Some Results On The Spectrum Of The Graph And Chromatic Number

In this paper, we mainly study "the spectrum of graph" of Algebraic Graph Theory. There are already many useful theories and techniques in this field. On the base of them, we determine the sharp upper bound of the spectral of a general graph and the lower and upper bounds of the spread of a unicyclic graph. Then, we study the bounds of the spectral of the graphs with k edge-cut. We also study the chromatic number of a 4-regular circulant graph.1. As the similar matrice have the same characteristic polynomial, we determine the sharp upper bound of 1 of a graph by this character.2 . Study the bounds of the spectral of the graphs with k edge-cut.3 . We discuss the bound of n of a unicyclic graph and use the result to study the spread of unicyclic graphs.4 . Analyze the structur of the vertice and edges of the 4-regular graphs and colour them to have the chromatic number.