We study the negative spectrum of the Laplacian on a metric graph with general vertex matching conditions and with two length scales: a compact core whose edges have length of order a small parameter $\epsilon$, together with finitely many edges of infinite length. As $\epsilon\to0$, some negative eigenvalues may escape to $-\infty$, and we describe precisely how. There are exactly two rates of escape, $\epsilon^{-1}$ and the fractional rate $\epsilon^{-2/3}$. We determine the number of eigenvalues of each rate, together with their leading coefficients, explicitly from the vertex conditions. The analysis rests on the Dirichlet-to-Neumann map of the graph and on an implicit Rellich-type theorem, that identifies the power-law rates of the solution branches of a nonlinear 2-parameter matrix pencil with the leading orders of a one-parameter Hermitian family.
We study discrete-time quantum walks on weighted graphs, where every edge in an edge cut set is assigned a small weight $\epsilon>0$. The parameter $\epsilon$ represents the strength of the connections through the cut edges: as $\epsilon \to 0$, these connections vanish, and the graph decomposes into the connected comp...
We establish the large-$n$ asymptotics of Hankel determinants for unitary random matrix ensembles possessing a higher-order edge singularity. We focus on ensembles where the equilibrium measure is supported on a single interval and the limiting eigenvalue density vanishes to order $2k+\frac{1}{2}$ for $k \in \mathbb{N}...
We study uniform high-frequency localization for the Laplacian on compact metric graphs through the least $L^2$-mass that eigenfunctions must place in a prescribed measurable observation set. We first identify this asymptotic localization constant with the minimum of a linear functional over the attainable edge-intensi...
In this paper, we resolve a 30-year-old conjecture of Spielman and Teng concerning the performance of the spectral partitioning method on graphs embeddable on an orientable surface of genus $g\ge 1$. In particular, for such a graph $G$ with $n$ vertices and maximum degree $\Delta$, we show that the second-smallest eige...
Benedikt Kolbe, Jack Spalding-Jamieson· 0 citations
We consider the adjacency matrix of a uniformly random simple $d$-regular graph on $N$ vertices. For every fixed $d\geq3$ and every fixed bulk energy, we prove that the rescaled eigenvalue point process converges to the $\mathrm{Sine}_1$ process with intensity $1/\pi$. We also establish universality of consecutive gaps...
This study investigates the spectral properties of the total zero-divisor graph associated with the ring $\mathbb{Z}_{p^2q^2}$, where $p$ and $q$ are distinct primes. Using an explicit equitable partition of the graph, we obtain a complete description of the adjacency spectrum. We show that the spectrum consists of two...
Meet J. Mashru, P. Lalchandani· Journal of Algebra Combinato...· 0 citations
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