Asymptotic behavior of Laplacian eigenvalues of subspace inclusion graphs (Q6618660)

Spectral graph theory deals with finite graphs, mostly the simple undirected ones, by considering the eigenvalues and eigenvectors of various kinds of naturally associated matrices. The paper's results fit into this subject, but address a relevant problem that arises in a broad context, which indeed provides a motivation that may be considered among the main ones to spectral graph theory itself. That context is very well outlined in the introduction, from which one may learn, for instance, what the Laplacian matrix of a graph has to do with the Laplace operator encountered in the early years of undergraduate courses (provided one has some knowledge of basic homology theories and the de Rham theorem).\N\NThe simple undirected graph under consideration is given by all nontrivial subspaces of a vector space of finite dimension \(n\) over a field of finite order \(q\), together with the (symmetrized, proper) inclusion relation. That is, two nontrivial subspaces are joined by an edge if and only if one of them is properly included in the other. This graph is (naturally identified with) the \(1\)-skeleton of the abstract simplicial complex \(\mathrm{Fl}_{n,q}\) given by all flags of nontrivial subspaces of the same space. The weighted Laplacian \(\Delta_0\left(\mathrm{Fl}_{n,q}\right)\) under consideration is the zero-dimensional instance of the weighted upper Laplacians \(\Delta^+_k\left(\mathrm{Fl}_{n,q}\right)\) of that simplicial complex, and the paper proves, in this case, a conjecture raised in [\textit{M. Papikian}, Eur. J. Math. 2, No. 3, 579--613 (2016; Zbl 1390.22018)], which is about the asymptotic behaviour of the eigenvalues with respect to the size of the base field. The main theorem of the paper gives indeed quite detailed estimates on the eigenvalues of \(\Delta_0\left(\mathrm{Fl}_{n,q}\right)\) for large \(q\).\N\NThe essential background is presented in a clear and concise form. Next, as a first step, the weighted Laplacian matrix of the graph under consideration is abandoned in favour of one that is similar, namely because represents the weighted Laplacian operator \(\Delta_0\left(\mathrm{Fl}_{n,q}\right)\) in a suitable basis of the space of \(0\)-cochains of \(\mathrm{Fl}_{n,q}\). The new matrix has a block decomposition with nontrivial blocks that are Kronecker products of identity matrices by matrices with size not depending on \(q\). To achieve this, it has been needed a careful analysis of subspace inclusion matrices \(A_{ij}\), which is a much-studied subject with independent interest. Let us also note that when \(i\ne j\), \(A_{ij}\) and \(A_{ji}\) are the two nontrivial blocks, transposed to each other, of the adjacency matrix of the bipartite subgraph that is induced on the set of all subspaces whose dimension is \(i\) or \(j\), whereas \(A_{ii}\) is the identity matrix (though someone may have expected it to vanish) by the definition reported at the beginning of Section~4, where the symbol `\(\subset\)' is meant to allow equality.\N\NIn this way, the problem of estimating the eigenvalues of the weighted Laplacian is reduced to the problem of estimating the eigenvalues of the mentioned matrices with size not depending on \(q\). Then, a relation between their characteristic polynomials and the sign-alternating Fibonacci polynomials studied in [\textit{R. G. Donnelly} et al., Enumer. Comb. Appl. 1, No. 2, Article ID S2R15, 13 p. (2021; Zbl 1510.11043)] is established. This leads to the main result.