Abstract Wiener index, which is defined as the sum of distances between all unordered pairs of vertices, is one of the oldest and most popular molecular descriptors. It is known that among graphs on n vertices that have just one block, the n-cycle has the biggest Wiener index. In a previous work of the same authors, it was shown that among all graphs on n vertices which have p ≥ 2 blocks, the maximum Wiener index is attained by a graph composed of two cycles Ca and Cb joined by a path of length p − 2 (possibly a and b are of size 2). In this paper we elaborate further this result and determine the sizes of a and b in the extremal graphs for each n and p. We distinguish six cases with crucial values being 5 p − 7 and 5 p − 8 . Roughly speaking, if n bigger than 5 p − 7 , then the extremal graphs is obtained for a = 2 . i.e. the graph is a path glued to a cycle. For values n = 5 p − 8 and 5 p − 7 , we have more than one extremal graph. And, when n 5 p − 8 , the extremal graphs is again unique with a and b being equal or almost equal depending of the congruence of n − p modulo 4. We also showed unimodal property of a behaviour of Wiener index.

中文翻译：

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具有指定块数的图的最大维纳指数

摘要 维纳指数定义为所有无序顶点对之间距离的总和，是最古老和最流行的分子描述符之一。已知在n个顶点上只有一个块的图中，n-循环的维纳指数最大。在同一作者之前的工作中，表明在具有 p ≥ 2 个块的 n 个顶点上的所有图中，最大维纳指数是通过由两个循环 Ca 和 Cb 组成的图获得的，该图由长度为 p 的路径连接而成 - 2（可能 a 和 b 的大小为 2）。在本文中，我们进一步阐述了这个结果，并确定了每个 n 和 p 的极值图中 a 和 b 的大小。我们区分了 6 个关键值分别为 5 p - 7 和 5 p - 8 的情况。粗略地说，如果 n 大于 5 p − 7 ，则得到 a = 2 的极值图。IE 该图是一条粘在循环上的路径。对于值 n = 5 p − 8 和 5 p − 7 ，我们有不止一个极值图。并且，当 n 5 p − 8 时，极值图又是唯一的，a 和 b 相等或几乎相等，这取决于 n − p 模 4 的同余。我们还展示了维纳指数行为的单峰特性。