The α-spectral Extremal Problem Of Graphs With No 4-cycle And 5-cycle

Spectral extremal graph theory mainly studies the various matrix representations of graphs,including the spectral properties of adjacency matrix,Laplacian matrix or signless Laplacian matrix,especially the extremal problem of the spectral radius of graphs without special substructure.In recent years,the extremal problems of adjacency spectrum,Laplacian spectrum and signless Laplacian spectrum of graphs have been widely studied.In order to track the change from adjacency matrix to signless Laplace matrix,Nikirofov proposed to study the spectral properties on a linear convex combination of adjacency matrix and degree diagonal matrix,that is,the matrix Aα(G)=αD(G)+(1-α)A(G),α∈E[0,1].At the same time,Nikiforov also proposed the following problem,that is,determining the extreme value of α-spectral radius in graphs without special substructure.In this thesis,the extremal problem of the α-spectral radius of graphs with no 4-cycle and 5-cycles is studied,and the following results are obtained:(1)If G is a graph of order n≥ 10 with no 4-cycle,and α∈[1/2,1),then λα(G)≤λa(Fn),with equality holding if and only if G=Fn.(2)Suppose that G is a graph of order n with no 5-cycle.If n>11/1-2α+4 andα∈[0,1/2),then λα(G)≤λα(T2(n)),with equality holding if and only if G=T2(n).(3)If G is a graph of order n≥ 12 with no 5-cycle,then there exists α0∈(1/2,1),such that for any α∈[1/2,α0),λα(G)≤λα(Sn,2),with equality holding if and only if G=Sn,2.These results not only partially answer the mentioned problem on the extreme value of α-spectral radius of graphs,but also generalize the existing results about adjacency spectrum and signless Laplacian spectrum of graphs.