The Partially Distance-regular Graphs With Bounded Local Eigenvalues

In this thesis,we classify the 2-partially distance-regular graphs such that all of its second largest local eigenvalues are at most 1.Also we will use this result to classify the 2-walk-regular graphs with smallest eigenvalue-(b1/2)-1.These extend the result of Koolen and Yu about the classification of distance-regular graph such that all of its second largest local eigenvalues are at most 1,and the result of Koolen,who classified the distance-regular graphs with smallest eigenvalue-(b1/2)-1.Let ? be a 2-partially distance-regular graphs such that all of its second largest local eigenvalues are at most one.The basic idea of our research is to find out all the possible local graphs and ?-graphs of ?,then we will bound the intersection numbers c2 and a2 by studying the structures of these local graphs and ?-graphs.At last,we give the nonexistence of some ? with the bounded intersection numbers.