combin. 43, 783–799 (2016)
Martin, J.L., Wagner, J.D.: On the spectra of simplicial rook graphs. Graphs Combin. 31, 1589–1611 (2015)
Nivasch, G., Lev, E.: Nonattacking queens on a triangle. Math. Mag. 78(5), 399–403 (2005)
Vermette, J.R.: Spectral and combinatorial properties of friendship graphs, simplicial rook graphs, and extremal expanders. Ph.D. thesis, University of Delaware (2015)
Funding
The author declares that no funds, grants, or other support were received during the preparation of this manuscript.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author has no relevant financial or non-financial interests to disclose.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Renteln, P. On the Spectra of Simplicial Rook Graphs. Graphs and Combinatorics 39, 95 (2023). https://doi.org/10.1007/s00373-023-02695-z
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00373-023-02695-z