71 research outputs found
Linear estimation in models based on a graph
AbstractTwo natural linear models associated with a graph are considered. The GaussāMarkov theorem is used in one of the models to derive a combinatorial formula for the MooreāPenrose inverse of the incidence matrix of a tree. An inequality involving the MooreāPenrose inverse of the Laplacian matrix of a graph and its distance matrix is obtained. The case of equality is discussed. Again the main tool used in the proof is the theory of linear estimation
Perron eigenvector of the Tsetlin matrix
AbstractWe consider the move-to-position k linear search scheme where the sequence of record requests is a Markov chain. Formulas are derived for the stationary distribution of the permutation chain for k=1,2,nā1 and n, where n is the number of records. Certain identities for the Perron complement are established in the process
Symmetric function means and permanents
AbstractWe define a function using permanents which generalizes the symmetric function means and show that it is monotonic. The function is conjectured to be superadditive. A special case of the conjecture is proved
Simple expressions for the long walk distance
The walk distances in graphs are defined as the result of appropriate
transformations of the proximity measures, where
is the weighted adjacency matrix of a connected weighted graph and is a
sufficiently small positive parameter. The walk distances are graph-geodetic,
moreover, they converge to the shortest path distance and to the so-called long
walk distance as the parameter approaches its limiting values. In this
paper, simple expressions for the long walk distance are obtained. They involve
the generalized inverse, minors, and inverses of submatrices of the symmetric
irreducible singular M-matrix where is the Perron
root of Comment: 7 pages. Accepted for publication in Linear Algebra and Its
Application
Nonnegative idempotent matrices and the minus partial order
AbstractWe describe the structure of nonnegative matrices dominated by a nonnegative idempotent matrix under the minus order
Reconstructing sparticle mass spectra using hadronic decays
Most sparticle decay cascades envisaged at the Large Hadron Collider (LHC) involve hadronic decays of intermediate particles. We use state-of-the art techniques based on the Kā„ jet algorithm to reconstruct the resulting hadronic final states for simulated LHC events in a number of benchmark supersymmetric scenarios. In particular, we show that a general method of selecting preferentially boosted massive particles such as WĀ±, Z0 or Higgs bosons decaying to jets, using sub-jets found by the Kā„ algorithm, suppresses QCD backgrounds and thereby enhances the observability of signals that would otherwise be indistinct. Consequently, measurements of the supersymmetric mass spectrum at the per-cent level can be obtained from cascades including the hadronic decays of such massive intermediate bosons
Safe sets, network majority on weighted trees
Let G = (V, E) be a graph and let w : V ā ā>0 be a positive weight function on the vertices of G. For every subset X of V, let w(X) ā āvāGw(v). A non-empty subset ā is a weighted safe set if, for every component C of the subgraph induced by S and every component D of G/S, we have w(C) ā„ w(D) whenever there is an edge between C and D. If the subgraph G(S) induced by a weighted safe set S is connected, then the set S is called a weighted connected safe set. In this article, we show that the problem of computing the minimum weight of a safe set is NP-hard for trees, even if the underlying tree is restricted to be a star, but it is polynomially solvable for paths. We also give an O(n log n) time 2-approximation algorithm for finding a weighted connected safe set with minimum weight in a weighted tree. Then, as a generalization of the concept of a minimum safe set, we define the concept of a parameterized infinite family of proper central subgraphs on weighted trees, whose polar ends are the vertex set of the tree and the centroid points. We show that each of these central subgraphs includes a centroid point. Ā© 2017 Wiley Periodicals, Inc
Combinatorial integer labeling theorems on finite sets with applications
Tuckerās well-known combinatorial lemma states that, for any given symmetric triangulation of the n-dimensional unit cube and for any integer labeling that assigns to each vertex of the triangulation a label from the set {Ā±1, Ā±2, Ā· Ā· Ā· , Ā±n} with the property that antipodal vertices on the boundary of the cube are assigned opposite labels, the triangulation admits a 1-dimensional simplex whose two vertices have opposite labels. In this paper, we are concerned with an arbitrary finite set D of integral vectors in the n-dimensional Euclidean space and an integer labeling that assigns to each element of D a label from the set {Ā±1, Ā±2, Ā· Ā· Ā· , Ā±n}. Using a constructive approach, we prove two combinatorial theorems of Tucker type. The theorems state that, under some mild conditions, there exists two integral vectors in D having opposite labels and being cell-connected in the sense that both belong to the set {0, 1} n +q for some integral vector q. These theorems are used to show in a constructive way the existence of an integral solution to a system of nonlinear equations under certain natural conditions. An economic application is provided
A bound for the permanent of the Laplacian matrix
AbstractIt is shown that if G is a simple connected graph on n vertices, then perL(G)ā©¾ 2(n ā 1)Īŗ(G), where L(G) is the Laplacian matrix of G and Īŗ(G) is the complexity of G
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