论文部分内容阅读
This thesis consists of two parts.In the first part,we study distance-regular graphs with diameter 3 and eigenvalue a2-c3.We consider the distance-regular graphs ’ whose distance-2 graphs Γ2 are strongly regular.We note that if Γ is bipartite,then its distance-2 graph is not connected.So,we are interested in the class of non-bipartite distance-regular graphs.This extends a paper by Brouwer[23]in 1984.He studied distance-regular graphs with diameter 3 such that their distance-2 graphs are strongly regular graphs.For example,he showed that for an antipodal non-bipartite distance-regular graph ’ with diameter 3,its distance-2 graph Γ2 is strongly regular if,and only if,the graph Γ has intersection array{st,s（t-1）,1;1,t-1,St｝for some integers s,t>2.Note that a strongly regular graph with parameter set（（St+1）（s+1）,s（t+1）,s-1,t+1）for positive integers s,t,is called pseudo-geometric.We will extend this result.Also,we will give several kinds of classifications of non-bipartite distance-regular graphs with diameter 3 and eigenvalue a2-C3 under various conditions,for example,the valency k is at most 2（a1+1）,c3≤9,a2≤7 and c≤26.In the second part,we will show that there does not exist a distance-regular graph Γwith diameter 3 and intersection array{80,54,12;1,6,60}.In order to do so,we will first study the claw-bound that was introduced by Bose in the case of pseudo geometric strongly regular graphs[18].We note that a similar bound for amply-regular graphs with an s-claw was shown by Godsil[44]and that Koolen and Park[64]also studied the claw-bound for distance-regular graphs and they mentioned that three known examples of Terwilliger graphs with c2≥2 meet equality in the claw-bound.After that,Gavrilyuk[39]showed that if an amply-regular Terwilliger graph with c≥ 2 attains equality in the claw-bound,then the graph is one of the known Terwilliger graphs.Next,we will show that a local graph △ of,does not contain a coclique with 5 vertices,and then we prove that the graph ’ is geometric by showing that △ consists of 4 disjoint cliques with 20 vertices.Then we apply a result of Koolen and Bang to the graph Γ,and we obtain that there is no such a distance-regular graph.