In all the paper we work on a graphG with maximal degree∆. For a set of vertices S ⊂ V (G), the graph G\S is the graph induced by V (G)\S. The distance d(u, v) between two vertices u and v is the number of edges in the shortest path from u to v. We say that v is a d-neighbor of u if d(u, v) = d. Let Nd(v) be the set of d-neighbors of v. We will generally use the common term neighbor instead of ...

Let i1 ≥ i2 ≥ i3 ≥ 1 be integers. An L(i1, i2, i3)-labelling of a graph G = (V, E) is a mapping φ : V → {0, 1, 2, . . .} such that |φ(u)− φ(v)| ≥ it for any u, v ∈ V with d(u, v) = t, t = 1, 2, 3, where d(u, v) is the distance in G between u and v. The integer φ(v) is called the label assigned to v under φ, and the difference between the largest and the smallest labels is called the span of φ. ...

This paper presents a graph neural network (GNN) technique for low-level reconstruction of neutrino interactions in Liquid Argon Time Projection Chamber (LArTPC). GNNs are still relatively novel technique, and have shown great promise similar tasks the LHC. In this paper, multihead attention message passing is used to classify relationship between detector hits by labelling edges, determining w...

We show that there exists an adjacency labelling scheme for planar graphs where each vertex of n -vertex graph G is assigned a (1 + o(1)) log 2 -bit label and the labels two vertices u v are sufficient to determine if uv edge . This optimal up lower order term first such asymptotically result. An alternative, but equivalent, interpretation this result that, every positive integer , U with 1+o(1...

We describe a robust approach for linking already existing lexical/semantic hierarchies. We use a constraint satisfaction algorithm (relaxation labelling) to select –among a set of candidates– the node in a target taxonomy that bests matches each node in a source taxonomy. In this paper we present the complete mapping of the nominal, verbal, adjectival and adverbial parts of WordNet 1.5 onto Wo...