Related papers: Evolutionary Hamiltonian Graph Theory
The notion of treewidth, introduced by Robertson and Seymour in their seminal Graph Minors series, turned out to have tremendous impact on graph algorithmics. Many hard computational problems on graphs turn out to be efficiently solvable in…
Recently, computational modelling became a very important research tool that enables us to study problems that for decades evaded scientific analysis. Evolutionary systems are certainly examples of such problems: they are composed of many…
Evolutionary scenarios describing the evolution of a family of genes within a collection of species comprise the mapping of the vertices of a gene tree $T$ to vertices and edges of a species tree $S$. The relative timing of the last common…
Graph contrastive learning (GCL) has achieved remarkable success by following the computer vision paradigm of preserving absolute similarity between augmented views. However, this approach faces fundamental challenges in graphs due to their…
Evolution algebras are non-associative algebras inspired from biological phenomena, with applications to or connections with different mathematical fields. There are two natural ways to define an evolution algebra associated to a given…
The Turing machine, as it was presented by Turing himself, models the calculations done by a person. This means that we can compute whatever any Turing machine can compute, and therefore we are Turing complete. The question addressed here…
A non-statistical theory of continuous, but irreversible, evolution can be constructed in terms of the Cartan calculus. The fundamental postulate, for an evolutionary theory which admits irreversible processes, is that the topology of the…
For quantum theories with a classical limit (which includes the large N limits of typical field theories), we derive a hierarchy of evolution equations for equal time correlators which systematically incorporate corrections to the limiting…
Prediction and control of network dynamics are grand-challenge problems in network science. The lack of understanding of fundamental laws driving the dynamics of networks is among the reasons why many practical problems of great…
Large language models (LLMs) possess vast semantic knowledge but often struggle with complex reasoning tasks, particularly in relational reasoning problems such as kinship or spatial reasoning. In this paper, we present Path-of-Thoughts…
In this paper we present the first deterministic polynomial time algorithm for determining the existence of a Hamiltonian cycle and finding a Hamiltonian cycle in general graphs. Our algorithm can also solve the Hamiltonian path problem in…
We pursue research leading towards the nature of causality in the universe. We establish the equation of the universe's evolution from the universe-state function and its series expansion, in which causes and effects connect together to…
Subgraph and homomorphism counting are fundamental algorithmic problems. Given a constant-sized pattern graph $H$ and a large input graph $G$, we wish to count the number of $H$-homomorphisms/subgraphs in $G$. Given the massive sizes of…
Ecosystems represent archetypal complex dynamical systems, often modelled by coupled differential equations of the form $$ \frac{d x_i}{d t} = x_i \varphi_i(x_1,\cdots, x_N)\ , $$ where $N$ represents the number of species and $x_i$, the…
For integers $k\geq 1$ and $n\geq 2k+1$, the Kneser graph $K(n,k)$ has as vertices all $k$-element subsets of an $n$-element ground set, and an edge between any two disjoint sets. It has been conjectured since the 1970s that all Kneser…
We consider models of growing multi-level systems wherein the growth process is driven by rules of tournament selection. A system can be conceived as an evolving tree with a new node being attached to a contestant node at the best hierarchy…
In 1999, Jacobson and Lehel conjectured that for $k \geq 3$, every $k$-regular Hamiltonian graph has cycles of at least linearly many different lengths. This was further strengthened by Verstra\"{e}te, who asked whether the regularity can…
We address the properties of continuous-time quantum walks with Hamiltonians of the form $\mathcal{H}= L + \lambda L^2$, being $L$ the Laplacian matrix of the underlying graph and being the perturbation $\lambda L^2$ motivated by its…
A long-standing conjecture of Kelly states that every regular tournament on n vertices can be decomposed into (n-1)/2 edge-disjoint Hamilton cycles. We prove this conjecture for large n. In fact, we prove a far more general result, based on…
The dynamical mechanisms that can stabilize the coexistence of species (or strategies) are of substantial interest for the maintenance of biodiversity and in sociobehavioural dynamics. We investigate the mean extinction time in the…