Characterizing Generalized Line Graphs By Star Complements

Let G be a connected graph of order n and μ an eigenvalue of G with multiplicity m.If μ isn’t an eigenvalue of some induce subgraph H of G of order n-m, then H is called a star complement of G about eigenvalue μ.The main aim of this dissertation is to discuss some properties of general-ized line graphs L(H) with the term of star complement. It consists of three chapters, and the arrangement is as follows.In Chapter one, we mainly introduce the background of our research and state our main results.In Chapter two, we investigate the maximal graphs having the cycle and isolated vertex as a star complement for the eigenvalue-2, and we obtain following results:(1)Suppose that t is an odd integer not less than1, and s is a nonnegative integer, then the generalized line graph L(H)=L(Kt+s;0,…,0,1,…,1)(the numbers of0and1are t and s, respectively) is the unique maximal graph having H=Ct+2sK1as a star complement for the eigenvalue-2.(2)Suppose that t is an even integer greater than8, and s is a nonneg-ative integer, and t=r+y, where r and y are odd integers greater than1, then L(H)=L(Kt+s;0,…,0,1,…,1)(the numbers of0and1are t and s, respectively) is the unique maximal graph having H=Cr UCy+2sK1as a star complement for the eigenvalue-2.In Chapter three, we investigate the maximal graphs having the complete graphs by removing a edge as a star complement for the eigenvalue-2, and we obtain following results:(1)Suppose that n is an integer greater than6, and r, z are nonnegative integers, then the generalized line graphs L(H)=L(Kr+1;0,…,0,z+1)(the number of0is r and r+z=n, z≠n-1) are n maximal graphs having H=Kn+2-e as star complement for the eigenvalue-2. (2)Suppose that n is an integer greater than6,t is an odd integer not less than3,and r,z are nonnegative integers,then the generalized line graphs L(H)=L(Kt+r+l;O,…,O:z+1)(the number of the O is t+r and r+z:n, z≠n-1)are n maximal graphs having H=Gt+Kn+2-e as star complement for the eigenvalue-2.