The Adjacency Spectral Radius Of Bicyclic Graphs With Given The Number Of Vertices And Maximal Degree

The theory of graph spectra is one of the important research fields of algebraic graph theory.It has a wide rang of application in quantum chemistry,computer science,communication network,physics and other disciplines.The study of the adjacency spectral of graphs is a very important content in the theory of graph spectra.The main contents of this paper are divided into two parts:1. we study the properties and structure of the maximal adjacency spectrum bicyclic graph in the bicyclic set with given the maximal degree and order;2. we study the question of estimating the upper bound on the adjacency spectral radius of bicyclic graphs with given the maximal degree and order.We get some results as follows:(1)Some properties about basic cycles、internal vertices and pendant vertices on the maximal adjacency spectrum bicyclic graphs in the bicyclic set with given the maximal degree △(△≥3) and order n( n ≥△+2) are given,meanwhile,the structure of the maximal adjacency spectrum bicyclic graph in this bicyclic set is described by these properties.(2)Two new upper bounds on the adjacency spectral radius of bicyclic graphs with given the maximal degree △(△ ≥3)and order n( n ≥△(10)2) are given.One is the best upper bound expressed by n and△ at present.The other is the best upper bound expressed only by△(△ ≥4).