本文選題：有向強正則圖　切入點：結(jié)合方案　出處：《河北師范大學(xué)》2016年博士論文　論文類型：學(xué)位論文

【摘要】：距離正則圖是代數(shù)組合論研究的主要對象之一,它對應(yīng)于一類特殊的結(jié)合方案,與有限幾何、組合設(shè)計以及編碼理論等有著密不可分的聯(lián)系.強正則圖作為直徑為2的距離正則圖,更是學(xué)者們研究的一個熱點Duval于1988年將強正則圖概念推廣到了有向強正則圖.2013年,Dam和Omidi引進(jìn)了強步正則圖這一概念,2015年他們又將此研究推廣到了有向強步正則圖,這使得強正則圖理論的研究更加深入,豐富.本文主要基于有限幾何構(gòu)作了四類有向強正則圖和一類結(jié)合方案,此結(jié)合方案中三個關(guān)系圖的并是所構(gòu)作的第二類有向強正則圖的補圖.本文計算了有向強正則圖的參數(shù)與結(jié)合方案的交叉數(shù),并得到第一類圖的全自同構(gòu)群和該結(jié)合方案的自同構(gòu)群.首先,我們利用有限域上秩為1的矩陣構(gòu)作了有向強正則圖Ⅰ.對于q元有限域Fq上任意n級方陣X,令[X]={tX|t∈Fq*],且我們用X代替[X].定義圖T(n,q)是以{X|X∈Fq(n×n),r(X)=1,X2=0}為頂點集的有向圖,對于兩個頂點A,B,A→B當(dāng)且僅當(dāng)AB≠0.類似地,定義圖T'(n,q)是以{X|X∈Fq(n×n),r(X)=1,X2≠0}為頂點集的有向圖,鄰接關(guān)系定義為A→B當(dāng)且僅當(dāng)AB=0.則T(n,q)和T'(n,q)均為有向強正則圖,我們分別確定了它們的參數(shù)和全自同構(gòu)群.其次,我們以F。上n-維射影空間中全體s-flats作為點集P,全體m-flats作為區(qū)組集召,其中0smn,建立一個關(guān)聯(lián)結(jié)構(gòu)T(s,m;n,q)=(P,B,∈),點與區(qū)組的關(guān)聯(lián)關(guān)系定義為flats之間的關(guān)聯(lián)關(guān)系.我們利用關(guān)聯(lián)結(jié)構(gòu)T(s,m;n,q)定義圖Ⅱ.圖T(s,m;n,q)是以{(X,Y)∈P×B|X ∈Y}為頂點集的有向圖,鄰接關(guān)系定義為(X1,Y1)→(X2,Y2)(?)(X1,Y1)≠(X2,Y2),X1 ∈Y2對于任意一個(s-1)-flat K,定義圖T(s,m;n,q,K)為T(s,m;n,q)中由{(X,Y)∈VT(s,m;,q)|K(?)X}誘導(dǎo)所得到的子圖.則T(s,m;n,q)是一個11/2-設(shè)計當(dāng)且僅當(dāng)s=0或m=n-1.T(0,m;n,q), T(s,n-1;n,q)和T(s,m;n,q,K)均為有向強正則圖,且T(s,m;n,q,K)的參數(shù)與K的選取無關(guān).然后,我們利用跡函數(shù)和1-維子空間分別構(gòu)作了有向強正則圖Ⅲ和Ⅳ.最后,我們利用有限域上n-維向量空間中1-維子空間與2-維子空間的有序?qū)Φ玫搅艘活惤Y(jié)合方案.令Mm為Fq上n-維向量空間中全體m-維子空間構(gòu)成的集合,其中n4.設(shè)X={(A,B)|A ∈M1,B ∈M2,(?)B}.Fq上n階一般線性群GLn(Fq)在X上的作用是可遷的,自然地誘導(dǎo)了一個類數(shù)為6的結(jié)合方案(?).我們計算了(?)的交叉數(shù),并確定了其自同構(gòu)群.
[Abstract]:Distance regular graph is one of the main objects in the study of algebraic combinatorial theory. It corresponds to a class of special associative schemes and finite geometry. Combinatorial design and coding theory are closely related. Strongly regular graphs are regarded as distance canonical graphs with a diameter of 2. In 1988, Duval extended the concept of strongly regular graphs to strongly regular graphs. In 2013, the concept of strongly regular graphs was introduced by Omidi and Dam. In 2015, they extended this study to strongly regular graphs. In this paper, four kinds of strongly regular graphs and a class of associative schemes are constructed based on finite geometry. The complements of three relational graphs of the second kind of strongly regular graphs are constructed. In this paper, the cross numbers of the parameters of the strongly regular graphs and the associative schemes are calculated. The total automorphism group of the graph of the first kind and the automorphism group of the associative scheme are obtained. In this paper, we construct a strongly regular graph by using a matrix with rank 1 over a finite field 鈪，

本文編號：1566044