Exercise 7. Let A be an n × n matrix such that the sum of every row is 0 and the sum of every column is 0. Let Aij be the (n − 1) × (n − 1) matrix obtained by removing row i and column j from A. Prove: det(Aij) = (−1) det(A11). (Note that this result applies in particular to the Laplacian L: the determinant of the reduced Laplacian, obtained by removing the i-th row and the i-th column from L, ...

The Laplacian and normalized Laplacian energy of G are given by expressions EL(G) = ∑n i=1 |μi − d|, EL(G) = ∑n i=1 |λi − 1|, respectively, where μi and λi are the eigenvalues of Laplacian matrix L and normalized Laplacian matrix L of G. An interesting problem in spectral graph theory is to find graphs {L,L}−noncospectral with the same E{L,L}(G). In this paper, we present graphs of order n, whi...

For a simple connected graph G with n-vertices having Laplacian eigenvalues μ1, μ2, . . . , μn−1, μn = 0, and signless Laplacian eigenvalues q1, q2, . . . , qn, the Laplacian-energy-like invariant(LEL) and the incidence energy (IE) of a graph G are respectively defined as LEL(G) = ∑n−1 i=1 √ μi and IE(G) = ∑n i=1 √ qi. In this paper, we obtain some sharp lower and upper bounds for the Laplacian...

It is shown that if L and D are the Laplacian matrix and the distance matrix of a tree respectively, then any minor of the Laplacian equals the sum of the cofactors of the complementary submatrix of D, upto a sign and a power of 2. An analogous, more general result is proved for the Laplacian and the resistance matrix of any graph. A similar identity is proved for graphs in which each block is ...

This paper presents a new voice activity detection (VAD) method using the Laplacian distribution and a uniformly most powerful (UMP) test. The UMP test is employed to derive the new decision rule based on likelihood ratio test (LRT). The proposed method provide the decision rule by comparing the sum of magnitude of real and imaginary parts of the noisy spectral component to the adaptive thresho...

A new spectral algorithm for reordering a sparse symmetric matrix to reduce its envelope size was described in [2]. The ordering is computed by associating a Laplacian matrix with the given matrix and then sorting the components of a specified eigenvector of the Laplacian. In this paper we provide an analysis of the spectral envelope reduction algorithm. We describe related 1and 2-sum problems;...

The PET and CT fusion images, combining the anatomical and functional information, have important clinical meaning. This paper proposes a novel fusion framework based on adaptive pulse-coupled neural networks (PCNNs) in nonsubsampled contourlet transform (NSCT) domain for fusing whole-body PET and CT images. Firstly, the gradient average of each pixel is chosen as the linking strength of PCNN m...

Let $T$ be a tree of order $n$ and $S_2(T)$ the sum two largest Laplacian eigenvalues $T$. Fritscher et al. proved that for any $n$, $S_2(T) \leq n+2-\frac{2}{n}$. Guan determined with maximum among all trees $n$. In this paper, we characterize \geq n+1$ except some trees. Moreover, also determine first $\lfloor\frac{n-2}{2}\rfloor$ according to their $S_2(T)$. This extends result