Distance Laplacian Research Articles

On distance and distance Laplacian spectra of corona of two graphs

Corona of two graphs has been defined in [F. Harary, Graph Theory (Addison-Wesley, 1969)]. In this paper, we study the distance and the distance Laplacian spectra of corona of two graphs and describe the complete distance (distance Laplacian) spectrum for some particular cases. As an application, we show that the corona operation can be used to create distance singular graphs. We also show that these results enable us to construct infinitely many pairs of distance (respectively, distance Laplacian) cospectral graphs. Last, we give a graph transformation and discuss its effect on the distance Laplacian spectral radius.

Accurate and Efficient Computation of Laplacian Spectral Distances and Kernels

This paper introduces the Laplacian spectral distances, as a function that resembles the usual distance map, but exhibits properties e.g. smoothness, locality, invariance to shape transformations that make them useful to processing and analysing geometric data. Spectral distances are easily defined through a filtering of the Laplacian eigenpairs and reduce to the heat diffusion, wave, biharmonic and commute-time distances for specific filters. In particular, the smoothness of the spectral distances and the encoding of local and global shape properties depend on the convergence of the filtered eigenvalues to zero. Instead of applying a truncated spectral approximation or prolongation operators, we propose a computation of Laplacian distances and kernels through the solution of sparse linear systems. Our approach is free of user-defined parameters, overcomes the evaluation of the Laplacian spectrum and guarantees a higher approximation accuracy than previous work.

Graph functions maximized on a path

Given a connected graph G of order n and a nonnegative symmetric matrix A=[ai,j] of order n, define the function FA(G) asFA(G)=∑1≤i<j≤ndG(i,j)ai,j, where dG(i,j) denotes the distance between the vertices i and j in G.In this note it is shown that FA(G)≤FA(P) for some path of order n. Moreover, if each row of A has at most one zero off-diagonal entry, then FA(G)<FA(P) for some path of order n, unless G itself is a path.In particular, this result implies two conjectures of Aouchiche and Hansen:–the spectral radius of the distance Laplacian of a connected graph G of order n is maximal if and only if G is a path;–the spectral radius of the distance signless Laplacian of a connected graph G of order n is maximal if and only if G is a path.

Proof for four conjectures about the distance Laplacian and distance signless Laplacian eigenvalues of a graph

The distance Laplacian matrix L(G) of a graph G is defined to be L(G)=diag(Tr)−D(G), where D(G) denotes the distance matrix of G and diag(Tr) denotes the diagonal matrix of the vertex transmissions in G. Similarly, the distance signless Laplacian matrix of G is defined as Q(G)=diag(Tr)+D(G). The eigenvalues of L(G) and Q(G) are called the distance Laplacian and distance signless Laplacian eigenvalues, respectively. In this paper, four conjectures proposed by M. Aouchche and P. Hansen about the largest and the second largest distance Laplacian eigenvalues and the second largest distance signless Laplacian eigenvalue of a graph are proved.

Some properties of the distance Laplacian eigenvalues of a graph

The distance Laplacian of a connected graph G is defined by \(\mathcal{L} = Diag(Tr) - \mathcal{D}\), where \(\mathcal{D}\) is the distance matrix of G, and Diag(Tr) is the diagonal matrix whose main entries are the vertex transmissions in G. The spectrum of \(\mathcal{L}\) is called the distance Laplacian spectrum of G. In the present paper, we investigate some particular distance Laplacian eigenvalues. Among other results, we show that the complete graph is the unique graph with only two distinct distance Laplacian eigenvalues. We establish some properties of the distance Laplacian spectrum that enable us to derive the distance Laplacian characteristic polynomials for several classes of graphs.

Three distance characteristic polynomials of some graphs

We give complete information about the distance, distance Laplacian and distance signless Laplacian characteristic polynomials of graphs obtained by a generalized join graph operation on families of graphs. As an application of these results, we construct many pairs of nonisomorphic distance, distance Laplacian and distance signless Laplacian cospectral graphs, and then give a negative answer to the question “Can every connected graph be determined by its distance Laplacian spectrum and/or distance signless Laplacian spectrum?” proposed in Aouchiche and Hansen (2013) [2].

Two Laplacians for the distance matrix of a graph

We introduce a Laplacian and a signless Laplacian for the distance matrix of a connected graph, called the distance Laplacian and distance signless Laplacian, respectively. We show the equivalence between the distance signless Laplacian, distance Laplacian and the distance spectra for the class of transmission regular graphs. There is also an equivalence between the Laplacian spectrum and the distance Laplacian spectrum of any connected graph of diameter 2. Similarities between n, as a distance Laplacian eigenvalue, and the algebraic connectivity are established.

Speaker independent speech recognition method utilizing multiple training interations

A method for recognizing spoken utterances of a speaker is disclosed, the method comprising the steps of providing a database of labeled speech data; providing a prototype of a Hidden Markov Model (HMM) definition to define the characteristics of the HMM; and parameterizing speech utterances according to one of linear prediction parameters or Mel-scale filter bank parameters. The method further includes selecting a frame period for accommodating the parameters and generating HMMs and decoding to specified speech utterances by causing the user to utter predefined training speech utterances for each HMM. The method then statistically computes the generated HMMs with the prototype HMM to provide a set of fully trained HMMs for each utterance indicative of the speaker. The trained HMMs are used for recognizing a speaker by computing Laplacian distances via distance table lookup for utterances of the speaker during the selected frame period; and iteratively decoding node transitions corresponding to the spoken utterances during the selected frame period to determine which predefined utterance is present.

Signal matching using wavelet correlation

AbstractThe problem of detecting corresponding points is studied in the case in which local deformations exist and a new method named “wavelet correlation” is proposed. There is a difficulty in that a reasonable window width cannot be designed in local correlation, which is one of the methods for a corresponding problem. the wavelet correlation is derived by extending the notion of local correlation and is considered to overcome difficulty. the fundamental concept is derived by the belief that a signal can be decomposed to several (sinusoidal) components and the window width can be varied according to each component. It is claimed that any algorithm using local correlation can be replaced by the one using wavelet correlation. In this paper, the wavelet correlation derived from local correlation is compared with the Laplacian distance and the local correlation itself by experiments. Further, a matching method that uses a narrow‐band property of a wavelet correlation function is proposed and the matching error is evaluated through experiments using one‐dimensional signals. Finally, an absolute mesure of matching by using normalized wavelet correlation is introduced and applied for detecting discontinuities of local deformations.

Speaker-independent connected digit recognition algorithm for a single chip signal processor

An algorithm was designed for a fixed point arithmetic signal processor chip to perform real-time speaker-independent English digit recognition. Each word was represented by a single 10-state Markov model, the states of which were 9-way mixtures of 24-dimensional Gaussian densities of cepstral features. Algorithms for feature extraction include autocorrelation, linear predictive coding, and the computation of both cepstra and differential cepstra. Algorithms for pattern matching include a Laplacian distance measure, viterbi decoder, best choice, and partial traceback. To achieve real-time operation, single precision arithmetic was employed for the Laplacian distance metric, which is the bottleneck in the recognizer. Memory storage was minimized by quantizing model parameters to 10 bits and dynamically pruning a tree of word candidates. Recognition accuracy of about 98% per word was obtained; this is approximately the same as that obtained with a floating point simulator as tested on a connected digits NIST database.