Sep 2026· Annales de l'Institute Henri Poincare. Physique theorique· 0 citations· 33 references
Abstract
We study the heat content for Laplacians on compact, finite metric graphs with Dirichlet conditions imposed at the “boundary” (i.e., a given set of vertices) and standard conditions imposed elsewhere. We prove a closed formula of combinatorial flavor, as it is expressed as a sum over all paths starting and ending at boundary vertices. By delivering a small-time asymptotic expansion, our approach yields information on crucial geometric quantities of the metric graph, much in the spirit of the celebrated corresponding result for manifolds due to Gilkey–van den Berg; but unlike other known formulae based on different methods, ours holds for all times
$$t>0$$
t
>
0
and it displays a stronger decay rate in the short time limit. Furthermore, we prove new surgery principles for the heat content and use them to derive comparison principles for the heat content between metric graphs of different topology.
This paper investigates spectral minimal partitions for weighted graphs, thus extending the extensive class of results that are currently available on domains and, to a lesser extent, manifolds and metric graphs. We provide a rigorous framework for analyzing graph Laplacians under Dirichlet, Neumann, and boundaryless e...
M. Hofmann, James B Kennedy, Delio Mugnolo et al.· 0 citations
We develop a Hopf-algebraic theory of the graph coefficients arising in the quantum Magnus expansion. At the classical level, the graph expansion is governed by directed trees, whereas its quantum counterpart involves loop graphs, multiple edges, and different types of edge data. We introduce a contraction Hopf algebra...
Li Guo, Joon-Hwi Kim, Jung-Wook Kim et al.· 1 citation
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...
We obtain explicit formulas for heat kernels of the normalized Laplacians for two infinite families of Cayley graphs. The first family corresponds to free products of finite groups $G=G_1*\dots*G_r$, with $r\ge2$, and where all factors $G_i$ have the same order $L\ge2$. The second family corresponds to free products of...
We develop a constructive method for solving semilinear elliptic equations $\Delta u(x)=f(x,u(x))$ on locally finite, connected infinite graphs with layered structure. Using Eidelheit's theorem, we establish coupling criteria ensuring that arbitrary initial-layer data extend to global solutions for every $f$. We apply...
We study an inverse boundary value problem for the Helmholtz equation on a smooth compact non-trapping Riemannian manifold with strictly convex boundary. We prove that, given two such metrics, for sufficiently large but fixed frequency $\lambda$, equality of their Dirichlet-to-Neumann maps implies equality of their len...
Mihajlo Cekić, Katya Krupchyk, S. K. Sahoo et al.· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
