Spectral and Energy Analysis of the Queen Hypergraph Derived from the 8 × 8 Chessboard
S. G. Jakkewad *
Department of Mathematics, K. B. P. College Vashi, Navi Mumbai, Maharashtra, India.
Y. A. Yadav
Department of Mathematics, K. B. P. College Vashi, Navi Mumbai, Maharashtra, India.
N. B. Nalawade
Department of Mathematics, S. G. M. College Karad, Satara, Maharashtra, India.
*Author to whom correspondence should be addressed.
Abstract
In this paper, we study the spectral properties of the Queen hypergraph HQ associated with the standard 8 × 8 chessboard. Each square of the board is represented as a vertex, and adjacency is defined according to the legal movements of the queen along rows, columns, and diagonals. The structural characteristics of this hypergraph are analyzed through three matrix representations: the adjacency, Laplacian, and Seidel matrices. The adjacency spectrum reflects the dense connectivity produced by the queen’s movement, while the Laplacian spectrum confirms the connectivity and describes the variation in vertex degrees. The computed energy values are adjacency energy EA ≈ 245.54, Laplacian energy EL ≈ 259.15, and Seidel energy ES ≈ 455.10. These results provide insight into the algebraic structure of chessboard-based graphs and offer a foundation for further studies on generalized n×n chessboard configurations and related combinatorial models.
Keywords: Queen hypergraph, spectral graph theory, adjacency matrix, laplacian matrix, seidel matrix, chessboard graphs