Foundations of Potential Theory on Finite Non-Symmetric Networks
- 1Department of Mathematics, College of Science, King Saud University, P. O. Box 2455, Riyadh 11451, Saudi Arabia
- 2Department of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, VA 22030, USA
Abstract
In this article, we develop a rigorous function theory for finite sets based on the properties of a non-symmetric Laplacian matrix Δ. We consider a finite connected network {X,t(x,y)} where the transition functions t(x,y) are non-negative and satisfy t(x,y)=0 if, and only if, t(y,x)=0. We do not assume the symmetry of the transition functions, thereby allowing for a broader application to directed networks and non-reversible Markov chains. Unlike symmetric networks, the non-symmetric case requires a careful treatment of the transition function t(x,y) using the Perron–Frobenius Theorem to establish existence and uniqueness results. We define and characterize classes of subharmonic and harmonic functions, establishing discrete analogs of fundamental principles, including the Minimum Principle and the Maximum Principle. A central result of the paper is the solvability of the general Dirichlet-Poisson problem, whose proof is based on showing that the Laplacian Δ of a connected network consisting of n vertices has rank n−1. This theoretical framework provides the necessary mathematical basis for solving boundary value problems and understanding the global behavior of functions on finite networks without the assumption of symmetry.
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BibTeX
@article{bajunaid2026foundations,
title = {{Foundations of Potential Theory on Finite Non-Symmetric Networks}},
author = {Ibtesam Bajunaid and Flavia Colonna},
journal = {Axioms},
year = {2026},
month = sep,
doi = {10.3390/axioms15100715},
url = {https://doi.org/10.3390/axioms15100715},
note = {CC BY},
}
RIS
TY - JOUR
TI - Foundations of Potential Theory on Finite Non-Symmetric Networks
AU - Ibtesam Bajunaid
AU - Flavia Colonna
PY - 2026
DA - 2026/09/01
JO - Axioms
DO - 10.3390/axioms15100715
UR - https://doi.org/10.3390/axioms15100715
AB - In this article, we develop a rigorous function theory for finite sets based on the properties of a non-symmetric Laplacian matrix Δ. We consider a finite connected network {X,t(x,y)} where the transition functions t(x,y) are non-negative and satisfy t(x,y)=0 if, and only if, t(y,x)=0. We do not assume the symmetry of the transition functions, thereby allowing for a broader application to directed networks and non-reversible Markov chains. Unlike symmetric networks, the non-symmetric case requires a careful treatment of the transition function t(x,y) using the Perron–Frobenius Theorem to establish existence and uniqueness results. We define and characterize classes of subharmonic and harmonic functions, establishing discrete analogs of fundamental principles, including the Minimum Principle and the Maximum Principle. A central result of the paper is the solvability of the general Dirichlet-Poisson problem, whose proof is based on showing that the Laplacian Δ of a connected network consisting of n vertices has rank n−1. This theoretical framework provides the necessary mathematical basis for solving boundary value problems and understanding the global behavior of functions on finite networks without the assumption of symmetry.
SN - 2075-1680
ER -