A Sharp Lower Bound of the Spectral Radius of Simple Graphs

Let G be a simple connected graph with vertex set V = {1, 2, . . . , n}. Let d(i, j) denote the distance between vertices i and j. For i ∈ V , the degree of i and the average of the degree of the vertices adjacent to i are denoted by di and mi, respectively. The 2-degree of vertex i is denoted by ti, which is the sum of degrees of the vertices adjacent to i, that is ti = midi. Let Ni be the sum of the 2-degree of vertices adjacent to i. Let A(G) be the adjacency matrix of G. By the Perron-Frobenius theorem [1, 2], the spectral radius ρ(G) is simple and there is a unique positive unit eigenvector. Since A(G) is a real symmetric matrix, its eigenvalues must be real, and may ordered as λ1(G) ≥ λ2(G) ≥ · · · ≥ λn(G). The sequence of n eigenvalues is called the spectrum of G, the largest eigenvalue λ1(G) is often called the spectral radius of G, denoted by ρ(G) = λ1(G). In this paper, we give a sharp lower bound on the spectral radius of simple graphs. For some recent surveys of the known results about this problem and related topics, we refer the reader to [3, 4, 7] and references therein.