Spectral and Energy Analysis of the Rook Hypergraph Derived from the 8 × 8 Chessboard
S. G. Jakkewad, Y. A. Yadav, N. B. Nalawade
Asian Research Journal of Mathematics, Vol. 22, Issue 4, pp. 15–31 (2026)
10.9734/arjom/2026/v22i41063
Abstract
Graph and hypergraph models derived from chessboard movements provide an effective framework for studying structural and spectral properties of discrete mathematical systems. In this work, we investigate the Rook hypergraph associated with the standard 8×8 chessboard. In this construction, each square of the chessboard is considered as a vertex, while hyperedges are formed by combining all vertices lying in the same row and column as a given square, reflecting the legal movement of a Rook. We develop the adjacency, Laplacian, and Seidel matrix representations corresponding to this hypergraph and examine their spectral characteristics. The eigenvalues and their multiplicities are obtained through numerical computation using Python. Based on these spectra, the adjacency energy, Laplacian energy, and Seidel energy of the Rook hypergraph are determined. The analysis shows that the structure is regular of degree 14 and highly symmetric due to the row–column configuration of the chessboard. In particular, the adjacency and Laplacian energies are both equal to 196, while the Seidel energy is 364. These results illustrate how chessboard-based constructions yield structured spectral behavior and provide a useful model for studying grid-based networks and combinatorial structures.
Topics
Field: Mathematics · Subfield: Geometry and Topology
Keywords
Hypergraph,Adjacency list,Adjacency matrix,Eigenvalues and eigenvectors,Laplacian matrix,Laplace operator,Row and column spaces
UN Sustainable Development Goals
- Peace, Justice and strong institutions (0.47)
All Available Versions
- Landing page — Asian Research Journal of Mathematics publishedVersion
- Landing page — HAL (Le Centre pour la Communication Scientifique Directe) submittedVersion
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