David A. Gregory

The spread of the spectrum of a graph

Upper and lower bounds are obtained for the spread λ1 − λn of the eigenvalues λ1 λ2 ··· λn of the adjacency matrix of a simple graph.

Algebraic multiplicity of the eigenvalues of a tournament matrix

Let F” denote the set of irreducible n X n tournament matrices. Here arc our main results: (1) For all n > 3, every matrix in K has at least three distinct eigenvalues; such a matrix has exactly three distinct eigenvalues if and only if it is a Hadamard tournament matrix. (2) For all n 2 3 there is a matrix in Y” having n distinct eigenvalues. (3) If cr, denotes the maximum algebraic multiplicity of 0 as an eigenvalue of the matrices in z, then ln /21-2 6. (4) If r,, is the minimum Perron value (i.e. spectral radius) of all matrices in r”, then 2 8.

Primes in the semigroup of Boolean matrices

Abstract Let A , B , C be n × n matrices of zeros and ones. Using Boolean addition and multiplication, we say that A is prime if it is not a permutation matrix and if A = BC implies that B or C must be a permutation matrix. Conditions sufficient for a matrix to be prime are provided, and a characterization of primes in terms of a nation of rank is given. Finally, an order property of primes is used to obtain a result on prime factors.

Matchings in regular graphs from eigenvalues

Let G be a connected k-regular graph of order n. We find a best upper bound (in terms of k) on the third largest eigenvalue that is sufficient to guarantee that G has a perfect matching when n is even, and a matching of order n-1 when n is odd. We also examine how other eigenvalues affect the size of matchings in G.

PRINCIPAL EIGENVECTORS OF IRREGULAR GRAPHS

Let G be a connected graph. Thispaper s extreme entriesof the principal eigenvector x of G, the unique positive unit eigenvector corresponding to the greatest eigenvalue λ1 of the adjacency matrix of G.I fG hasmaximum degree ∆, the greates t entry xmax of x isat mos t 1/ 1+ λ 2 /∆. This improves a result of Papendieck and Recht. The least entry xmin of x aswell asthe principal ratio xmax/xmin are studied. It is conjectured that for connected graphs of order n ≥ 3, the principal ratio isalwaysattained by one of the lollipop graphsobtained by attaching a path graph to a vertex of a complete graph.

Biclique decompositions and Hermitian rank

The Hermitian rank, h(A), of a Hermitian matrix A is defined and shown to equal max{n+(A),n−(A)}, the maximum of the numbers of positive and negative eigenvalues of A. Properties of Hermitian rank are developed and used to obtain results on the minimum number, b(G), of complete bipartite subgraphs needed to partition the edge set of a graph G. Witsenhausens inequality b(G)⩾max{n+(G),n−(G)} is reproved and conditions necessary for equality to hold are given. The results are then used to estimate b(G) for several classes of graphs. For example, if G is the complement of a path then b(G)=⌊23(n−1)⌋, while if G is the complement of a cycle then b(G)=2⌊13(n−1)⌋ or ⌊13(2n−1)⌋.

Nonnegative Rank-Preserving Operators

Abstract Analogues of characterizations of rank-preserving operators on field-valued matrices are determined for matrices witheentries in certain structures S contained in the nonnegative reals. For example, if S is the set of nonnegative members of a real unique factorization domain (e.g. the nonnegative reals or the nonnegative integers), M is the set of m×n matrices with entries in S , and min(m,n)⩾4, then a “linear” operator on M preserves the “rank” of each matrix in M if and only if it preserves the ranks of those matrices in M of ranks 1, 2, and 4. Notions of rank and linearity are defined analogously to the field-valued concepts. Other characterizations of rank-preserving operators for matrices over these and other structures S are also given.

Pick's inequality and tournaments

Abstract Let T be a tournament on n nodes, and let A be its (adjacency) matrix. A. Brauer and I. Gentry observed that an inequality due to G. Pick implies that | Im λ|⩽ 1 2 cot ( π 2n ) for all eigenvalues λ of A. We say that T is a Pick (or P -) tournament if equality holds for some λ. We determine when equality holds in Picks inequality for arbitrary real matrices and use this to show that the P -tournaments can all be constructed from the transitive tournament M by reversing the arcs between the sets of certain node partitions or cuts {U, Ū}. The cuts are specified by ±1 n-vectors u such that utw=0, where w =[ 1 , σ,…, σ n− 1 ] t , σ=e iπ n . This links the cuts to cyclotomic polynomials. There is at least one P -tournament on n nodes if and only if n≠2k, k⩾1. Up to isomorphism, there is precisely one P -tournament on n nodes if (and only if) n=p or n=2p for some odd prime p. For odd n, up toisomorphism, the only regular P -tournament on n nodes has matrix Zn = Circ(0, 1,…, 1, 0,…, 0). A composition rule is used on the matrices Zp to form all P -tournaments on n=2kpl nodes, l⩾1.

Biclique coverings of regular bigraphs and minimum semiring ranks of regular matrices

We study the minimum number of complete bipartite subgraphs needed to cover and partition the edges of a k-regular bigraph on 2n vertices. Bounds are determined on the minima of these numbers for fixed n and k. Exact values of the minima are found for all n and k ≤ 4. The same results hold for directed graphs. Equivalently, we have determined bounds on the minimum value of the Boolean and nonnegative integer ranks of binary n × n matrices with constant row and column sum k for fixed n and k, obtaining the exact values of the minimum for k ≤ 4.

Inertia and biclique decompositions of joins of graphs

We characterize the inertia of A + B for Hermitian matrices A and B when the rank of B is one. We use this to characterize the inertia of a partial join of two graphs. We then provide graph joins G for which the minimum number of complete bipartite graphs needed in a partition of the edge multi-set of G is equal to the maximum of the number of positive and negative Eigenvalues of G.