Minimum higher eigenvalues of Laplacians on graphs (Q1918368)

If \(G\) is an undirected graph, let \(A\) be the adjacency matrix of \(G\), \(D\) be the diagonal matrix whose \((v,v)\) entry is the degree of \(v\), and \(\Delta=D-A\) be the Laplacian of \(G\). If \(G\) is a connected graph, it is known that the eigenvalues of \(\Delta\) satisfy the condition \(0=\mu_1<\mu_2\leq\cdots\leq \mu_n\). The author finds the smallest possible \(i\)th eigenvalue \(\mu_i\) of \(\Delta\) in the class of all connected graphs on \(n\) vertices. He gives examples of graphs which achieve this value, and determines when the graph achieving this value is unique.