We investigate several related measures of peripherality and centrality for vertices and edges in networks, including the Mostar index which was recently introduced as a measure of peripherality for both edges and networks. We refute a conjecture on the maximum possible Mostar index of bipartite graphs from (Do\v{s}li\'{c} et al, Journal of Mathematical Chemistry, 2018) and (Ali and Do\v{s}li\'{c}, Applied Mathematics and Computation, 2021). We also correct a result from the latter paper, where they claimed that the maximum possible value of the terminal Mostar index among all trees of order $n$ is $(n-1)(n-2)$. We show that this maximum is $(n-1)(n-3)$ for $n \ge 3$, and that it is only attained by the star. We asymptotically answer another problem on the maximum difference between the Mostar index and the irregularity of trees from (F. Gao et al, On the difference of Mostar index and irregularity of graphs, Bulletin of the Malaysian Mathematical Sciences Society, 2021). We also prove a number of extremal bounds and computational complexity results about the Mostar index, irregularity, and measures of peripherality and centrality. We discuss graphs where the Mostar index is not an accurate measure of peripherality. We construct a general family of graphs with the property that the Mostar index is strictly greater for edges that are closer to the center. We also investigate centrality and peripherality in two graphs which represent the SuperFast and MOZART-4 systems of atmospheric chemical reactions by computing various measures of peripherality and centrality for the vertices and edges in these graphs. For both of these graphs, we find that the Mostar index is closer to a measure of centrality than peripherality of the edges. We also introduce some new indices which perform well as measures of peripherality on the SuperFast and MOZART-4 graphs.
翻译:我们调查网络中脊椎和边缘的外围和核心相关度度,包括最近作为边缘和网络的外围度度度而引入的莫斯塔尔指数。我们驳斥了来自(Do\v{s}li\'{c}等)和(Ali和Do\v{s}li\\{c})以及(Ali和Do\v{sli\{c},应用数学和计算,2021年)。我们还纠正了后一篇论文的结果,该论文声称,所有顺序树的Ostar 4 直径指数的最大可能值为$(n-1) (n-2)。我们反驳了关于双partite 图形图的最大值的莫斯塔尔指数值最高值的推测。我们证明,美元3美元的最大值是美元(n-1) (n-3),而只有恒星才能达到这一点。我们从Otrial指数和树的异常值之间的最大差, 也是(F. Gael, 关于Oestal Indentality 和Mestrial Centrial) 中, 最差和最不定期的中间度的数值, 也是马雅数。