292 papers
This paper solves a novel nonlocal discrete isoperimetric problem by characterizing the minimizers of a generalized "bi-axial" perimeter that accounts for all internal and external components of a polyomino, thereby establishing a rigorous link to the metastable behavior of a long-range bi-axial Ising model.
This paper investigates the spectral properties of degree-based weighted adjacency matrices for complete and crown multipartite graphs, characterizes families with three distinct eigenvalues and integral spectra, corrects previous findings on energy changes after edge deletion, and resolves an open problem regarding ISI energy in multipartite graphs.
This paper investigates Max-Cut instances on complete graphs with geometrically decreasing edge weights, establishing a sharp phase diagram for the optimality of isolated cuts based on threshold polynomials and conjecturing their global optimality for sufficiently large .
This paper investigates the domination polynomial of the co-maximal graph by deriving explicit formulas for specific cases of , establishing structural connections for general , and analyzing the unimodality, log-concavity, and modulus bounds of its roots.
This paper establishes the existence and combinatorial characterization of the asymptotic v-number for Noetherian graded families of homogeneous ideals via Newton-Okounkov regions, while also proving its eventual quasi-linearity and strict inequalities with regularity and multiplicity.
This paper extends the study of crystal melting models for toric Calabi-Yau 4-folds by developing an algorithm to construct crystals from periodic quivers, analyzing their behavior and partition functions under triality cascades, and introducing stable variables that reveal stabilization patterns to guide the search for generalized cluster algebras in 2d (0,2) quiver theories.
This paper systematically determines the circular chromatic indices of small graphs and multigraphs with maximum degrees up to 6, constructs infinite families of graphs with specific fractional indices, and refutes edge-connectivity variants of the "Upper Gap Conjecture."
This paper establishes a structure theorem for -labeled graphs that exclude a fixed -labeled immersion, demonstrating that such graphs admit a tree-cut decomposition where each bag contains either few high-degree vertices or is nearly signed over a proper subgroup of the finite group .
This paper argues that conserved quantities in mass-action networks are dual to internal cycles and proposes conjectured duality relationships between preclusters, maximal invariant polyhedral supports, and siphons.
This paper proves that randomly initialized, polynomially over-parameterized convolutional neural networks contain structured subnetworks capable of approximating smaller networks without training, by developing new mathematical tools to overcome previous limitations in analyzing the Strong Lottery Ticket Hypothesis for structured pruning.
This paper introduces AlphaEvolve, an LLM-based code mutation agent that serves as a unified meta-algorithm to improve the lower bounds of five classical Ramsey numbers and successfully recover or match existing bounds across various cases, demonstrating a shift from bespoke search methods to a single, adaptable framework for combinatorial structure generation.
This paper demonstrates that a lightweight, automated AI pipeline integrating next-generation large language models with citation-based verification can successfully generate and solve sophisticated, research-grade mathematical problems, including previously unpublished questions, with verified results and open-sourced tools.
This paper investigates the state complexity of automata recognizing relations and operations on -automatic sequences, establishing a link between the subword complexity of interior sequences and the state complexity of their linear subsequences while resolving a recent question by Zantema and Bosma regarding most-significant-digit-first inputs.
This paper extends the existing bijection between ranked tree shapes and F-matrices to fully heterochronous cases by establishing an explicit correspondence that enables efficient enumeration and the development of probabilistic models for phylogenetic trees with leaves at varying sampling times.
This paper extends the known concentration of the largest induced tree size in random graphs to all vanishing edge probabilities and demonstrates that for sparser graphs where , this size fails to concentrate around the standard expectation threshold.
This paper provides a complete proof for the previously proposed upper bound of on the double domination number of maximal outerplanar graphs, correcting an earlier incomplete proof by Abd Aziz, Rad, and Kamarulhaili.
This paper details a previously unrecorded error in Guy and Kelly's 2004 heuristic argument for the no-three-in-line problem, identified by Gabor Ellmann, and presents the resulting correction to their conjectured upper bound.
This paper establishes sharp bounds and asymptotic equivalences among the Albertson, Sombor, and Sigma indices for trees, demonstrating that the Sombor index serves as an intermediate descriptor linking global degree dispersion and local edge irregularity through precise -relationships.
This paper investigates the binomial sum by expressing it in terms of powers of for specific sequences like Fibonacci numbers and various polynomials, while also establishing its properties, probabilistic interpretations, and generating functions in relation to and Appell polynomials.
This paper surveys recent developments in the arithmetic of power monoids—structures formed by finite subsets of a monoid—and highlights their role in advancing factorization theory, particularly within non-cancellative and non-commutative settings.