The Study Of Some Problems On The Adjacency Spectral Radius Of Unicyclic Graphs

The theory of graph spectra is one of the important research directions in algebraic graph theory, and it mainly studies the spectral of the graphs, the characterization of the graphs and the relationships between the spectra and the invariants of graphs. The theory of graph spectra has been widely applied in communication networks, computer science and quantum chemistry. The study of the adjacency spectral radius of unicyclic graphs is one of the most attractive research subjects in the theory of graph spectra. Therefore,the study of some problems on the adjacency spectral radius of unicyclic graphs has important theoretical and practical significance.The main results of this paper are listed as follows:Firstly, we give some properties of the maximal adjacency spectrum unicyclic graphs in the set of unicyclic graphs with given the number of the vertices n, the maximal degree △(△ ≥3) and the girth l(l ≥4).Secondly, we obtain a new upper bound on the adjacency spectral radius in the set of unicyclic graphs with given the number of the vertices n, the maximal degree △(△ ≥3) and the girth l(l ≥4).Thirdly, we give some properties and possible structures of the maximal adjacency spectrum unicyclic bipartite graphs in the set of unicyclic graphs with given the number of the vertices n and the maximal degree △(△ ≥3).Finally, we obtain a new upper bound on the adjacency spectral radius in the set of unicyclic bipartite graphs with given the number of the vertices n and the maximal degree △(△ ≥3). When n and △ meet the condition of n ≤3Δ-2, the upper bound given in this paper is better than the results obtained in the current related literature. When n and △ meet the condition of n >3Δ-2, the upper bound given in this paper is also better than the results obtained in the current related literature in some cases.