👉 The problem of finding the shortest path in a weighted graph, a common application of graph theory, introduces a rich set of mathematical concepts. We leverage techniques from linear algebra and optimization to model the graph as a matrix, where each entry represents the weight (or cost) between nodes. By applying methods such as Dijkstra's algorithm or the Floyd-Warshall algorithm, we can efficiently compute the shortest paths between all pairs of nodes. This involves solving systems of linear equations and using matrix operations to update path costs iteratively, ultimately leading to a comprehensive understanding of the minimum-cost traversal through the graph. This problem not only showcases the power of linear algebra but also highlights the practical application of optimization techniques in solving real-world problems.
