【摘要】：在化學(xué)理論中,拓?fù)渲笜?biāo)可以用來理解混合物的物理和化學(xué)性質(zhì),不同的指標(biāo)反映了分子的不同性能.分子拓?fù)渲笜?biāo)以及分子圖的不變量的研究是化學(xué)圖論中的研究領(lǐng)域之一.簡單無向圖G=(V,E)的每個頂點(diǎn)代表分子中的一個原子,每條邊代表原子間形成的化學(xué)鍵,這種圖就叫做分子圖.分子圖中頂點(diǎn)數(shù)、邊數(shù)等都可以作為分子中的一些穩(wěn)定不變量.在實(shí)際應(yīng)用中,通常用不同的數(shù)值來描述分子的不同的可測量的物理化學(xué)性質(zhì),所以為了將分子的拓?fù)湫再|(zhì)與分子的可測量的物理化學(xué)性質(zhì)聯(lián)系起來,很有必要引入一些可以用數(shù)值表示的量,并且他們與分子圖中的某些性質(zhì)有關(guān),分子拓?fù)渲笜?biāo)在物理、化學(xué)和生物等許多學(xué)科有重要的用途.Wiener指標(biāo)是被廣泛研究的拓?fù)渲笜?biāo)之一,它不僅是與有機(jī)化合物的物理化學(xué)性質(zhì)關(guān)系緊密的早期拓?fù)渲笜?biāo),更是許多數(shù)學(xué)家和化學(xué)家的一個研究主題.Wiener指標(biāo)[25]是連通圖G中所有無序點(diǎn)對的距離和,即W(G)=∑{u,v}?VdG(u, v) =12∑(u,v)∈V×VdG(u,v),其中dG(u,v)表示G中u到v的距離.連通圖G的hyper-Wiener指標(biāo)定義為WW(G)=12W(G)+1在文獻(xiàn)[12,13]中,Gutman等人提出了乘積版本的Wiener指標(biāo)(也叫做π指標(biāo)):π(G)=Π{u,v}?V(G)dG(u,v).令Hn是所有n點(diǎn)Hamilton圖組成的集合.本文刻畫了Hn中具有第i小(1≤i≤n-2)Wiener,hyper-Wiener和π指標(biāo)的圖.更進(jìn)一步,我們也刻畫了Hn中具有最大,第二大,第三大Wiener,hyper-Wiener和π指標(biāo)的圖,并給出了相應(yīng)的指標(biāo)計(jì)算公式.
[Abstract]:In chemical theory, topological indices can be used to understand the physical and chemical properties of mixtures, and different indices reflect the different properties of molecules. The study of molecular topological indices and invariants of molecular graphs is one of the research fields in chemical graph theory. Each vertex of a simple undirected graph G = (V, E) represents an atom in the molecule, and each edge represents the chemical bonds formed between the atoms. This graph is called a molecular graph. The number of vertices and edges in the molecular graph can be regarded as some stable invariants in the molecule. In practical applications, different values are usually used to describe the different measurable physical and chemical properties of molecules, so in order to relate the topological properties of molecules to the measurable physical and chemical properties of molecules, It is necessary to introduce some quantities that can be expressed numerically, and they are related to some of the properties of the molecular graph, and the molecular topological indices are in physics, Wiener index is one of the topological indexes widely studied. It is not only an early topological index closely related to the physical and chemical properties of organic compounds, but also an early topological index which is closely related to the physical and chemical properties of organic compounds. Wiener index [25] is the distance sum of all disordered point pairs in connected graph G, that is, W (G) = 鈭，

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