Graphs and Hermitian matrices: Exact interlacing

We prove conditions for equality between the extreme eigenvalues of a matrix and its quotient: In particular, we give a lower bound on the largest singular value of a matrix and generalize a result of Finck and Grohmann about the largest eigenvalue of a graph. Keywords: extreme eigenvalues, tight interlacing, graph Laplacian, singular values, nonnegative matrix 1 Introduction Our notation is standard (e.g., see [1], [3], and [6]); in particular, all graphs are de…ned on the vertex set [n] = f1; : : : ; ng and G (n) stands for a graph of order n. Given a graph G = G (n) ; 1 (G) ::: n (G) are the eigenvalues of its adjacency matrix A (G), and 0 = 1 (G) ::: n (G) are the eigenvalues of its Laplacian L (G). If X; Y V (G) are disjoint sets, we write G [X] for the graph induced by X; and G [X; Y ] for the bipartite graph induced by X and Y ; we set e (X) = e (G [X]) and e (X; Y ) = e (G [X;Y ]). We assume that partitions consist of nonempty sets. In this note we study conditions for …nding exact eigenvalues using interlacing. As proved in [2], if G = G (n) and [n] = [i=1Pi is a partition, then 1 (G) + : : :+ k (G) k X i=1 2e (Pi) jPij ; (1) n k+2 (G) + : : :+ n (G) k X i=1 2e (Pi) jPij 2e (G) n ; (2) Department of Mathematical Sciences, University of Memphis, Memphis TN 38152, USA yDepartment of Pure Mathematics & Mathematical Statistics University of Cambridge Centre for Mathematical Sciences Wilberforce Road Cambridge CB3 0WB zResearch supported in part by NSF grants CCR-0225610, DMS-0505550 and W911NF-06-1-0076.