Modular Sumset Labelling of Graphs

Graph labelling is an assignment of labels or weights to the vertices and/or edges of a graph. For a ground set X of integers, a sumset labelling of a graph is an injective map f:VG→PX such that the induced function f⊕:EG→PX is defined by f+uv=fu+fv, for all uv∈EG, where fu+fv is the sumset of the set-label, the vertices u and v. In this chapter, we discuss a special type of sumset labelling of a graph, called modular sumset labelling and its variations. We also discuss some interesting characteristics and structural properties of the graphs which admit these new types of graph labellings.

Sumset Valuations of Graphs and Their Applications

Graph labelling is an assignment of labels to the vertices and/or edges of a graph with respect to certain restrictions and in accordance with certain predefined rules. The sumset of two non-empty sets A and B, denoted by A+B, is defined by A+B=\{a=b: a\inA, b\inB\}. Let X be a non-empty subset of the set \Z and \sP(X) be its power set. An \textit{sumset labelling} of a given graph G is an injective set-valued function f: V(G)\to\sP_0(X), which induces a function f+: E(G)\to\sP_0(X) defined by f+(uv)=f(u)+f(v), where f(u)+f(v) is the sumset of the set-labels of the vertices u and v. This chapter discusses different types of sumset labeling of graphs and their structural characterizations. The properties and characterizations of certain hypergraphs and signed graphs, which are induced by the sumset-labeling of given graphs, are also done in this chapter.

Fuzzy Magic and Bi-magic Labelling of Intuitionistic Path Graph

A graph labelling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Fuzzy labelling models precision, flexibility and compatibility to the classical models. The objective is to discuss about the magic and bi-magic labelling of fuzzy graphs. In the beginning the magic labelling of fuzzy path graph is discussed followed by the bi-magic labelling of fuzzy path graph. The next main purpose is to introduce the Intuitionistic path graph and examine the existence of fuzzy magic and bi-magic labelling.

Link Prediction Evaluation Using Palette Weisfeiler-Lehman Graph Labelling Algorithm

Link prediction is gaining interest in the community of machine learning due to its popularity in the applications such as in social networking and e-commerce. This paper aims to present the performance of link prediction using a set of predictive models. In link prediction modelling, feature extraction is a challenging issue and some simple heuristics such as common-neighbors and Katz index were commonly used. Here, palette weisfeiler-lehman graph labelling algorithms have been used, which has a few advantages such as it has order-preserving properties and provides better computational efficiency. Whereas, other feature extraction algorithms cannot preserve the order of the vertices in the subgraph, and also take more computational time. The features were extracted in two ways with the number of vertices in each subgraph, say K = 10 and K = 15. The extracted features were fitted to a range of classifiers. Further, the performance has been obtained on the basis of the area under the curve (AUC) measure. Comparative analysis of all the classifiers based on the AUC results has been presented to determine which predictive model provides better performance across all the networks. This leads to the conclusion that ADABoost, Bagging and Adaptive Logistic Regression performed well almost on all the network. Lastly, comparative analysis of 12 existing methods with three best predictive models has been done to show that link prediction with predictive models performs well across different kinds of networks.