Abstract: This paper investigates Hyper G-matrices and their applications in network models and communication systems. We establish fundamental properties of Hyper G-matrices, including spectral invariance under similarity transformations, canonical block decomposition, and preservation of positive semidefiniteness under specific conditions. New theorems demonstrate that Hyper G-pairs maintain eigenvalues under diagonal similarity and admit interlacing bounds in associated block constructions. Illustrative examples, including visualizations of block matrices and Laplacian matrices of path graphs, highlight the impact of Hyper G-transformations on spectral and structural properties. Connections with ternary logic B3 are also explored through decomposition and spectral projection techniques, suggesting potential applications in logic, computation, and network science. Future directions include algorithmic computation of Hyper G-spectra, visualization of large-scale matrices, and application to data-driven communication networks. The results provide a versatile algebraic framework bridging classical matrix theory, graph theory, and network modeling.

Keywords: Hyper G-matrix, ternary logic $$B_{3}$$), signed graphs, spectral graph theory, structured matrices, quantum computation, network resilience

Pages: 220-230