WIENER-TYPE INVARIANTS OF TREES AND THEIR RELATION
WIENER-TYPE INVARIANTS OF TREES AND THEIR RELATION
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发表时间：
2004
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通讯作者：
I. Gutman;A. Dobrynin;S. Klavžar
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作者：
I. Gutman;A. Dobrynin;S. Klavžar
（连通）图G的顶点u和v之间的距离d（u;vjG）是连接u和v的最短路径的长度（=边数）。G的Wiener数W（G）是G的所有顶点对之间的距离之和。考虑一类Wiener{型不变量W（G），定义为G的所有顶点对上的项d（u;vjG）的和. W <$（G）的几种特殊情况，即<$= +1（原始维纳数）以及<$= i2;i1;+1 = 2;+2和+3的不变量，以前在化学文献中研究过，并发现了作为分子结构描述符的应用。我们修改W <$（G）的定义，使它也推广到非连通图，然后推出恒等式W <$+1（T）=（ni 1）W <$（T）i P W <$（T i e），对任意n-顶点树T都有效，其和包含T的所有边e.
The distance d(u;vjG) between the vertices u and v of a (connected) graph G is the length (= number of edges) of a shortest path connecting u and v . The Wiener number W(G) of G is the sum of distances between all pairs of vertices of G. We consider a class of Wiener{type invariants W‚(G), deflned as the sum of the terms d(u;vjG) ‚ over all pairs of vertices of G. Several special cases of W‚(G), namely the invariants for ‚ = +1 (the original Wiener number) as well as for ‚ = i2;i1;+1=2;+2 and +3, were previously studied in the chemical literature, and found applications as molecular structure descriptors. We modify the deflnition of W‚(G) so that it extends also to non-connected graphs and then deduce the identity W‚+1(T) = (n i 1)W‚(T) i P W‚(T i e), valid for any n-vertex tree T , with the summation embracing all edges e of T .