Gutman极值六角链猜想的证明

THE PROOF OF GUTMAN'S CONJECTURES CONCERNING EXTREMAL HEXAGONAL CHAINS

六角系统是理论化学中苯碳氢化合物的自然图表示．六角链是一个六角系统满足任意一个顶点至多属于两个六角形，并且每个六角形至多与两个六角形相邻．Gutman提出了两个猜想：1）含有相同六角形个数、具有点独立集总数（Hosoya指数）最小的六角链是唯一的，且为锯齿链；2）含有相同六角形个数、具有边独立集总数（Merrifield－Simmons指数）最大的六角链是唯一的且为锯齿链．

A hexagonal chain is a hexagonal system with the properties that it has no vertex belonging to three hexagons, and it has no hexagon with more than two adjacent hexagons. Denote by Cn the set of all hexagonal chains with n hexagons. Two conjectures proposed by Gutman are: 1) The element of the class Cn with the largest Hosoya index is unique and is the zig-zag polyphene graph, 2) The element of the class C. with the smallest Merrifield-Simmons index is unique and is the zig-zag polyphene graph. The conjectures are proved in this paper.