On the second eigenvalue of hypergraphs
We first define the second eigenvalue of 3-uniform hypergraph, and then discuss the general notion, as it applies to other uniform hypergraphs and graphs. Second, just as ordinary directed graphs can have multiple edges and self-loops, we can accomodate such notions here. Thirdly, we can also handle the notion of directed edges. We say that a directed 3-uniform hypergraph is a hypergraph where each edge \(\{i, j, k\}\) has a specified order. Now the trilinear form \(\tau\) is defined by \(\tau_{i, j, k}\) being the multiplicity if the edge \(\{i, j, k\}\). Fourth, we define for any trilinear form on \(L^2(V)\), \(\mu\), its first eigenvalue to be its norm with respect to \(L^2(V)\). Fifth, it becomes clear how we want to define the second eigenvalue in the more general case. Sixth, the first and second eigenvalue of multilinear forms (as defined in this paper) are always non-negative numbers. Lastly, we could define the eigenvalues and/or norms of multilinear forms with respect to other norms on the space of functions on \(V\), such as \(L^p(V)\) for any \(1\leq p\leq \infty\).
scientific article; zbMATH DE number 750658
scientific article