This article discusses a mixed graph called mixed hourglass graph. The adjacency of a mixed hourglass graph is coined from an hourglass matrix in which the matrix is obtained from quadratic interlocking factorization. Mixed hourglass graph is a weighted graph with loops, but replacing the weights of the graph with 1’s produce an unweighted mixed hourglass graph or simply mixed hourglass graph. Then the total number of arcs and edges and the spanning subgraph of mixed hourglass graph are examined to infer that the graph is a complete mixed graph and acyclic graph.

Graph theory is one of the neglected branches of mathematics in Nigeria but with the most applications in other fields of research. This article shows the paucity, importance, and necessity of graph theory in the development of Nigeria. The adjacency matrix and dual graph of the Nigeria map were presented. The graph spectrum and energies (graph energy and Laplacian energy) of the dual graph were computed. Then the chromatic number, maximum degree, minimum spanning tree, graph radius, and diameter, the Eulerian circuit and Hamiltonian paths from the dual graph were obtained and discussed.

No AbstractKeywords: Cramer’s rule; determinant; system of linear equation

This paper presents hourglass matrix, a dense square matrix which is obtained from Quadrant Interlocking Factorization (QIF) of nonsingular matrix. We discuss the condition to generate its nonzero entries and formulate its total number of zero and nonzero entries. Furthermore, the determinant of even and odd order of the matrix as well as the eigenvalue of the odd order of the matrix are examined. In all, we give some deductions to differentiate between hourglass matrix and classical Z-matrix.

Noise and vibration are pervasive challenges in built-up structures, impacting structural integrity, operational efficiency, and occupant well-being. These issues are particularly pronounced in urban and industrial settings, where traditional materials often struggle to deliver effective mitigation across the broad range of relevant frequencies. This paper introduces an integrated mathematical modeling and virtual design framework for the development of advanced metamaterials aimed at reducing noise and vibration in such complex structures. The approach combines finite element analysis, dynamic energy analysis, and optimization algorithms to design metamaterials with frequency-selective properties that create targeted barriers to acoustic and vibrational disturbances. The study not only develops a systematic methodology for designing these metamaterials but also validates their efficacy through comprehensive simulations and benchmarking against established solutions. The results highlight the advantages of the proposed metamaterials in terms of adaptability, efficiency, and performance robustness across various operating conditions. Sensitivity analyses and comparative evaluations further underscore the superiority of the framework in addressing frequency-dependent challenges, offering significant improvements over conventional materials. A unique aspect of this research is the inclusion of natural metamaterials (NMs) as a sustainable alternative for mitigating ground vibrations. The study reviews the potential of NMs for diverse functionalities, particularly in attenuating ground vibrations in urban environments. These findings emphasize the versatility and eco-friendliness of natural materials, providing a roadmap for their development and application in achieving clean and quiet environments. The proposed framework, therefore, bridges theoretical advancements with practical applications, paving the way for resilient and sustainable solutions to noise and vibration challenges in built-up structures.

This paper reviews the theory of matrices and determinants. Matrix and determinant are nowadays considered inseparable to some extent, but the determinant was discovered over two centuries before the term matrix was coined. Our review associate determinant with the matrix as part of linear systems but not with polynomials. Thus, the paper first gives the background on matrix with vast applications in all fields of study and then reviews the history of determinants which is based on its major contributors in chronological order from the sixteenth century to the twenty-first century