for a simple digraph $g$ of order $n$ with vertex set${v_1,v_2,ldots, v_n}$, let $d_i^+$ and $d_i^-$ denote theout-degree and in-degree of a vertex $v_i$ in $g$, respectively. let$d^+(g)=diag(d_1^+,d_2^+,ldots,d_n^+)$ and$d^-(g)=diag(d_1^-,d_2^-,ldots,d_n^-)$. in this paper we introduce$widetilde{sl}(g)=widetilde{d}(g)-s(g)$ to be a new kind of skewlaplacian matrix of $g$, where $widetilde{d}(g...

‎let $g$ be a graph without an isolated vertex‎, ‎the normalized laplacian matrix $tilde{mathcal{l}}(g)$‎‎is defined as $tilde{mathcal{l}}(g)=mathcal{d}^{-frac{1}{2}}mathcal{l}(g) mathcal{d}^{-frac{1}{2}}$‎, where ‎$‎mathcal{‎d}‎$ ‎is a‎ diagonal matrix whose entries are degree of ‎vertices ‎‎of ‎$‎g‎$‎‎. ‎the eigenvalues of‎‎$tilde{mathcal{l}}(g)$ are ‎called ‎ ‎ as ‎the ‎normalized laplacian ...

In this paper, we investigate some properties of eigenvalues and eigenfunctions of boundary value problems with separated boundary conditions. Also, we obtain formal series solutions for some partial differential equations associated with the second order differential equation, and study necessary and sufficient conditions for the negative and positive eigenvalues of the boundary value problem....

let $g$ be a simple graph with vertex set $v(g) = {v_1, v_2,ldots, v_n}$ and edge set $e(g) = {e_1, e_2,ldots , e_m}$. similar tothe randi'c matrix, here we introduce the randi'c incidence matrixof a graph $g$, denoted by $i_r(g)$, which is defined as the$ntimes m$ matrix whose $(i, j)$-entry is $(d_i)^{-frac{1}{2}}$ if$v_i$ is incident to $e_j$ and $0$ otherwise. naturally, therandi'c incidenc...

The eigenvalues of a nonhermitian Toeplitz matrix A are usually highly sensitive to perturbations, having condition numbers that increase exponentially with the dimension N. An equivalent statement is that the resolvent ( ZZ A)’ of a Toeplitz matrix may be much larger in norm than the eigenvalues alone would suggest-exponentially large as a function of N, even when z is far from the spectrum. B...

the stability analysis of islanded inverter-based microgrids (ibmgs) is increasingly an important and challenging topic due to the nonlinearity of ibmgs. in this paper, a new linear model for such microgrids as well as an iterative method to correct the linear model is proposed. using the linear model makes it easy to analyze the eigenvalues and stability of ibmgs due to the fact that it derive...

The purpose of this paper is to study the higher order asymptotic distributions of the eigenvalues associated with a class of Sturm-Liouville problem with equation of the form w??=(?2f(x)?R(x)) (1), on [a,b, where ? is a real parameter and f(x) is a real valued function in C2(a,b which has a single zero (so called turning point) at point 0x=x and R(x) is a continuously differentiable function. ...