Spectral graph theory has been a prominent area of research in theoretical computer science and serves as a backbone to various complex frameworks in machine learning like graph neural networks. The core idea of spectral graph theory is to derive graph properties from the eigenvalues and eigenvectors of the adjacency matrix or Laplacian matrix. In this work, we define a novel metric for all edges called "edge scores", which is a function of the eigenvector of the Laplacian matrix. This edge score captures important information, and we use it to solve three well-known problems: graph isomorphism, hypergraph partitioning, and hyperedge prediction.

The spectral framework-based algorithms for graph isomorphism utilize all the eigenvectors of the Laplacian matrix to certify isomorphism for graphs with non-repeating eigenvalues. Using the edge-scores proposed in this work, we propose a practical graph isomorphism algorithm for graphs with at least one positive non-repeating eigenvalue, which uses only a few eigenvectors. We utilize "edge scores" to extend the use of the spectral framework for graph partitioning to k-uniform hypergraphs. Specifically, we propose the ratio-cut for hypergraphs and prove that the solution of relaxed optimization minimizing ratio-cut can be obtained from the Fiedler vector computed from the Laplacian tensor of hypergraphs. We show the improvement in ratio-cut value by the proposed method on synthetic hypergraphs generated by stochastic block models. The same idea of using hyperedge scores is utilized in the hyperedge prediction problem. We construct new hyperedges with lower hyperedge scores. We report approximately 20% improvement on four real hypergraph datasets over three existing baselines.