Research On The Spectral Radii Of Weighted Graphs And Some Related Problems

This paper which stands on the basis of previous results, do further research on the spectral radii of weighted graphs and some related problems. The concrete content is in the following.? In the first two sections, we introduce the background and significance of the research, including the development of a representative at home and abroad regarding this aspect. Based on this research background and profound dis-cussion, by using deep-going analysis, it fully shows the main work's necessity and innovation.? In Section3, We use a simpler and original method to determine the extremal graph whose adjacent spectral radius is the largest in the weighted trees with q-matchings and a positive weight set, which has been gotten by Tan in [S.W. Tan, On the sharp upper bound of spectral radius of weighted trees, J. Math. Res. Exposition29(2009)293-301].? In Section4, We determine the extremal graph whose adjacent spectral radius is the largest in the weighted trees with fixed pendant vertices and a positive weight set. First we know the extremal graph is in the smaller kind of weighted trees with fixed pendant vertices and a positive weight set, then we determine the extremal graph by the graph whose adjacent spectral radius is the largest in weighted trees with fixed the number of vertices of degree2.? In Section5, We determine the graph whose adjacent spectral radius is the largest in the weighted unicyclic graphs with a special positive weight set. First, we get the structure of the extremal graph by some known information, and then we get the optimal distribution of nonnegative weights in the extremal graph. So we get the unique extremal graph.? In Section6, first we give a new upper bound on p(G) for an unweighted triangulation G, then we get some new bounds on q1(G) and q1(G)+q1(Gc) for an unweighted graph G. For these new bounds, we compare them with some known bounds, then we can give an account of their significance by some examples.