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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.

While Cramer's rule allows complete substitution of constant terms to the coefficient matrix in the system of linear equations, the modified methods of Cramer's rule consider the constant terms as well as the coefficients of the matrix at the same time. The methods are derived from one of the properties of determinants. Furthermore, we prove the two methods to be equivalent and provide MATLAB codes for the modified methods. However, the methods are not practically suitable for higher system of linear equations and instability of Cramer's rule.

Goldreich-Goldwasser-Halevi (GGH) encryption scheme is lattice-based cryptography with its security based on the shortest vector problem (SVP) and closest vector problem (CVP) with immunity to almost all attacks, including Shor's quantum algorithm and Nguyen's attack of higher lattice dimension. To improve the efficiency and security of the GGH Scheme by reducing the size of the public basis to be transmitted, we use an hourglass matrix obtained from quadrant interlocking factorization as a public key. The technique of quadrant interlocking factorization to yield a nonsingular hourglass matrix compensates the encryption scheme with better efficiency and security.