In this article, we compute the Hilbert series of the generalized binomial edge ideal associated with the join of two graphs. Consequently, we compute the Hilbert series of complete $r$-partite graphs. Further, we show that if a vertex satisfies a certain degree condition, then some Hilbert coefficients remain unchanged upon its removal. This provides a reduction technique for computing them. As an application, we compute the dimension and multiplicity of several classes of graphs.
We compute the diagonal $F$-threshold of binomial edge ideals associated to graphs in the case of cycles, cycles with whiskers, bipartite graphs, and block graphs. Moreover, we provide general bounds for any graph.
Classes of squarefree monomial ideals were intensively studied being an important connection between two main areas in mathematics: commutative algebra and combinatorics. Many algebraic invariants of the squarefree monomial ideals are given in terms of properties of the combinatorial associated objects (graphs, simplic...
The theory of combinatorial Hopf algebras is a powerful framework for studying algebraic invariants of combinatorial objects. The aim of this work is to incorporate chromatic invariants of signed graphs into this context. We define the combinatorial Hopf algebra of signed graphs. Through this definition, algebraic inva...
The concept of irreducible submodule of a module over a ring is well known. Analogously, in this paper, we define $e$-irreducibility and prove several properties that involve $e$-irreducibility, the complement in a lattice. Furthermore, we establish an isomorphism between the graph constructed on the equivalence class...
Sridhara Kumar Barik, Tapatee Sahoo, P. Harikrishnan et al.· Boletim da Sociedade Paranae...· 0 citations
We generalize a construction of Herzog and Takayama pertaining to differential graded algebra structures on mapping cones. Via this generalization, we provide the minimal free resolution of the edge ideal of a new graph built from suspension over a vertex cover of an old graph, and we discuss when this process preserve...
Hugh Geller, Desiree Martin, Matthew Mastroeni et al.· 0 citations
For decades, a fundamental question about the difficulty of computing in high-dimensional spaces has resisted proof. New research resolves this question, removing a major theoretical obstacle to understanding the efficiency of randomized algorithms used across science and computing. The Kannan–Lovász–Simonovits (KLS) conjecture was proposed in 1995. It asks whether, for a broad class of high-dimensional […] The post Solving a 31-year-old grand challenge in mathematics – the KLS conjecture appear…
MIT News · Artificial Intelligence· news.mit.eduOct 15, 2026
A new computational approach reveals how cells communicate to assemble into tissues. This could accelerate stem cell engineering, improve disease models, and inform therapies.
Microsoft Research Blog· microsoft.comOct 15, 2026
Zero-knowledge proofs can verify computing results without rerunning workloads or exposing sensitive data. Magnetar helps bring them into the mainstream with support for standard programming languages, major speed gains, and quantum-resistant security. The post Proof-based general-purpose computing with Magnetar appeared first on Microsoft Research.
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