With any (not necessarily proper) edge k-colouring γ : E(G) −→ {1, . . . , k} of a graph G, one can associate a vertex colouring σγ given by σγ(v) = ∑ e∋v γ(e). A neighbour-sumdistinguishing edge k-colouring is an edge colouring whose associated vertex colouring is proper. The neighbour-sum-distinguishing index of a graph G is then the smallest k for which G admits a neighbour-sum-distinguishin...

Factor graphs are a very general knowledge representation, subsuming many existing formalisms in AI. Sum-product networks are a more recent representation, inspired by studying cases where factor graphs are tractable. Factor graphs emphasize expressive power, while sum-product networks restrict expressiveness to get strong guarantees on speed of inference. A sum-product network is not simply a ...

The idea of integral sum graphs was introduced by Harary (1994). A graph G is said to be an integral sum graph if its nodes can be given a labeling f with distinct integers, so that for any two distinct nodes u and v of G, uv is an edge of G if and only if f(u) + f(v) = f(w) for some node w in G. A tree is said to be a generalized star if it can be obtained from a star by extending each edge to...

A graph G is called a sum graph if there is a so-called sum labeling of G, i.e. an injective function l : V (G) → N such that for every u, v ∈ V (G) it holds that uv ∈ E(G) if and only if there exists a vertex w ∈ V (G) such that l(u) + l(v) = l(w). We say that sum labeling l is minimal if there is a vertex u ∈ V (G) such that l(u) = 1. In this paper, we show that if we relax the conditions (ei...

A graph is f -choosable if for every collection of lists with list sizes specified by f there is a proper coloring using colors from the lists. We characterize f -choosable functions for block graphs (graphs in which each block is a clique, including trees and line graphs of trees). The sum choice number is the minimum over all choosable functions f of the sum of the sizes in f . The sum choice...