Marcos Masip | Searcharxiv

arXiv2026

Graph Theoretic Framework of Dynamical Systems Through the Boundary Polynomial

This work is devoted to the study of the relationships between graph theory and the qualitative analysis of ordinary differential equations, with a special focus on two-dimensional systems. In particular, we reinterpret classical results through the lens of boundary polynomials of graphs. The theory naturally leads to questions about limit cycles, which arise in many processes in nature and remain a central object of study in dynamical systems. The significance of limit cycles is underscored by their place in Hilbert's 16th problem, one of the unsolved challenges from his famous list of 23 problems posed in 1900.

math.DS↗

arXiv2025

On the boundary polynomial of a graph

In this work, we introduce the boundary polynomial of a graph $G$ as the ordinary generating function in two variables $B(G;x,y):= \displaystyle\sum_{S\subseteq V(G)} x^{|B(S)|}y^{|S|}$, where $B(S)$ denotes the outer boundary of $S$. We investigate this graph polynomial obtaining some algebraic properties of the polynomial. We found that some parameters of $G$ are algebraically encoded in $B(G;x,y)$, \emph{e.g.}, domination number, Roman domination number, vertex connectivity, and differential of the graph $G$. Furthermore, we compute the boundary polynomial for some classic families of graphs. We also establish some relationships between $B(G;x,y)$ and $B(G^\prime;x,y)$ for the graphs $G^\prime$ obtained by removing, adding, and subdividing an edge from $G$. In addition, we prove that a graph $G$ has an isolated vertex if and only if its boundary polynomial has a factor ($y+1$). Finally, we show that the classes of complete, complete without one edge, empty, path, cycle, wheel, star, double-star graphs, and many others are characterized by the boundary polynomial.

math.CO↗