The Research Of The Problem Of Signless Laplacian Spectral Radius

Spectral graph theory is a very active and important field of graph theory.It is mainly studied by using the methods and techniques in matrix theory and algebraic theory.Spectral extremal graph theory is an important research direction of spectral graph theory.It mainly studies the properties of the spectrum of the adjacency matrix,Laplacian matrix,and signless Laplacian matrix of a graph,especially the eigenvalues or the corresponding eigenvectors of the graph that does not contain a specific structure.In this paper,let H_k be a graph consisting of quadrangles which intersect in exactly a common vertex,which is called an intersecting quadrangle.Let B_k be a graph consisting of k triangles which share one common edge,whcih is called a book.These two graphs H_k and B_k both contain C₄ as a subgraph.In 2013,Freitas,Nikiforov and Patuzzi（[1]（Abreu et al.,2013））proposed the following problem:given a graph H,if a graph G on n vertices does not contain H as a subgraph,what is the maximum signless Laplacian spectral radius of G?In the same year,Freitas,Nikiforov and Patuzzi（[13]（Freitas et al.,2013））proved the maximum signless Laplacian spectral radius of a graph that does not contain C₄ as a subgraph and characterized all the extremal graphs.As an extension,in this paper,we give the maximum signless Laplacian spectral radius of all graphs not containing B₂ as a subgraph and characterized all the extremal graphs.In addition,we also give the upper bound of the signless Laplacian spectral radius of a graph not containing H_k as subgraph.Finally,we made a summary and outlook,and put forward two conjectures that can be further studied.