Sharp Lower Bound of the Least Eigenvalue of Graphs with Given Diameter

The eigenvalues of graphs play an important role in the fields of quantum chemistry,physics, computer science, communication network, and information science. Particularly,they can be interpreted in some situations as the energy levels of an electron in a molecule or as the possible frequencies of the tone of a vibrating membrane.The diameter of a graph,the maximum distance between any two vertices of a graph, has great impact on the service quality of communication networks. So we were motivated to investigate the sharp lower bound of the least eigenvalue of graphs with given diameter. Let n, d be the set of graphs on n vertices with diameter d. For any graph G∈n,d,by considering the least eigenvalue of its connected spanning bipartite subgraph, we obtained the sharp lower bound of the least eigenvalue of graph G. Furthermore,an upper bound of Laplacian spectral radius of graph G was given.

The eigenvalues of graphs play an important role in the fields of quantum chemistry, physics, computer science, communication network, and information science. Particularly, they can be interpreted in some situations as the energy levels of an electron in a molecule or as the possible frequencies of the tone of a vibrating membrane. The diameter of a graph, the maximum distance between any two vertices of a graph, has great impact on the service quality of communication networks. So we were motivated to investigate the sharp lower bound of the least eigenvalue of graphs with given diameter. Let gn. d be the set of graphs on n vertices with diameter d. For any graph G ∈ gn, d, by considering the least eigenvalue of its connected spanning bipartite subgraph, we obtained the sharp lower bound of the least eigenvalue of graph G. Furthermore, an upper bound of Laplacian spectral radius of graph G was given.