Let G = (V,E) be an undirected graph, LG ∈ R ×V be the associated Laplacian matrix, and b ∈ R V be a vector. Solving the Laplacian system LGx = b has numerous applications in theoretical computer science, machine learning, and network analysis. Recently, the notion of the Laplacian operator LF : R V → 2 V for a submodular transformation F : 2 → RE+ was introduced, which can handle undirected gr...

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^s be a signed graph, where G = (V;E) is the underlying simple graph and s : E(G) to {+, -} is the sign function on E(G). In this paper, we obtain k-th signed spectral moment and k-th signed Laplacian spectral moment of Gs together with coefficients of their signed characteristic polynomial and signed Laplacian characteristic polynomial are calculated.

Last time, we talked about a very simple solver for Laplacian-based systems of linear equations, i.e., systems of linear equations of the form Ax = b, where the constraint matrix A is the Laplacian of a graph. This is not fully-general—Laplacians are SPSD matrices of a particular form—but equations of this form arise in many applications, certain other SPSD problems such as those based on SDD m...

The theory of dynamical networks is concerned with systems of dynamical units coupled according to an underlying graph structure. It therefore investigates the interplay between dynamics and structure, between the temporal processes going on at the individual units and the static spatial structure linking them. In order to analyse that spatial structure, formalized as a graph, we discuss an ess...