Spectral Properties and Energy Analysis of the Bishop Hypergraph on the 8×8 Chessboard
S. G. Jakkewad *
Department of Mathematics, K. B. P. College Vashi, Navi Mumbai, Maharashtra, India.
G. A. Dhanorkar
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.
*Author to whom correspondence should be addressed.
Abstract
In this paper, we investigate the Bishop hypergraph HB obtained from an 8×8 chessboard, where each square is treated as a vertex and two vertices are adjacent if a Bishop can move between them in a single move. We construct the adjacency and Laplacian matrices of the Bishop hypergraph HB and analyze their spectral properties. The eigenvalues of these matrices are computed numerically using Python, and the corresponding graph energies are evaluated. The results highlight the influence of diagonal-based connectivity on the spectrum and energy of the Bishop hypergraph HB and provide a comparative perspective with other chessboard based graphs.
Keywords: Bishop Hypergraph HB, Adjacency Energy, Laplacian Energy, Siedal Energy