The Upper And Lower Bounds On The Adjacency Spectral Radius Of Graphs

The set of all the eigenvalues of adjacency matrix are called the spectrum of graph, sothe spectrum of graph are determined by the given graph. In resent years, researching therelations between the distribution of spectrum and the structures of graphs is an activeresearch field. In the paper, we focus to research some characters of graphs by spectrumof graph. But for most graphs, we can't compute the spectrum directly. So the estimateson the eigenvalues of graphs is an important respect in spectrum of graph, especially forthe bounds on the spectral radius of graphs.Nowadays, there are many results for the bounds on the spectral radius of graphs.The bounds presented by the number of vertices were given (see, for instance, [6]), thebounds presented by the degree of vertices (see, [5]), the bound presented by the numberof edges (see, [7], [8]), the bound presented by the maximum degree,minimum degree ofvertices (see, [9], [11], [12], [15], [16], [17]).With the developing of spectrum theory, there are bounds presented by the 2-degree[13]and average 2-degree were given (see, for instance, [10], [12], [14]), presented by largerneighbor of vertices (see, [18]). But for the bound presented by average k-degree (see thedefinition 6 of section 2 in chapter 1) of the vertices, there are few results on this. Thus,in the chapter 2 of the paper, we give the upper and lower bounds for spectral radius ofgraphs presented by average k-degree of the vertices.With the developing of spectrum theory, the graph develop from undirected graphto directed graph (short for digraph), from unweighted graph to weighted graph. Forthe digraph, There are many results for the bounds on the spectral radius. The boundspresented by the out-degree and in-degree of vertices were given (see, for instance, [19],[20]), presented by the 2-outdegree or average 2-outdegree of vertices (see, [4], [21]). Forthe spectral radius of weighted graph (see [22]-[28]). But, the bound on the generalizedweighted digraph (see the definition 8 of section 2 in chapter 1)are began to research.Thus, in the chapter 3 of the paper, we give a upper bound for the spectral radius ofgeneralized weighted digraph.In the paper, in chapter 1, introduction, we present the background of the spectrumtheory and give the basic definitions, symbols and notations about graph spectra. We alsogive the definitions of weighted graph and digraph. We also give some new definitions, forexample, k-degree, average k-degree, generalized weighted digraph. List some importantknown results about the spectral radial of adjacency spectra and the spectral radial of weighted graphs and the spectral radial of digraphs, in the section 4, we list the importantresults of the paper.In chapter 2, we give the sharp bounds for spectral radius of graph presented byaverage k-degree of the vertices (see theorem 2.2.1, 2.2.2). From which, we can get someknown results (see [5], [10], [12]) and our result is better than others usually by someexamples.In chapter 3, we give a sharp upper bound for the spectral radius of generalizedweighted digraph (see Theorem 3.2.1), which generalizes some other results on the spectralradius of digraphs and weighted graphs (see [5], [19], [23]).