Degrees And The Bound On The Spectral Radius Of Graphs

The set of all the eigenvalues is called the spectrum of graphs. For most graphs, we can't comput their eigenvalues directly, so the estimate on the eigenvalues of graphs is an active question for discussion. Nowadays, there are many results about the question, especially for the bound on the spectral radius of graphs. But for the low bound on the spectral radius, there are few about it. In this paper, we obtain that(1) the low bound depending on the k-degree sequence:Let G be a connected graph with k-degree sequence (dk(1),...,dk(n)). Then with equality if and only if G is pseudoregular or strictly psedosemiregular, and Finally we make apply of the result, and obtain an upper bound of the energyThat is the main part of this paper and has published in《MATCH Commun. Math. Comput. Chem.》.(2)the bound on the spectral radius of especial graphs—unicyclic graphs.Firstly, we give the concept of an especial unicyclic graph. Then we obtain some results about the eigenvalues of the especial unicyclic graph. By using these results, an upper bound on the spectral radius of unicyclic graphs is obtained.