Total Distance, Wiener Index and Opportunity Index in Wreath Products of Star Graphs

Abstract

In the last decades much attention has turned towards centrality measures on graphs. The Wiener index and the total distance are key tools to investigate the median vertices, the distance-balanced property and the opportunity index of a graph. This interest has recently been addressed to graphs obtained via classical graph products like the Cartesian, the direct, the strong and the lexicographic product. We extend this study to a relatively new graph product, that is, the wreath product. In this paper, we compute the total distance for the vertices of an arbitrary wreath product graph $G\wr H$ in terms of the total distances in $H$ and of some distance-based indices of $G$. We explicitly compute these indices for the star graph $S_n$, providing a closed formula for the total distances in $S_n\wr H$ when $H$ is complete or a star. As a consequence, we obtain the Wiener index of these graphs, we characterize the median and the central vertices, and finally we give an upper and a lower bound for the opportunity index of $S_n\wr S_m$ in terms of tail conditional expectations of an associated binomial distribution.

Extremal results on stepwise transmission irregular graphs

“…[1,22,[24][25][26]. The transmission also led to the Wiener complexity (the number of different transmissions) [3], is closely related to other topological indices [26], and characterizes the distance-balanced property and the opportunity index [9,13].…”

Distance-balanced graphs and travelling salesman problems

“…The investigation of distance-balanced graphs began in [13], though an explicit definition was provided later in [15,18]. Such graphs generated a certain degree of interest also by virtue of their connection with centrality measures [2,8] and with some well known and largely studied distance-based invariants, such as the Wiener and the Szeged index [2,14,15,16]. For instance, it was proven in [14] that, in the bipartite case, distancebalanced graphs maximize the Szeged index.…”

“…From another point of view, that we will develop in the last part of the paper, our work can be interpreted as the investigation of the distance-balancedness in a wreath product of graphs. In this sense, it is the natural continuation of [8]. The wreath product of graphs represents the graph analogue of the classical wreath product of groups, as it is true that the wreath product of the Cayley graphs of two finite groups is the Cayley graph of the wreath product of the groups.…”

Wiener index of unicycle graphs with given number of even degree vertices

“…The Wiener index is one of the most studied topological indices in mathematical chemistry and it is still a very active research topic (see [1,2,3,6,7,8,9,10,11]). One of the fundamental problems pertaining to the Wiener index is finding extremal graphs in a graph family that achieve maximum and/or minimum values of the index in the family.…”