Related papers: Evolutionary Hamiltonian Graph Theory
Graph plays a significant role in representing and analyzing complex relationships in real-world applications such as citation networks, social networks, and biological data. Recently, Large Language Models (LLMs), which have achieved…
We present progress on three old conjectures about longest paths and cycles in graphs. The first pair of conjectures, due to Lov\'{a}sz from 1969 and Thomassen from 1978, respectively, states that all connected vertex-transitive graphs…
Lotka's seminal work (A.J. Lotka A., Proc. Natl. Acad. Sci. U.S.A. 6 (1920) 410) "on certain rhythmic relations'' is already one hundred years old, but the research activity about pattern formations due to cyclical dominance is more vibrant…
We consider the population dynamics of a set of species whose network of catalytic interactions is described by a directed graph. The relationship between the attractors of this dynamics and the underlying graph theoretic structures like…
A first-principles statistical theory is constructed for the evolution of two dimensional interfaces in Laplacian fields. The aim is to predict the pattern that the growth evolves into, whether it becomes fractal and if so the…
Despite the advances in large language models (LLMs), how they use their knowledge for reasoning is not yet well understood. In this study, we propose a method that deconstructs complex real-world questions into a graph, representing each…
One of the central models in distributed computing is Linial's LOCAL model [SIAM J. Comp. 1992]. Over time, researchers have studied distributed graph problems in the LOCAL model under slightly different assumptions, such as whether nodes…
Many decision-making algorithms draw inspiration from the inner workings of individual biological systems. However, it remains unclear whether collective behavior among biological species can also lead to solutions for computational tasks.…
To perform effective causal inference in high-dimensional datasets, initiating the process with causal discovery is imperative, wherein a causal graph is generated based on observational data. However, obtaining a complete and accurate…
Recent studies have discovered that Chain-of-Thought prompting (CoT) can dramatically improve the performance of Large Language Models (LLMs), particularly when dealing with complex tasks involving mathematics or reasoning. Despite the…
A digraph is attached to any evolution algebra. This graph leads to some new purely algebraic results on this class of algebras and allows for some new natural proofs of known results. Nilpotency of an evolution algebra will be proved to be…
The classical Dirac theorem asserts that every graph $G$ on $n$ vertices with minimum degree $\delta(G) \ge \lceil n/2 \rceil$ is Hamiltonian. The lower bound of $\lceil n/2 \rceil$ on the minimum degree of a graph is tight. In this paper,…
Algebraic Combinatorics originated in Algebra and Representation Theory, studying their discrete objects and integral quantities via combinatorial methods which have since developed independent and self-contained lives and brought us some…
We study Loewner chains in $\mathcal{H}_0(\mathbb{D})$ without assuming univalence of each element. We prove a decomposition: every chain admits a factorization $f_t=F\circ g_t$, where $F$ is analytic on $\mathbb{D}(0,r)$ with $r=\lim_{t…
Graph Convolutional Networks (GCNs) suffer from severe performance degradation in deep architectures due to over-smoothing. While existing studies primarily attribute the over-smoothing to repeated applications of graph Laplacian operators,…
We study the evolution, under convex Hamiltonian flows on cotangent bundles of compact manifolds, of certain distinguished subsets of the phase space. These subsets are generalizations of Lagrangian graphs, we call them pseudographs. They…
In this paper we consider generalization of classical and quantum mechanics that directly follows from the causality principle and topology of a system state space. In generalized mechanics, the Hamiltonian/Schrodinger equations remain the…
The relevance of chaos to evolution is discussed in the context of the origin and maintenance of diversity and complexity. Evolution to the edge of chaos is demonstrated in an imitation game. As an origin of diversity, dynamic clustering of…
In this paper, we investigate the well-studied Hamiltonian cycle problem (HCYCLE), and present an interesting dichotomy result on split graphs. T. Akiyama et al. (1980) have shown that HCYCLE is NP-complete in planar bipartite graphs with…
In the complete system of equations of evolution of the classical system of charges and the electromagnetic field generated by them, the field variables are excluded. An exact closed relativistic non-Hamiltonian system of nonlocal kinetic…