A tournament matrix and its corresponding directed graph both arise as a record of the outcomes o... more A tournament matrix and its corresponding directed graph both arise as a record of the outcomes of a round robin competition. Ann × ncomplex matrixAis calledh-pseudo-tournament if there exists a complex or real nonzero column vectorhsuch thatA+A* =hh* −I. This class of matrices is a generalisation of well-studied tournament-like matrices such ash-hypertournament matrices, generalised tournament matrices, tournament matrices, and elliptic matrices. We discuss the eigen-properties of anh-pseudo-tournament matrix, and obtain new results when the matrix specialises to one of these tournament-like matrices. Further, several results derived in previous articles prove to be corollaries of those reached here.
There is a very large literature devoted to counting structures, e.g., spanning trees, Hamiltonia... more There is a very large literature devoted to counting structures, e.g., spanning trees, Hamiltonian cycles, independent sets, acyclic orientations, in the n × m grid graph G(n, m). In particular the problem of counting the number of structures in fixed height graphs, i.e., fixing m and letting n grow, has been, for different types of structures, attacked independently by many different authors, using a transfer matrix approach. This approach essentially permits showing that the number of structures in G(n, m) satisfies a fixed-degree constant-coefficient recurrence relation in n. In contrast there has been surprisingly little work done on counting structures in grid-cylinders (where the left and right, or top and bottom, boundaries of the grid are wrapped around and connected to each other) or in grid-tori (where the left edge of the grid is connected to the right and the top edge is connected to the bottom one). The goal of this paper is to demonstrate that, with some minor modifications, the transfer matrix technique can also be easily used to count structures in fixed height grid-cylinders and tori.
Science in China Series a Mathematics Physics Astronomy, 1999
The asymptotic properties of the numbers of spanning trees and Eulerian trails in circulant digra... more The asymptotic properties of the numbers of spanning trees and Eulerian trails in circulant digraphs and graphs are studied. Let C (p , sl, sz, ..., s ,) be a directed circulant graph. Let T (C (p , sl, sz. '.-, s*)) and E (C (p. sl, ~2 , .-., s &)) be the numbers of spanning trees and of Eulerian trails, respectively. Then
In this note, we show that the number of digraphs with n vertices and with cycles of length k, 0 ... more In this note, we show that the number of digraphs with n vertices and with cycles of length k, 0 ≤ k ≤ n, is equal to the number of n × n (0,1)-matrices whose eigenvalues are the collection of copies of the entire kth unit roots plus, possibly, 0's. In particular, 1) when k = 0, since the digraphs reduce to be acyclic, our result reduces to the main theorem obtained recently in [1] stating that, for each n = 1, 2, 3, …, the number of acyclic digraphs is equal to the number of n × n (0,1)-matrices whose eigenvalues are positive real numbers; and 2) when k = n, the digraphs are the Hamiltonian directed cycles and it, therefore, generates another well-known (and trivial) result: the eigenvalues of a Hamiltonian directed cycle with n vertices are the nth unit roots [2].
A preconditioning technique based on the application of a fixed but arbitrary number of I + S max... more A preconditioning technique based on the application of a fixed but arbitrary number of I + S max steps is proposed. A reduction of the spectral radius of the Gauss-Seidel iteration matrix is theoretically analyzed for diagonally dominant Z-matrices. In particular, it is shown that after a finite number of steps this matrix reduces to null matrix. To illustrate the performance of the proposed technique numerical experiments on a wide variety of matrices are presented. Point and block versions of the preconditioner are numerically studied.
Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
2009 Proceedings of the Sixth Workshop on Analytic Algorithmics and Combinatorics (ANALCO), 2009
A double fixed-step loop network, C p,q , is a digraph on n vertices 0, 1, 2, ..., n − 1 and for ... more A double fixed-step loop network, C p,q , is a digraph on n vertices 0, 1, 2, ..., n − 1 and for each vertex i (0 < i ≤ n − 1), there are exactly two arcs leaving from vertex i to vertices i + p, i + q (mod n). In this paper, we first derive an expression formula of elementary symmetric polynomials as polynomials in sums of powers then, by using this, for any positive integers p, q, n with p < q < n, an explicit formula for counting the number of spanning trees in a class of double fixed-step loop networks with constant or nonconstant jumps. We allso find two classes of networks that share the same number of spanning trees and we, finally, prove that the number of spanning trees can be approximated by a formula which is based on the mth order Fibonacci numbers. In some special cases, our results generate the formulas obtained in [15],[19],[20]. And, compared with the previous work, the advantage is that, for any jumps p, q, the number of spanning trees can be calculated directly, without establishing the recurrence relation of order 2 q−1 .
The maxima-finding is a fundamental problem in computational geometry with many applications. In ... more The maxima-finding is a fundamental problem in computational geometry with many applications. In this paper, a volume first maxima-finding algorithm is proposed. It is proved that the expected running time of the algorithm is N+ o (N) when choosing points from CI ...
The asymptotic properties of the numbers of spanning trees and Eulerian trails in circulant digra... more The asymptotic properties of the numbers of spanning trees and Eulerian trails in circulant digraphs and graphs are studied. Let C (p , sl, sz, ..., s ,) be a directed circulant graph. Let T (C (p , sl, sz. '.-, s*)) and E (C (p. sl, ~2 , .-., s &)) be the numbers of spanning trees and of Eulerian trails, respectively. Then
Physica A: Statistical Mechanics and its Applications, 2010
Discovering a community structure is fundamental for uncovering the links between structure and f... more Discovering a community structure is fundamental for uncovering the links between structure and function in complex networks. In this paper, we discuss an equivalence of the objective functions of the symmetric nonnegative matrix factorization (SNMF) and the maximum optimization of modularity density. Based on this equivalence, we develop a new algorithm, named the so-called SNMF-SS, by combining SNMF and a semi-supervised clustering approach. Previous NMF-based algorithms often suffer from the restriction of measuring network topology from only one perspective, but our algorithm uses a semi-supervised mechanism to get rid of the restriction. The algorithm is illustrated and compared with spectral clustering and NMF by using artificial examples and other classic real world networks. Experimental results show the significance of the proposed approach, particularly, in the cases when community structure is obscure.
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