本文關(guān)鍵詞： 電阻距離 Wiener指標(biāo) Krichhoff指標(biāo) 單圈圖 Nordhaus-Gaddum型結(jié)果　出處：《煙臺(tái)大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：設(shè)G是一個(gè)連通圖,其頂點(diǎn)集合為V(G).對(duì)G中任意兩個(gè)頂點(diǎn)i和j,i和j之間的距離定義為連接這兩個(gè)頂點(diǎn)之間的最短路的長(zhǎng)度,而i和j之間得電阻距離定義為用單位電阻代替G中的每條邊后所得的電網(wǎng)絡(luò)中這兩個(gè)節(jié)點(diǎn)之間的等效電阻.圖G的Wiener指標(biāo),記作W(G),定義為G中所有頂點(diǎn)之間的距離的和.圖G的Kirchhoff指標(biāo),記作Kf(G),定義為G中所有頂點(diǎn)之間的電阻距離之和.圖的Wiener指標(biāo)和Kirchhoff指標(biāo)是圖的重要不變量,并且在化學(xué)里QSAR和QSPR的研究中具有廣泛的應(yīng)用.本文主要研究圖的Wiener指標(biāo)和Kirchhoff指標(biāo),具體研究?jī)?nèi)容如下所述.首先,我們刻畫具有第四大和第四小Wiener指標(biāo)的單圈圖.只含有一個(gè)圈的圖稱作單圈圖.在之前的研究中,具有前三大和前三小Wiener指標(biāo)的單圈圖已經(jīng)刻畫清楚.沿著這個(gè)方向,我們進(jìn)一步刻畫具有第四大和第四小Wiener指標(biāo)的單圈圖.本文證明了在所有頂點(diǎn)數(shù)≥8的單圈圖中,C_5(S_(n-4)和C_2~(u1,u2)(S_3,S_(n-4))具有第四小的Wiener指標(biāo),而C_3~(u1,u2)(P_3,P_(n-4))具有第四大的Wiener指標(biāo).其次,我們完全解決了關(guān)于Kirchhoff指標(biāo)的Nordhaus-Gaddum型結(jié)果的一個(gè)猜想.在2011年,Yang,Zhang和Klein[Y.Yang,H.Zhang,D.J.Klein,New Nordhaus-Gaddum-type results for the Kirchhoff index,J.Math Chem.49(2011)1587-1598]提出了關(guān)于Kirchhoff指標(biāo)的NordhausGaddum型結(jié)果的一個(gè)猜想.他們猜想一個(gè)圖G及其補(bǔ)圖(?)的Kirchhoff指標(biāo)的和達(dá)到最大當(dāng)且僅當(dāng)G是路P_n或者其補(bǔ)圖(?).在文本中,應(yīng)用圖論和電網(wǎng)絡(luò)的方法和技巧,我們完全解決了該猜想.
[Abstract]:Let G be a connected graph, and the vertex set of G is VG. The distance between any two vertices I, J and j in G is defined as the length of the shortest path between the two vertices. And the resistance distance between I and j is defined as the equivalent resistance between the two nodes in the electric network after replacing each edge of G with the unit resistor. The Wiener index of figure G. The Kirchhoff index of a graph G is defined as the sum of the distances between all vertices in G. it is defined as the sum of resistance distances between all vertices in G. the Wiener index and the Kirchhoff index of a graph are important invariants of a graph. And it is widely used in the study of QSAR and QSPR in chemistry. This paper mainly studies the Wiener index and Kirchhoff index of the graph, the specific research contents are as follows. We characterize unicyclic graphs with 4th and 4th small Wiener indices. Graphs with only one cycle are called unicyclic graphs. In previous studies, unicyclic graphs with the first three largest and the first three small Wiener indices have been clearly characterized. We further characterize unicyclic graphs with 4th large and 4th small Wiener indexes. In this paper, we prove that in all unicycle graphs with vertex number 鈮，

本文編號(hào)：1504915