本文選題：樹(shù) + 單峰性��； 參考：《江蘇師范大學(xué)》2017年碩士論文

【摘要】：圖的獨(dú)立多項(xiàng)式是代數(shù)圖論研究中的一個(gè)重要組成部分,對(duì)其單峰型性質(zhì)的研究是代數(shù)圖論中的一個(gè)熱點(diǎn)問(wèn)題.1987年,Erd?os等人猜想任意一棵樹(shù)或者森林的獨(dú)立多項(xiàng)式都是單峰的.這個(gè)猜想吸引了許多圖論研究者的興趣,雖然已取得了部分進(jìn)展,但是至今仍未解決.本文通過(guò)對(duì)獨(dú)立多項(xiàng)式進(jìn)行因式分解,將復(fù)合圖的獨(dú)立多項(xiàng)式分解為幾個(gè)多項(xiàng)式因子的乘積來(lái)研究,進(jìn)而證明復(fù)合圖的獨(dú)立多項(xiàng)式的對(duì)數(shù)凹性,從而為Erd?os等人的上述猜想,提供了更多的例子.具體內(nèi)容如下:第一章圖論的基本概念以及獨(dú)立多項(xiàng)式的已知結(jié)果.第二章本章仿照蜈蚣圖,定義了新的復(fù)合圖,取名為廣義蜈蚣圖.通過(guò)求解遞推關(guān)系式,給出了其獨(dú)立多項(xiàng)式的顯式表達(dá).進(jìn)一步,考慮了樹(shù)圖中三類(lèi)特殊的廣義蜈蚣圖,并證明了它們的獨(dú)立多項(xiàng)式都是對(duì)數(shù)凹的,從而是單峰的.第三章本章定義了一類(lèi)樹(shù),取名為雙燈樹(shù).利用某類(lèi)無(wú)爪圖,給出了雙燈樹(shù)的獨(dú)立多項(xiàng)式的表達(dá)式,并證明了其僅有實(shí)零點(diǎn),從而是對(duì)數(shù)凹和單峰的.第四章總結(jié)。
[Abstract]:The independent polynomial of a graph is an important part in the study of algebraic graph theory, and the study of its unimodal property is a hot issue in the theory of algebraic graph. In 1987, Erdfos et al assumed that the independent polynomial of any tree or forest is unimodal. This conjecture has attracted the interest of many graph theorists. Although some progress has been made, it has not been solved. In this paper, by factorizing independent polynomials, the independent polynomials of complex graphs are decomposed into the product of several polynomial factors, and the logarithmic concave of independent polynomials of composite graphs is proved, which is the conjecture of Erd?os et al. More examples are provided. The main contents are as follows: the first chapter is the basic concept of graph theory and the known results of independent polynomials. In the second chapter, a new compound graph is defined, named the generalized centipede graph, following the centipede graph. The explicit expression of its independent polynomial is given by solving the recursive relation. Furthermore, three special generalized centipede graphs in tree graphs are considered, and it is proved that their independent polynomials are logarithmic concave and thus unimodal. The third chapter defines a kind of tree, named double lamp tree. In this paper, we give the expression of independent polynomial of double lamp tree by using some kind of claw free graph, and prove that it has only real zero point, so it is logarithmic concave and unimodal. Chapter IV Summary.
【學(xué)位授予單位】：江蘇師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O157.5

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