Minimizing the least eigenvalue of bicyclic graphs with fixed diameter

Let B(n, d) be the set of bicyclic graphs with both n vertices and diameter d, and let theta* consist of three paths u(0)w(1)v(0), v(0)w(2)v(0) and u(0)w(3)v(0). For four nonnegative integers n, d, k, j satisfying n >= d + 3, d = k + j + 2, we let B(n,d;k, j) denote the bicyclic graph obtained from theta* by attaching a path of length k to u(0), attaching a path of length j to vertex v(0) and n - d - 3 pedant edges to v(0), and let B(n, d; k, j) = {B(n,d; k, j)vertical bar k + j >= 1}. In this paper, the extremal graphs with the minimal least eigenvalue among all graphs in B(n, d; k, j) are well characterized, some structural characterizations about the extremal graphs with the minimal least eigenvalue among all graphs in B(n, d) are presented as well.