Explore The Arc Transitive Of Kneser Graph And Automorphism Group Of Switching Graph

In graph theory algebraic properties of graph concern in recent years, because the combination of traditional methods, the use of algebraic methods to solve a problem in graph theory unparalleled simplicity and rigor. In particular the past three decades, with the fast developing of the computer technology,this subject is being widely studied by most people home or abroad. Except for the traditional spectral theory,the symmetric of the graph and the using for net theory, code theory are being studied. In the paper, some algebraic properties of two kinds of particular graph are studied. There are two parts in this paper:In the first part, the s -arc transitive problem of odd graph K_2r+1,r is investigated. Godsil and Royle proved that each odd graph K_2r+1,r was 2-arc transitive at least, then it yield the problem: what is the maximal s for the K_2r+1,r? We obtain that the maximal s is three, where r≥2, and the necessary and sufficient condition of 3-arc regular by analyzing the structure of arc for K_2r+1,r.In the second part, we studied the automorphisms, vertex-transitive and spectrum of switching graph, and the Cayley graph is invariant on the switched action by the algebraic method.