Miriam Farber

On totally positive matrices and geometric incidences

A matrix is called totally positive if every minor of it is positive. Such matrices are well studied and have numerous applications in Mathematics and Computer Science. We study how many times the value of a minor can repeat in a totally positive matrix and show interesting connections with incidence problems in combinatorial geometry. We prove that the maximum possible number of repeated d × d -minors in a d × n totally-positive matrix is O ( n d - d d + 1 ) . For the case d = 2 we also show that our bound is optimal. We consider some special families of totally positive matrices to show non-trivial lower bounds on the number of repeated minors. In doing so, we arrive at a new interesting problem: How many unit-area and axis-parallel rectangles can be spanned by two points in a set of n points in the plane? This problem seems to be interesting in its own right especially since it seems to have a flavor of additive combinatorics and relates to interesting incidence problems where considering only the topology of the curves involved is not enough. We prove an upper bound of O ( n 4 3 ) and provide a lower bound of n 1 + 1 O ( log ? log ? n ) .

A LOWER BOUND FOR THE SECOND LARGEST LAPLACIAN EIGENVALUE OF WEIGHTED GRAPHS

Let G be a weighted graph on n vertices. Letn 1(G) be the second largest eigenvalue of the Laplacian of G. For n� 3, it is proved thatn 1(G)� dn 2(G), where dn 2(G) is the third largest degree of G. An upper bound for the second smallest eigenvalue of the signless Laplacian of G is also obtained. 1. Introduction. Let G = E(G),V (G) � be a simple graph (a graph without loops or multiple edges) with |V (G)| = n. We say that G is a weighted graph if it has a weight (a positive number) associated with each edge. The weight of an edge � i,j � ∈ E(G) will be denoted by wij. We define the adjacency matrix A(G) of G to be a symmetric matrix which satisfies

Conjectured bounds for the sum of squares of positive eigenvalues of a graph

A well known upper bound for the spectral radius of a graph, due to Hong, is that µ 1 2 ź 2 m - n + 1 if ź ź 1 . It is conjectured that for connected graphs n - 1 ź s + ź 2 m - n + 1 , where s + denotes the sum of the squares of the positive eigenvalues. The conjecture is proved for various classes of graphs, including bipartite, regular, complete q -partite, hyper-energetic, and barbell graphs. Various searches have found no counter-examples. The paper concludes with a brief discussion of the apparent difficulties of proving the conjecture in general.

Nonpositive Eigenvalues of Hollow, Symmetric, Nonnegative Matrices

Among hollow, symmetric

The Number of Interlacing Equalities Resulting from Removal of a Vertex from a Tree

n

Interlacing networks

-by-

Exact results for perturbation to total positivity and to total nonsingularity

n

Weak separation, pure domains and cluster distance

nonnegative matrices, it is shown that any number

Arrangements of minors in the positive Grassmannian and a triangulation of the hypersimplex

k

The structure of Schur complements in hollow, symmetric nonnegative matrices with two nonpositive eigenvalues

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