ScholarBank@NUShttps://scholarbank.nus.edu.sgThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Sat, 28 May 2022 08:35:58 GMT2022-05-28T08:35:58Z5091- Connected Sproutshttps://scholarbank.nus.edu.sg/handle/10635/103044Title: Connected Sprouts
Authors: Lam, T.K.
Sat, 01 Feb 1997 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1030441997-02-01T00:00:00Z
- Hook immanantal inequalities for trees explainedhttps://scholarbank.nus.edu.sg/handle/10635/103386Title: Hook immanantal inequalities for trees explained
Authors: Chan, O.; Lam, T.K.
Abstract: Let d̄k denote the normalized hook immanant corresponding to the partition (k, 1n-k) of n. P. Heyfron proved the family of immanantal inequalities det A = d̄1(A) ≤ d̄2(A) ≤ ⋯ ≤ d̄n(A) = per A (1) for all positive semidefinite Hermitian matrices A. Motivated by a conjecture of R. Merris, it was shown by the authors that (1) may be improved to d̄k-1(L(.T)) ≤ k - 2 / k - 1d̄k(L(T)) (2) for all 2 ≤ k ≤ n whenever L(T) is the Laplacian matrix of a tree T. The proof of (2) relied on rather involved recursive relations for weighted matchings in the tree T as well as identities of hook characters. In this work, we circumvent this tedium with a new proof using the notion of vertex orientations. This approach makes (2) immediately apparent and more importantly provides an insight into why it holds, namely the absence of certain vertex orientations for all trees. As a by-product we obtain an improved bound, 0 ≤ 1 / k-1[d̄k(L(T)) - d̄k(L(S(n)))] ≤ k - 2 / k - 1d̄k(L(T)) - d̄k - 1(L(T)), where S(n) is the star with n vertices. The ease with which the inequality in (2) and its improvement are derived points to the value of the concept of vertex orientation in the study of immanantal inequalities on graphs. © 1998 Elsevier Science Inc.
Wed, 01 Apr 1998 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1033861998-04-01T00:00:00Z
- Hook immanantal inequalities for Laplacians of treeshttps://scholarbank.nus.edu.sg/handle/10635/103385Title: Hook immanantal inequalities for Laplacians of trees
Authors: Chan, O.; Lam, T.K.
Abstract: For an irreducible character χλ of the symmetric group Sn, indexed by the partition λ, the immanant function dλ, acting on an n × n matrix A = (aij), is defined as dλ(A) = ∑σ ∈ Sn χλ(σ)Πn i = 1_ aiσ(i). The associated normalized immanant d̄λ is defined as d̄λ = dλ/χλ(identity) where identity is the identity permutation. P. Heyfron has shown that for the partitions (k, 1n - k), the normalized immanant d̄k satisfies det A = d̄1(A) ≤ d̄2(A) ≤ ⋯ ≤ d̄n(A) = per A (1) for all positive semidefinite Hermitian matrices A. When A is restricted to the Laplacian matrices of graphs, improvements on the inequalities above may be expected. Indeed, in a recent survey paper, R. Merris conjectured that d̄n - 1(A) ≤ n - 2/n - 1 d̄n(A) (2) whenever A is the Laplacian matrix of a tree. In this note, we establish a refinement for the family of inequalities in (1) when A is the Laplacian matrix of a tree, that includes (2) as a special case. These inequalities are sharp and equality holds if and only if A is the Laplacian matrix of the star. This is proved via the inequalities d̄k(A) - d̄k - 1(A) ≤ d̄k + 1(A) - d̄k(A) for k = 2,3, . . ., n - 1, where A is the Laplacian matrix of a tree. © Elsevier Science Inc., 1997.
Fri, 01 Aug 1997 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1033851997-08-01T00:00:00Z
- Estimating structured correlation matrices in smooth Gaussian random field modelshttps://scholarbank.nus.edu.sg/handle/10635/103215Title: Estimating structured correlation matrices in smooth Gaussian random field models
Authors: Loh, W.-L.; Lam, T.-K.
Abstract: This article considers the estimation of structured correlation matrices in infinitely differentiable Gaussian random field models. The problem is essentially motivated by the stochastic modeling of smooth deterministic responses in computer experiments. In particular, the log-likelihood function is determined explicitly in closed-form and the sieve maximum likelihood estimators are shown to be strongly consistent under mild conditions.
Sat, 01 Jan 2000 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1032152000-01-01T00:00:00Z
- Wiener number as an immanant of the Laplacian of molecular graphshttps://scholarbank.nus.edu.sg/handle/10635/104480Title: Wiener number as an immanant of the Laplacian of molecular graphs
Authors: Chan, O.; Lam, T.K.; Merris, R.
Abstract: In this work, we show that the Wiener number of an acyclic molecule may be expressed as a linear function of a hook immanant of the Laplacian matrix associated with the molecule. This relation leads to a new interpretation of the Wiener number as a weighted sum of matchings in the molecular graph of acyclic compounds. The connection between the Wiener number and immanants may pave the way for more use of algebraic tools in the study of the Wiener number.
Tue, 01 Jul 1997 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1044801997-07-01T00:00:00Z
- Vexillary Elements in the Hyperoctahedral Grouphttps://scholarbank.nus.edu.sg/handle/10635/104452Title: Vexillary Elements in the Hyperoctahedral Group
Authors: Billey, S.; Lam, T.K.
Abstract: In analogy with the symmetric group, we define the vexillary elements in the hyperoctahedral group to be those for which the Stanley symmetric function is a single Schur Q-function. We show that the vexillary elements can be again determined by pattern avoidance conditions. These results can be extended to include the root systems of types A, B, C, and D. Finally, we give an algorithm for multiplication of Schur Q-functions with a superfied Schur function and a method for determining the shape of a vexillary signed permutation using jeu de taquin.
Thu, 01 Jan 1998 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1044521998-01-01T00:00:00Z
- Algebraic connections between topological indiceshttps://scholarbank.nus.edu.sg/handle/10635/102799Title: Algebraic connections between topological indices
Authors: Chan, O.; Gutman, I.; Lam, T.-K.; Merris, R.
Abstract: A relation has been recently established between the Wiener number W and an immanant of the Laplacian matrix of the molecular graph [Chan, O.; Lam, T. K.; Merris, R. J. Chem. Inf. Comput. Sci. 1997, 37, 762-765]. On the basis of this result we now show that there exist algebraic connections between W and certain molecular-graph-based structure descriptors which, until now, were believed not to be related to W, namely the Hosoya index and quantities derived from it and the simple topological index of Narumi.
Thu, 01 Jan 1998 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1027991998-01-01T00:00:00Z
- Immanant inequalities for Laplacians of treeshttps://scholarbank.nus.edu.sg/handle/10635/103399Title: Immanant inequalities for Laplacians of trees
Authors: Chan, O.; Lam, T.K.
Abstract: Let Tn be the collection of trees on n vertices. Let Tn (b; p, q), Tn (m; k), and Tn (d; k) be subsets of Tn comprising trees, each whose vertex set has bipartition (p, q), trees whose maximum matching has size k, and trees of diameter k, respectively. Brualdi and Goldwasser [Discrete Math., 48 (1984), pp. 1-21] obtained lower bounds on the permanent of the Laplacian matrix of a tree from each of these subsets. They characterized the tree in each of these subsets whose Laplacian matrix has the smallest permanent as the "double star" in Tn (b; p, q), the "spur" in Tn (m; k), and the "broom" in Tn (d; k). In this work, the concept of vertex orientations and a new interpretation of the matching numbers in a tree allow us to formulate a unified approach to extending these results to all other immanant functions besides the permanent. It turns out that the "double star" and the "spur" remain the tree in Tn (b; p, q) and the tree in Tn (m; k), respectively, whose Laplacian matrix has the smallest immanant value for all immanants. For Tn (d; k) the tree that has the smallest immanant value varies with the immanant function, but it belongs to a small family of "caterpillars" whose legs are all concentrated on a single vertex.
Sun, 01 Aug 1999 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1033991999-08-01T00:00:00Z
- Lifting Markov chains to random walks on groupshttps://scholarbank.nus.edu.sg/handle/10635/103488Title: Lifting Markov chains to random walks on groups
Authors: Chan, O.; Lam, T.K.
Abstract: The determination of a Markov chain which can be lifted to a random walk on an abelian group or a group whose probability measure is a class function, is described. The analysis is facilitated whenever the probability measure on the group is a class function. A special case is that of an abelian group where any probability measure is a class function. The results show that the Fourier analysis using group representations simplifies to computations involving the irreducible characters of the group.
Sun, 01 May 2005 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1034882005-05-01T00:00:00Z