Publication: Wh Factorizationanditsapplication In Graphenergies
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Date
2025-01
Authors
Mohamed, Bashir Dlal Juma
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Abstract
Hourglass matrix is a square matrix obtained from quadrant interlocking factorization, known as WH factorization, which can be represented in mixed graph. Recent establishment of hourglass matrix and its factorization technique does not show the all the properties of the matrix. In this thesis, properties of WH factorization were put forward which include its existence for every strict dominant diagonal matrix, that its HSYSTEM forms a lower triangular invertible matrix and its counterpart W-matrix forms an upper triangular invertible matrix. To optimize WH factorization, a variant cramer’s rule is used to show that it has an advantage over the traditional cramer’s rule. It can be concluded that the block WH factorization technique is able to partition the matrix for effective factorization. The matrix obtained from WH factorization is known to be well represented as mixed graph, but not as topological index graph. There are over 50 types of graph energies obtained from degree based topological indices, only eight types are considered for the mixed hourglass graph: first and second zagreb energy, first and second hyper-zagreb energy, first and second guorava energy, and first and second hyper-guorava energy. Furthermore, all considered energies are compared to conclude that the energies give value of even numbers while the graph energy ratio (ratio between laplacian energy and the mixed energy of mixed hourglass graph) is odd.