We determine the exact value of the Wiener index, the edge-Wiener index, and the vertex-edge-Wiener index of the Basilica graphs, i.e., the sequence of finite Schreier graphs associated with the ...action of the Basilica group on the rooted binary tree. Moreover, we give a formula for the total distance of every vertex in the Basilica graphs, and we are able to make it explicit for some special vertices. We finally introduce the notions of asymptotic Wiener index and asymptotic total distance, which are compatible with that of convergence of the sequence of finite Basilica graphs to an infinite orbital limit graph in the Gromov–Hausdorff topology: the asymptotic values are explicitly computed.
Topological indices have an important role in molecular chemistry, network theory, spectral graph theory and several physical worlds. Most of the topological indices are defined in a crisp graph. As ...fuzzy graphs are more generalization of crisp graphs, those indices have more application in fuzzy graphs also. In this article, we introduced the fuzzy hyper-Wiener index (FHWI) and studied this index for various fuzzy graphs like path, cycle, star, etc and provided some interesting bounds of FHWI for that fuzzy graph. A lower bound of FHWI is established for n-vertex connected fuzzy graph depending on strength of a strong edges. A relation between FHWI of a tree and its maximum spanning tree is established and this index is calculated for the saturated cycle. Also, at the end of the article, an application in the share market of this index is presented.
The usefulness of different topological indices is inevitable in various fields such as Chemistry, Electronics, Economics and Business studies, medical and social sciences. The “purpose of this paper ...is to study” Wiener index for the Intuitionistic fuzzy graphs (IFG) in some detail while keeping it in parallel with the connectivity index. Some of the other distance and degree based topological indices have been defined for an IFG. Moreover, Average Wiener index has also been defined for an IFG. At the end, an application of Wiener index and Average Wiener index has also been presented in water a pipeline network which is very integral in water supply management.
In this paper, the Wiener index and the hyper-Wiener index of the Kragujevac trees is computed in term of its vertex degrees. As application, we obtain an upper bond and a lower bound for the Wiener ...index and the hyper-Wiener index of these trees.
The Wiener index W(G) of a connected graph G is a sum of distances between all pairs of vertices of G. In 1991, Šoltés formulated the problem of finding all graphs G such that for every vertex v the ...equality W(G)=W(G−v) holds. The cycle C11 is the only known graph with this property. In this paper we consider the following relaxation of the original problem: find a graph with a large proportion of vertices such that removing any one of them does not change the Wiener index of a graph. As the main result, we build an infinite series of graphs with the proportion of such vertices tending to 12.
In this article, we have examined the Wiener index in neutrosophic graphs. Wiener index is one of the most important topological indices. This index is a distance-based index that is calculated based ...on the geodesic distance between two vertices. Here, after defining the Wiener index in neutrosophic graphs, we calculated this index for some special modes such as the complete neutrosophic graph, cycle, and tree. In the following, by presenting a several theorems, we compared this index with the connectivity index, which is one of the most important degree-based indicators. Keywords: Wiener index; partial Wiener index; totally Wiener index; neutrosophic graph; neutrosophic tree; strong spanning tree; connectivity index
Social networks are becoming more indispensable in our lives, which leads researchers to seek to understand and analyze them. Social network analysis has become a specialty of sociology in network ...theory and graph theory as well. The principal role of this analysis is to focus on the entities and the structure of relationships between them in a specific context, which is going to highlight the most influential (important) person in a social network, in terms of the number of direct and indirect relationships.
Centrality indices are measures used to capture the notion of importance in a graph, by identifying the most significant person(s) in a social network. In this paper, we will propose and prove the generality of some centrality measures. In other words, the concept of these social networks analysis is part of sociological circles that study the relationships and social representations that we can analyze by levels using the dGu(k).
Distance-based topological indices, as a class of graph invariants, have received much attention. In particular, the Wiener index (sum of distances between all pairs of vertices) and terminal Wiener ...index (sum of distances between all pairs of leaves) are two of the most well known such indices in Chemical Graph Theory. Inspired by these concepts, the peripheral Wiener index is recently introduced as the sum of distances between all pairs of peripheral vertices (vertices with maximum eccentricity). In this note we consider a number of interesting problems related to this new distance-based index and propose some potential topics for further study.
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