Prime-level Factorial Designs And Several Families Of M-Cayley Graphs

Experimental design has rapidly developed to be one of the most important branches of mathematical statistics ever since R.A.Fisher established the modern discipline,and it has been successfully applied to many areas of scientific investigation,including agriculture,medicine,industry,biology,economics and so on.Row-column designs which provide unconfounded estimation of all main effects and the maximum number of two-factor interactions(2fi’s)are called 2fi-optimal.Godolphin(2019)constructed 2fi-optimal 2-level full factorial row-column designs;Zhou and Zhou(2023)proposed a method for constructing 2fi-optimal 3-level full factorial row-column designs.However,constructions of 2fi-optimal fractional factorial row-column designs for any prime levels have not been investigated.In this work,our first result solves this problem.Specifically,we present theoretical constructions of 2fi-optimal sn full factorial row-column designs for any odd prime level s and any parameter combination,and theoretical constructions of 2fi-optimal sn-1 and sn-2 fractional factorial row-column designs for any prime level s and any parameter combination.The existence problem and constructions of symmetric BIBDs are fundamental research areas in design theory.It is notoriously difficult to construct a new family of symmetric BIBDs.There are many unsolved conjectures about symmetric BIBDs,and one important conjecture is that a BIBD(n2+n+1,n+1,1)exists if and only if n is a prime power.Finding all possible orders of Hadamard designs is another difficult problem.We already know that for a symmetric BIBD(v,k,λ),the parameter v satisfies 4n-1 ≤v≤n2+n+1,where n=k-λ is the order of the BIBD.The upper bound of v corresponds to BIBD(n2+n+1,n+1,1),the projective plane of order n,and the lower bound of v corresponds to BIBD(4n1,2n-1,n-1),the Hadamard design of order n.Thus we can see that any symmetric BIBDs are "bounded by" the projective planes and the Hadamard designs.On one hand,the incidence graph of the classical projective plane of order q is a Cayley graph on the dihedral group D2(q2+q+1)and a bi-Cayley graph on the cyclic group Zq2+q+1.On the other hand,if n=(q+1)/4,where q≡3(mod 4)and q is a prime power,then the incidence graph of the Hadamard design of order n is a bi-Cayley graph on an elementary abelian group.In summary,the incidence graphs of symmetric BIBDs are "bounded by" two bi-Cayley graphs,one of which is a Cayley graph as well.Then a natural question is whether we can find a new family of symmetric BIBDs by using classification results of Cayley and bi-Cayley graphs.As the first step towards this problem,we investigate some Cayley graphs and bi-Cayley graphs on some important group families.In Chapter 4 we classify Cayley graphs on solvable groups(solvable Cayley graphs in short),which is a core object in the study of Cayley graphs.In the past 30 years,arc-transitive Cayley graphs on some specific groups(cyclic groups,dihedral groups and abelian groups)have been well characterized.But it was not until recently that Li and Xia(2022)and Zhou(2021)made a breakthrough for arc-transitive solvable Cayley graphs.They proved that nonbipartite s-arc-transitve solvable Cayley graphs are at most 2-arc-transitive(the sharp upper bound on s is 2).But they did not solve the bipartite case.Our second main result in this thesis answers this question:we characterize s-arc-transitve(s≥ 3)solvable Cayley graphs.In particular,we prove the sharp upper bound on s is 4 except cycles,and we point out that the incidence graph of the classical projective plane or order q is a 4-arc-transitive Cayley graph on the dihedral group D2(q2+q+1).Furthermore,we study Cayley graphs and bi-Cayley graphs on nonabelian simple groups:we characterize 2-arc-transitive hexavalent Cayley graphs;we completely determine symmetric and semisymmetric cubic bi-Cayley graphs,and give the structure of their full automorphism groups,and thus complete the study of cubic edge-transitive bi-Cayley graphs on nonabelian simple groups;we generalize some results about cubic symmetric bi-Cayley graphs to any prime-valent case.We will investigate the graphs in the classification results of Chapter 4 and Chapter 5 in a sequel,and further determine which Cayley graph and bi-Cayley graph is the incidence graph of a symmetric BIBD,with the aim of finding a new family of symmetric BIBDs.