On Spectral Radius Of Adjacency Matrix Of Weighted Bicyclic Graphs And Distance (laplacian) Matrix Of Graphs (Digraphs)

In recent years spectral theory is being rapidly developed and extremely important areaof research topic in graph theory. Based on the previous work we study some problemsabout the spectral radius of weighted bicyclic graph, the distance signless laplacian spec-tral radius of strongly connected digraphs and the distance signless laplacian spectral radiusand distance laplacian spectral radius of connected graphs. In this paper we first introducethe research background of spectral theory and the results attained, and then dertemine theeigenvalues of weighted graphs and the spectrum of distance matrix, distance signless Lapla-cian matrix and distance Laplacian matrix of graphs. Finially we introduce our main resultsin the next four sections. Main outcome is as follows:Firstly, let BWn,n+1denote the set of bicyclic weighted graphs of order n with the weightset W. In Section II, we determine the structure and some weight distribution of theweighted bicyclic graph with the largest spectral radius in BWn,n+1with a fixed weight setW={w1, w2,···, wn+1}, where w1≥w2≥···≥wn+1>0;Secondly, in Section III, we first give sharp upper and lower bounds for the distancesignless laplacian spectral radius of strongly connected digraphs; we then characterize thedigraphs having the maximal and minimal distance signless laplacian spectral radii amongall strongly connected digraphs; we also determine the extremal digraph with the minimaldistance signless laplacian spectral radius with given number;Thirdly, in Section IV, we first give bounds for the distance signless laplacian spectralradius of connected graphs; we then determine the extremal graph with the minimal distancesignless laplacian spectral radius with given chromatic number;Fourthly, in Section V, we give some upper bounds for the distance laplacian spectralradius of connected graphs.