Related papers: Second extra differential on odd graph complexes
An antimagic labeling of a graph $G$ with $m$ edges is a bijection from $E(G)$ to $\{1,2,\ldots,m\}$ such that for all vertices $u$ and $v$, the sum of labels on edges incident to $u$ differs from that for edges incident to $v$. Hartsfield…
Khovanov introduced a bigraded cohomology theory of links whose graded Euler characteristic is the Jones polynomial. The theory was subsequently applied to the chromatic polynomial of graph, resulting in a categorification known as the…
We study the cohomology of forested graph complexes with ordered and unordered hairs whose cohomology computes the cohomology of a family of groups $\Gamma_{g,r}$ that generalize the (outer) automorphism group of free groups. We give…
A graph is odd if all of its vertices have odd degrees. In particular, an odd spanning tree in a connected graph is a spanning tree in which all vertices have odd degrees. In this paper we establish a unified technique to enumerate odd…
Most Graph Neural Networks (GNNs) cannot distinguish some graphs or indeed some pairs of nodes within a graph. This makes it impossible to solve certain classification tasks. However, adding additional node features to these models can…
For each graph and each positive integer $n$, we define a chain complex whose graded Euler characteristic is equal to an appropriate $n$-specialization of the dichromatic polynomial. This also gives a categorification of $n$-specializations…
Graphs with given k vertices generate an (acyclic) simplicial complex. We describe the homology of its quotient complex, formed by all connected graphs, and demonstrate its applications to the topology of braid groups, knot theory,…
It is known that non-isomorphic strongly regular graphs with the same parameters must be cospectral (have the same eigenvalues). In this paper, we investigate whether the spectra of higher order Laplacians associated with these graphs can…
We define two new families of invariants for (3-manifold, graph) pairs which detect the unknot and are additive under connected sum of pairs and (-1/2)-additive under trivalent vertex sum of pairs. The first of these families is closely…
We obtain several new upper bounds of the odd graceful chromatic number of a graph $G$, which must be bipartite. Some of our bounds depend only on the number of the vertices of $G$ or the chromatic number of some graphs related to the…
Anomaly analytics is a popular and vital task in various research contexts, which has been studied for several decades. At the same time, deep learning has shown its capacity in solving many graph-based tasks like, node classification, link…
A connected graph can be associated with two distinct evolution algebras. In the first case, the structural matrix is the adjacency matrix of the graph itself. In the second case, the structural matrix is the transition probabilities matrix…
For vertex and edge connectivity we construct infinitely many pairs of regular graphs with the same spectrum, but with different connectivity.
In this paper we introduce the concept of characteristic number that are proven to be useful in the study of the combinatorics of graph cohomology. We claim that it is a good combinatorial counterpart for geometric Betti numbers. We then…
Graphs constructed to translate some graph problem into another graph problem are usually called auxiliary graphs. Specifically total graphs of simple graphs are used to translate the total colouring problem of the original graph into a…
The symplectic derivation Lie algebras defined by Kontsevich are related to various geometric objects including moduli spaces of graphs and of Riemann surfaces, graph homologies, Hamiltonian vector fields, etc. Each of them and its…
We prove that the Kontsevich graph complex $GC_d^{2}$ and its oriented version $OGC_{d+1}^2$ are quasi-isomorphic as dg Lie algebras.
We propose an approach to learning with graph-structured data in the problem domain of graph classification. In particular, we present a novel type of readout operation to aggregate node features into a graph-level representation. To this…
In a paired threshold graph, each vertex has a weight, and two vertices are adjacent if their weight sum is large enough and their weight difference is small enough. It generalizes threshold graphs and unit interval graphs, both very well…
Processing data on multiple interacting graphs is crucial for many applications, but existing approaches rely mostly on discrete filtering or first-order continuous models, dampening high frequencies and slow information propagation. In…