富勒烯图是3-连通3-正则平面图,并且恰好具有12个五边形面,其余的面都是六边形面。本文研究的富勒烯图是由六个同心的六边形层组成,两端都由一个六边形以及与这个六边形相邻的六个五边形面构成的顶盖封口。我们把该类富勒烯图称为管状富勒烯图。完美匹配计数在量子化学领域以及统计物理领域中具有广泛的应用,并且已被证实完美匹配计数问题是一个NP-难的问题。本文主要通过划分、求和以及嵌套递推的方式求出管状富勒烯图的完美匹配数。A fullerene graph is 3-connected cubic planar graph, and has exactly 12 pentagonal faces, the rest of which are hexagonal faces. The fullerene graph studied in this paper is composed of six concentric layers of hexagons, capped on each end by a cap formed by a hexagon and six pentagonal faces adjacent to the hexagon. This kind of fullerene graphs is called tubular fullerene graphs. The problem of counting the number of perfect matching is widely used in the field of quantum chemistry and statistical physics, and it has been proved that the problem of counting the number of perfect matching is NP-hard. In this paper, the number of perfect matching of tubular fullerene graphs is obtained by means of partition, summation and nested recursion.
Sombor指标是一种离散数学图论中的拓扑指标,能够清晰地反应图的特征。讨论拓扑指标的极值问题能够分析图的基本性质。本文讨论了在完美匹配的单圈图当中,指数型Sombor指标的极值问题。其中指数型Sombor指标定义为: eSO(G) =uv∈E(G)∑e√d2G(u)+d2G(v) 本文的主要结论是:若G∈U2m,m,则eSO(G) ≤ eSO(U2m,m)且eSO(U2m,m) ≤ (m - 2)e√5 + me√(m+1)2+4+e2√2+e√(m+1)2+1等号成立当且仅当G≅U2m,m,其中m为图G的匹配数。The Sombor Index is a Topological Index in Discrete Mathematical Graph Theory which can clearly reflect the characteristics of the graph. While the Extreme value of Topological Index is the key to analyse the basic properties of the graph. This paper discusses the Extreme Value of Exponential Sombor Index in Unicyclic Graph with Perfect Matching. The exponential Sombor index is defined as:eSO(G) =uv∈E(G)∑e√d2G(u)+d2G(v) The main result of this paper is:If G∈U2m,m,Then eSO(G) ≤ eSO(U2m,m),eSO(U2m,m) ≤ (m - 2)e√5 + me√(m+1)2+4+e2√2+e√(m+1)2+1If and only if G≅U2m,m the equal sign is established, Where m is the matching number of Graph G.
