Finiteness and exponential growth of full graph $p$-spectra

Abstract

We prove that the full spectrum of the graph $p$-Laplacian is finite for every finite graph and every real $p>1$. This resolves the problem of finiteness of the full graph spectrum posed by Amghibech (2003). More generally, for signed weighted graph $p$-Schrödinger operators with arbitrary real potentials, we obtain bounds in terms of the numbers of vertices and positive-weight edges, uniform in all coefficients and in $p$. On $n$ vertices with $m$ positive-weight edges, the logarithms of the spectral cardinality and of the total number of connected components of the normalised eigenvector sets are $O((n+m{)}^2)$; the sum over the full spectrum of their rational Betti numbers is at most $\exp (C(n+m{)}^2)$ for an absolute constant $C$. These topological bounds extend to homogeneous eigenproblems built from arbitrary finite families of linear forms, including generalised $p$-eigenvalue problems for matrix pairs. For the uniformly weighted $K_n$ and $p\ne 2$, we identify the nonconstant eigenlines with the barycentres of the coordinate-hyperplane cell decomposition of $\mathbb{R}{P}^{n-2}$ and determine their local Morse data on either side of $p=2$. A renormalised logarithmic limit describes the transition at $p=2$. Positive integer weights then separate these critical values, giving at least $(3^n-2^{n+1}+3)/2$ distinct spectral values for every fixed $p\ne 2$ and every $n\ge 2$. By contrast, the maximal spectral cardinality at $p=2$ is $n$; at $p=4$, the maximal spectral cardinalities in the unsigned and general classes both have exponential growth rate exactly $3$. For $n\ge 3$ the same examples resolve Amghibech's extremal question. The proof combines o-minimal and Pfaffian geometry with projective $L^p$-duality, Morse theory, critical groups, and tensor eigenvalue bounds.