4,368 research outputs found

### Randi\'c energy and Randi\'c eigenvalues

Let $G$ be a graph of order $n$, and $d_i$ the degree of a vertex $v_i$ of
$G$. The Randi\'c matrix ${\bf R}=(r_{ij})$ of $G$ is defined by $r_{ij} = 1 /
\sqrt{d_jd_j}$ if the vertices $v_i$ and $v_j$ are adjacent in $G$ and
$r_{ij}=0$ otherwise. The normalized signless Laplacian matrix $\mathcal{Q}$ is
defined as $\mathcal{Q} =I+\bf{R}$, where $I$ is the identity matrix. The
Randi\'c energy is the sum of absolute values of the eigenvalues of $\bf{R}$.
In this paper, we find a relation between the normalized signless Laplacian
eigenvalues of $G$ and the Randi\'c energy of its subdivided graph $S(G)$. We
also give a necessary and sufficient condition for a graph to have exactly $k$
and distinct Randi\'c eigenvalues.Comment: 7 page

### On the sphericity test with large-dimensional observations

In this paper, we propose corrections to the likelihood ratio test and John's
test for sphericity in large-dimensions. New formulas for the limiting
parameters in the CLT for linear spectral statistics of sample covariance
matrices with general fourth moments are first established. Using these
formulas, we derive the asymptotic distribution of the two proposed test
statistics under the null. These asymptotics are valid for general population,
i.e. not necessarily Gaussian, provided a finite fourth-moment. Extensive
Monte-Carlo experiments are conducted to assess the quality of these tests with
a comparison to several existing methods from the literature. Moreover, we also
obtain their asymptotic power functions under the alternative of a spiked
population model as a specific alternative.Comment: 37 pages, 3 figure

### On singular values distribution of a large auto-covariance matrix in the ultra-dimensional regime

Let $(\varepsilon_{t})_{t>0}$ be a sequence of independent real random
vectors of $p$-dimension and let
$X_T=\sum_{t=s+1}^{s+T}\varepsilon_t\varepsilon^T_{t-s}/T$ be the lag-$s$ ($s$
is a fixed positive integer) auto-covariance matrix of $\varepsilon_t$. This
paper investigates the limiting behavior of the singular values of $X_T$ under
the so-called {\em ultra-dimensional regime} where $p\to\infty$ and
$T\to\infty$ in a related way such that $p/T\to 0$. First, we show that the
singular value distribution of $X_T$ after a suitable normalization converges
to a nonrandom limit $G$ (quarter law) under the forth-moment condition.
Second, we establish the convergence of its largest singular value to the right
edge of $G$. Both results are derived using the moment method.Comment: 32 pages, 2 figure

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