The Research On The Laplacian Spectral Radius Of Unicyclic Graphs

The theory of graph spectra has a wide range of applications in physics, chemistry, biology, computer networks and other subjects. A lot of scientists have launched a number of studies about it in recent years. In this theory, extremal graphs and estimating the upper bounds on the Laplacian spectral radius of unicyclic graph are very important research contents. Hence, the study of problem on the Laplacian spectral radius of unicyclic graph has certain theoretical and practical significance.In this paper, we study the problem on the Laplacian spectral radius of unicyclic graphs, and get some new results.Firstly, we study some properties of graphs with maximal Laplacian spectral radius in the set of unicyclic bipartite graphs with the maximal degree Δ and order n. The girth of graphs with maximal Laplacian spectral radius in the set of unicyclic bipartite graphs with the maximal degree Δ and order n is obtained.Secondly, we study some properties of graphs with maximal Laplacian spectral radius in the set of unicyclic graphs with the maximal degree Δ and order n. We obtain some necessary conditions on the graphs with maximal Laplacian spectral radius in the set of unicyclic graphs with the maximal degree Δ and order n. Based on the results we discussed, we completely characterize the structures of graphs with maximal Laplacian spectral radius in the set of the unicyclic graphs with the maximal degree Δ and order n for n ≤3Δ-2, meanwhile, we give the three possible structures of graphs with maximal Laplacian spectral radius in the set of the unicyclic graphs with the maximal degree Δ and order n for n >3Δ-2.Finally, we study the problem of estimating the upper bounds on the Laplacian spectral radius of unicyclic graphs in the set of unicyclic graphs with the maximal degree Δ and order n. A new upper bound of the Laplacian spectral radius of the unicyclic graphs with the maximal degree Δ and order n is given, and this new upper bound is expressed only by two parameters with the number of vertices n and maximum degree Δ.