https://doi.org/10.1140/epjp/s13360-026-07300-9
Regular Article
Green functions and the green operator on the graphs: algebraic derivation via woodbury and probabilistic interpretation through absorbing Markov Chains in network transport theory
1
School of Mathematical Sciences, Jiangsu University, 212013, Zhenjiang, Jiangsu Province, China
2
C. K. Tedam University of Technology and Applied Sciences, 00233, Navrongo, Ghana
a
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Received:
24
October
2025
Accepted:
7
January
2026
Published online:
20
January
2026
Abstract
Green’s functions on graphs play a central role in statistical mechanics, where transport, relaxation, and response emerge from the interplay between geometry and spectral structure. In this work, we analyze the Dirichlet Green operator associated with subgraphs of complete graphs and show that, despite arising from a linear Laplacian, its behavior is driven by a fundamentally nonlinear boundary–bulk interaction. Using the Sherman–Morrison–Woodbury identity, we obtain closed-form Green functions for cliques viewed as low-rank perturbations of the complete graph, revealing how nonlinear dependence on defect size controls effective resistance, potential profiles, and energy dissipation. From a dynamical perspective, we construct absorbing random walks whose fundamental matrix provides a stochastic realization of the Green operator as a nonlinear response kernel encoding normalized visit densities before absorption. This dual algebraic–probabilistic treatment clarifies the distinction between intrinsic and ambient Green operators, shows how scalar shifts deform the spectrum, and uncovers the nonlinear scaling regime that arises as the subgraph density approaches the ambient limit. Together, these results provide a unified transport-theoretic framework for Green’s functions on subgraphs, connecting low-rank perturbation theory with nonequilibrium diffusion on networked systems.
Mathematics Subject Classification: 05C81 / 35K08 / 60G25 / 35P99 / 60C05
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2026
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
