Abstract

Fuzzy graph theory is a useful and well-known tool to model and solve many real-life optimization problems. Since real-life problems are often uncertain due to inconsistent and indeterminate information, it is very hard for an expert to model those problems using a fuzzy graph. A neutrosophic graph can deal with the uncertainty associated with the inconsistent and indeterminate information of any real-world problem, where fuzzy graphs may fail to reveal satisfactory results. The concepts of the regularity and degree of a node play a significant role in both the theory and application of graph theory in the neutrosophic environment. In this work, we describe the utility of the regular neutrosophic graph and bipartite neutrosophic graph to model an assignment problem, a road transport network, and a social network. For this purpose, we introduce the definitions of the regular neutrosophic graph, star neutrosophic graph, regular complete neutrosophic graph, complete bipartite neutrosophic graph, and regular strong neutrosophic graph. We define the d m - and t d m -degrees of a node in a regular neutrosophic graph. Depending on the degree of the node, this paper classifies the regularity of a neutrosophic graph into three types, namely d m -regular, t d m -regular, and m-highly irregular neutrosophic graphs. We present some theorems and properties of those regular neutrosophic graphs. The concept of an m-highly irregular neutrosophic graph on cycle and path graphs is also investigated in this paper. The definition of busy and free nodes in a regular neutrosophic graph is presented here. We introduce the idea of the &mu -complement and h-morphism of a regular neutrosophic graph. Some properties of complement and isomorphic regular neutrosophic graphs are presented here. Document type: Article

Abstract

This paper deals with the issue of road network vulnerability and describes the implementation of a methodology which ranks the links of a network according to their importance for maintaining a proper connectivity between all origin-destination pairs. Such a ranking can be easily used by practitioners and decision-makers for prioritising maintenance investments along the links of a road transport network. In this regard, following a conceptual approach well consolidated in transport literature, vulnerability is assumed to be related to the concept of importance, i.e. a measure of the consequences of the collapse of a network's element. In the present study, the definition of importance – with respect to a given link –simultaneously includes two aspects: the level of usage, i.e. how many people typically use the link for their trips, and the impact that the closure of the link itself can have on the general functionality of the network as a whole. These two aspects are considered in the link importance index formulation, as two different functions that can be properly weighted by means of different coefficients. The methodology proposed has been implemented in the framework of Paramount EU project, to obtain a ranking of importance for the links of a real-scale network, i.e. the road network of Bolzano, a highly mountainous province located in the Italian Alps. The application of the methodology led to satisfactory results represented by a ranking of links, in decreasing order importance scores. Furthermore, Spearman's rank correlation coefficient has been used to quantify the variation of the importance ranking caused by the variation of its coefficients.