Some Results On The Upper Bounds For The Adjacent Specral Radius Of Graphs

The estimate of the bound of the radius is a hot topic in spectral theory of graphs. There are already many theories and techniques in this field. In this paper, we mainly study the spectrum of simple connected graphs and the the double cyclic graph, achieving some new results. On the other hand, we investigate the special type of graph k tree. Some results will be given in this thesis.1 . By using the similar transfer of adjacent matrix, we determine the sharp upper bound of λ₁ of a graph:Morever, equality holds if and only if G ≌ G₁â–½ G₂, where G₁ is a p-regular graph with i - 1 vertices, G₂ is a (â–³₂ -q)-regular graph with n - i + 1 vertices.2 . we study the tree graphs T_G of the double cyclic graph. Morever, we get the bounds on the spectral radius of tree graphs T_G:where l are co-edges of the two basic cycles of G(n,n + 1).3 . we characterize the k trees that maximize the second and the third largest eigenvalues.