On ideal sumset labelled graphs

The sumset of two sets A and B of integers, denoted by A + B, is defined as A+B = {a+b : a ∈ A, b ∈ B}. Let X be a non-empty set of non-negative integers. A sumset labelling of a graph G is an injective function f : V (G) → P(X) − {∅} such that the induced function f+ : E(G) → P(X)−{∅} is defined by f+(uv) = f(u) +f(v) ∀uv ∈ E(G). In this paper, we introduce the notion of ideal sumset labelling of graph and discuss the admissibility of this labelling by certain graph classes and discuss some structural characterization of those graphs.

Regular Expressions and Transducers Over Alphabet-Invariant and User-Defined Labels

We are interested in regular expressions and transducers that represent word relations in an alphabet-invariant way — for example, the set of all word pairs [Formula: see text] where [Formula: see text] is a prefix of [Formula: see text] independently of what the alphabet is. Current software systems of formal language objects do not have a mechanism to define such objects. We define transducers in which transition labels involve what we call set specifications, some of which are alphabet invariant. In fact, we give a more broad definition of automata-type objects, called labelled graphs, where each transition label can be any string, as long as that string represents a subset of a certain monoid. Then, the behavior of the labelled graph is a subset of that monoid. We do the same for regular expressions. We obtain extensions of a few classic algorithmic constructions on ordinary regular expressions and transducers at the broad level of labelled graphs and in such a way that the computational efficiency of the extended constructions is not sacrificed. For transducers with set specs we obtain further algorithms that can be applied to questions about independent regular languages as well as a decision question about synchronous transducers.

MET: a Java package for fast molecule equivalence testing

An important task in cheminformatics is to test whether two molecules are equivalent with respect to their 2D structure. Mathematically, this amounts to solving the graph isomorphism problem for labelled graphs. In this paper, we present an approach which exploits chemical properties and the local neighbourhood of atoms to define highly distinctive node labels. These characteristic labels are the key for clever partitioning molecules into molecule equivalence classes and an effective equivalence test. Based on extensive computational experiments, we show that our algorithm is significantly faster than existing implementations within , and . We provide our Java implementation as an easy-to-use, open-source package (via GitHub) which is compatible with . It fully supports the distinction of different isotopes and molecules with radicals.

Understanding the Success of Graph-based Semi-Supervised Learning using Partially Labelled Stochastic Block Model

With the proliferation of learning scenarios with an abundance of instances, but limited amount of high-quality labels, semi-supervised learning algorithms came to prominence. Graph-based semi-supervised learning (G-SSL) algorithms, of which Label Propagation (LP) is a prominent example, are particularly well-suited for these problems. The premise of LP is the existence of homophily in the graph, but beyond that nothing is known about the efficacy of LP. In particular, there is no characterisation that connects the structural constraints, volume and quality of the labels to the accuracy of LP. In this work, we draw upon the notion of recovery from the literature on community detection, and provide guarantees on accuracy for partially-labelled graphs generated from the Partially-Labelled Stochastic Block Model (PLSBM). Extensive experiments performed on synthetic data verify the theoretical findings.

A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas

A signed graph is a graph whose edges are labeled by signs. This is a bibliography of signed graphs and related mathematics.Several kinds of labelled graph have been called "signed" yet are mathematically very different. I distinguish four types:Group-signed graphs: the edge labels are elements of a 2-element group and are multiplied around a polygon (or along any walk). Among the natural generalizations are larger groups and vertex signs.Sign-colored graphs, in which the edges are labelled from a two-element set that is acted upon by the sign group: - interchanges labels, + leaves them unchanged. This is the kind of "signed graph" found in knot theory. The natural generalization is to more colors and more general groups — or no group.Weighted graphs, in which the edge labels are the elements +1 and -1 of the integers or another additive domain. Weights behave like numbers, not signs; thus I regard work on weighted graphs as outside the scope of the bibliography — except (to some extent) when the author calls the weights "signs".Labelled graphs where the labels have no structure or properties but are called "signs" for any or no reason.