The classification problem of Q-polynomial distance-regular graphs was proposed by E.Bannai in the late 1970s,which is one of the core problems in distance-regular graphs.The distance-regular graphs with classical parameters(d,b,α,β)have Q-polynomial structures.And the classification problem for b=1 has been completely solved.E.R.van Dam et al.proposed a subproblem of this classification problem of Q-polynomial distanceregular graphs in 2016:determine the distance-regular graphs with classical parameters(d,b,α,β)with b≠1.In this paper,we study this subproblem and the applications of Q-polynomial distance-regular graphs to magic labelings and metric dimensions.When d≥3,b is an integer but neither 0 nor-1.Therefore,we divide our discussion into two cases:b<-1 and b>1.The main results are as follows:For b<-1,there are four cases of the classification problem that have not yet been studied,namely:(ⅰ)d≥3,c=a1=1;(ⅱ)d≥3,c2=1,a2=a1>1;(ⅲ)d=3,c2=1,a2>a1>1;(ⅳ)d=3,c2>1,a1≥1.We give a complete classification of the cases(ⅰ)-(ⅲ)with the help of tools such as strongly closed subgraphs and regular near polygons.The classification of the case(ⅳ)contains four subcases,the first three of which are specific graphs,and the last subcase gives a specific expression for the classical parameters of the graph.To further investigate the existence of the graph corresponding to the last subcase,we use the Terwilliger algebra and its representation theory to give two necessary conditions for the existence of such graph.For b>1,we give the classification of distance-regular graphs satisfying α=0,and we also give some properties of parameters for distance-regular graphs with α≠ 0.For the applications of Q-polynomial distance-regular graphs,this paper mainly discusses the magic labelings of the folded n-cube with n>3,the magic labelings of the halved folded n-cube with even n≥8,as well as the metric dimension of the halved folded n-cube with even n≥10.The main results are as follows:For the magic labelings of the folded n-cube,we give some nonsingular matrix with the help of tools such as balance sets.And then,by using the corresponding bijection of the matrix and the eigenvalues of the graph,we obtain the necessary and sufficient condition for the graph to be a {1}-magic graph and a {0,1}-magic graph,respectively.For the magic labelings of the halved folded n-cube,we give a special rectangular matrix,and show the necessary and sufficient condition for the mapping corresponding to the matrix to be a bijection.Thus,with the help of tools such as balanced sets and the eigenvalues of the graph,we obtain the necessary and sufficient condition for the graph to be a {1}-magic graph and a {0,1}-magic graph,respectively.With respect to the metric dimension of the halved folded n-cube,we construct a subset of the set of vertices of the graph for n=10,n=4d and n=4d+2,where d≥3,respectively.Then we prove that the set is a resolving set or a minimal resolving set of the graph.Thus,we obtain an upper bound on the metric dimension.Finally,we compare our bounds with Babai’s bounds. |