We recall definitions of linking numbers and Wu--Simon numbers for spatial graphs. We expose a `converse'to the Conway--Gordon--Sachs theorem (i.e. description of linking functions for embeddings $K_6\to\mathbb{R}^3$), and some results on Wu--Simon numbers. We conjecture and discuss a generalization of the Conway--Gordon--Sachs theorem to multiple linking. The exposition is based on plane diagrams, so no knowledge of spatial geometry is required.
We give a new proof by flow decomposition of the known Grigor'yan type integral criterion for stochastic completeness of weighted graphs. The main ingredient is a finite-domain estimate relating the mass of a killed $1$-resolvent and the current reaching the boundary to relative capacities of intrinsic balls. This esti...
Qing-Song Gu, Lu Hao, Xue-Ping Huang et al.· 0 citations
We investigate the behavior of arboricity under two fundamental graph operations, namely wedges and joins, proving an exact formula for wedges and establishing general upper and lower bounds for joins. Using the characterization of the simplicial geometric category of connected graphs in terms of arboricity, we derive...
In this article, we prove the Defect Conjecture for Cayley graphs of abelian groups, dihedral groups, and groups of the form $\mathbb{Z}/p\mathbb{Z} \rtimes \mathbb{Z}/(p-1)\mathbb{Z}$ where $p\geq 3$ is a prime. We also compute the Iwasawa invariants of the Bowen--Franks groups associated with these graphs in several...
Debanjana Kundu, Katharina Müller, A. Newton et al.· 0 citations
We investigate a combinatorial notion of orientability for abstract GKM graphs and its connections to graph cohomology in the sense of Guillemin--Zara. In particular, we prove that orientability of the GKM graph is equivalent to Poincar\'e duality of the rational (non-equivariant) graph cohomology algebra. As an applic...
Oliver Goertsches, Panagiotis Konstantis, L. Zoller· 0 citations
Let $X$ be a closed hyperbolic surface. We prove that the number of topological types of primitive closed geodesics of length at most $L$ is asymptotic to \[ \frac{1}{|\Isom(X)|}\frac{e^L}{2L}. \] as $L$ grows. Thus Huber's asymptotic remains unchanged after quotienting by topological type, up to the finite symmetry fa...
Albertson's conjecture states that every graph of chromatic number r has crossing number at least that of the complete graph K_r. We prove the conjecture for all r. The proof combines new clique-sensitive edge bounds for critical graphs, maximum-clique sampling, and a reserved immersion construction. For graphs of orde...
A. Sadhu· 1 citation
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