Related papers: Graph Theoretic Framework of Dynamical Systems Thr…
The dynamics of a linear dynamical system over a finite field can be described by using the elementary divisors of the corresponding matrix. It is natural to extend the investigation to a general finite commutative ring. In a previous…
The central nervous system is composed of many individual units -- from cells to areas -- that are connected with one another in a complex pattern of functional interactions that supports perception, action, and cognition. One natural and…
Modern methods of graph theory describe a graph up to isomorphism, which makes it difficult to create mathematical models for visualizing graph drawings on a plane. The topological drawing of the planar part of a graph allows representing…
Graph polynomials encode fundamental combinatorial invariants of graphs. Their computation is investigated using tree and path decomposition frameworks, with formal definitions of treewidth, k-trees, and pathwidth establishing the…
Using graphs to model irregular information domains is an effective approach to deal with some of the intricacies of contemporary (network) data. A key aspect is how the data, represented as graph signals, depend on the topology of the…
In this paper we study a construction of algebraic curves from combinatorial data. In the study of algebraic curves through degeneration, graphs usually appear as the dual intersection graph of the central fiber. Properties of such graphs…
We develop a clear connection between deFinetti's theorem for exchangeable arrays (work of Aldous--Hoover--Kallenberg) and the emerging area of graph limits (work of Lovasz and many coauthors). Along the way, we translate the graph theory…
We study functional graphs generated by quadratic polynomials over prime fields. We introduce efficient algorithms for methodical computations and provide the values of various direct and cumulative statistical parameters of interest. These…
We analyze the dynamics of a 4-parameter family of planar ordinary differential equations, given by a polynomial of degree 5 that is equivariant under a symmetry of order 6. We obtain the number of limit cycles as a function of the…
These notes develop aspects of perturbation theory of matrices related to so-called diagonalisation schemes. Primary focus is on constructive tools to derive asymptotic expansions for small/large parameters of eigenvalues and…
We study finite graphs embedded in oriented surfaces by associating a polynomial to it. The tools used in developing a theory of such graph polynomials are algebraic topological while the polynomial itself is inspired from ideas arising in…
In the 20th century, newly invented technical artifacts were connected to form large-scale complex engineering systems. Furthermore, the interactions found within these networked systems has grown in both degree as well as heterogeneity.…
Dynamic graph learning has gained significant attention as it offers a powerful means to model intricate interactions among entities across various real-world and scientific domains. Notably, graphs serve as effective representations for…
This paper deals with the maximum value of the difference between the determining number and the metric dimension of a graph as a function of its order. Our technique requires to use locating-dominating sets, and perform an independent…
We treat the problem of characterizing in a systematic way the qualitative features of two-dimensional dynamical systems. To that end, we construct a representation of the topological features of phase portraits by means of diagrams that…
For family $x'=(a_0+a_1\cos t+a_2 \sin t)|x|+b_0+b_1 \cos t+b_2 \sin t$, we solve three basic problems related with its dynamics. First, we characterize when it has a center (Poincar\'e center focus problem). Second, we show that each…
A book Chapter consisting of some of the main areas of research in graph theory applied to physics. It includes graphs in condensed matter theory, such as the tight-binding and the Hubbard model. It follows the study of graph theory and…
In this paper it is shown that it is possible to associate several polynomial ideals to a directed graph $D$ in order to find properties of it. In fact by using algebraic tools it is possible to give appropriate procedures for automatic…
In this paper, using geometric properties of the field rotation parameters, we present a solution of Smale's Thirteenth Problem on the maximum number of limit cycles for Li\'{e}nard's polynomial system. We also generalize the obtained…
\emph{Uncertain Graph} (also known as \emph{Probabilistic Graph}) is a generic model to represent many real\mbox{-}world networks from social to biological. In recent times analysis and mining of uncertain graphs have drawn significant…