The maximum nullity over a collection of matrices associated with a graph has been attracting the attention of numerous researchers for at least three decades. Along these lines various zero forcing parameters have been devised and utilized for bounding the maximum nullity. The maximum nullity and zero forcing number, and their positive counterparts, for general families of line graphs associated with graphs possessing a variety of specific properties are analysed. Building upon earlier work, where connections to the minimum rank of line graphs were established, we verify analogous equations in the positive semidefinite cases and coincidences with the corresponding zero forcing numbers. Working beyond the case of trees, we study the zero forcing number of line graphs associated with certain families of unicyclic graphs., Shaun Fallat, Abolghasem Soltani., and Obsahuje seznam literatury
We give a classification of all linear natural operators transforming p-vectors (i.e., skew-symmetric tensor fields of type (p, 0)) on n-dimensional manifolds M to tensor fields of type (q, 0) on TAM, where TA is a Weil bundle, under the condition that p ≥ 1, n ≥ p and n ≥ q. The main result of the paper states that, roughly speaking, each linear natural operator lifting p-vectors to tensor fields of type (q, 0) on TA is a sum of operators obtained by permuting the indices of the tensor products of linear natural operators lifting p-vectors to tensor fields of type (p, 0) on TA and canonical tensor fields of type (q − p, 0) on TA., Jacek Dębecki., and Obsahuje seznam literatury
Let M_{m,n} be the set of all m × n real matrices. A matrix A \in M_{m,n} is said to be row-dense if there are no zeros between two nonzero entries for every row of this matrix. We find the structure of linear functions T: M_{m,n} \rightarrow M_{m,n} that preserve or strongly preserve row-dense matrices, i.e., T(A) is row-dense whenever A is row-dense or T(A) is row-dense if and only if A is row-dense, respectively. Similarly, a matrix A \in M_{m,n} is called a column-dense matrix if every column of A is a column-dense vector. At the end, the structure of linear preservers (strong linear preservers) of column-dense matrices is found., Sara M. Motlaghian, Ali Armandnejad, Frank J. Hall., and Obsahuje seznam literatury
We propose a new localization result for the leading eigenvalue and eigenvector of a symmetric matrix A. The result exploits the Frobenius inner product between A and a given rank-one landmark matrix X. Different choices for X may be used, depending on the problem under investigation. In particular, we show that the choice where X is the all-ones matrix allows to estimate the signature of the leading eigenvector of A, generalizing previous results on Perron-Frobenius properties of matrices with some negative entries. As another application we consider the problem of community detection in graphs and networks. The problem is solved by means of modularity-based spectral techniques, following the ideas pioneered by Miroslav Fiedler in mid-’70s. We show that a suitable choice of X can be used to provide new quality guarantees of those techniques, when the network follows a stochastic block model., Dario Fasino, Francesco Tudisco., and Obsahuje seznam literatury
Consider the n×n matrix with (i, j)’th entry gcd (i, j). Its largest eigenvalue \lambda _{n} and sum of entries s_{n} satisfy \lambda _{n} > s_{n}/n. Because sn cannot be expressed algebraically as a function of n, we underestimate it in several ways. In examples, we compare the bounds so obtained with one another and with a bound from S.Hong, R.Loewy (2004). We also conjecture that \lambda _{n} > 6\Pi ^{-2}n log n for all n. If n is large enough, this follows from F.Balatoni (1969)., Jorma K. Merikoski., and Obsahuje seznam literatury