There is a wealth of applied problems that can be posed as a dynamical system defined on a network with both attractive and repulsive interactions. Some examples include: understanding synchronization properties of nonlinear oscillator;, the behavior of groups, or cliques, in social networks; the study of optimal convergence for consensus algorithm; and many other examples. Frequently the probl...

A Laplacian matrix, L = (lij) ∈ R , has nonpositive off-diagonal entries and zero row sums. As a matrix associated with a weighted directed graph, it generalizes the Laplacian matrix of an ordinary graph. A standardized Laplacian matrix is a Laplacian matrix with − 1 n ≤ lij ≤ 0 at j 6= i. We study the spectra of Laplacian matrices and relations between Laplacian matrices and stochastic matrice...

An oriented hypergraph is an incidence structure that allows for the generalization of graph theoretic concepts to integer matrices through its locally signed graphic substructure. The behaviors are formalized in subobject classifier hypergraphs. Moreover, injective envelope calculated and shown contain class uniform hypergraphs -- providing a combinatorial framework entries matrices. A multiva...

Dear Editor, This letter investigates the target enclosing control problem of multi-agent systems. A signed graph-based strategy is presented, where agents are steered to enclose dynamic from both sides as they move. inspired by phenomenon that networks exhibit bipartite clustering if underlying graph structurally balanced, so may naturally zero point opposite (+ and -) proper controllers appli...

We will consider the process {Z(t) df = X(Y (t)), t ≥ 0} which we will call “iterated Brownian motion” or simply IBM. Funaki (1979) proved that a similar process is related to “squared Laplacian.” Krylov (1960) and Hochberg (1978) considered finitely additive signed measures on the path space corresponding to squared Laplacian (there exists a genuine probabilistic approach, see, e.g., Ma̧drecki ...

In this paper, we present a novel volumetric mesh representation suited for parallel computing on modern GPU architectures. The data structure is based on a compact, ternary sparse matrix storage of boundary operators. Boundary operators correspond to the first-order top-down relations of k-faces to their (k−1)-face facets. The compact, ternary matrix storage format is based on compressed spars...

I discuss the work of many authors on various matrices used to study signed graphs, concentrating on adjacency and incidence matrices and the closely related topics of Kirchhoff (‘Laplacian’) matrices, line graphs, and very strong regularity.

Let $D$ be a diameter and $d_G(v_i, v_j)$ be the distance between the vertices $v_i$ and $v_j$ of a connected graph $G$. The complementary distance signless Laplacian matrix of a graph $G$ is $CDL^+(G)=[c_{ij}]$ in which $c_{ij}=1+D-d_G(v_i, v_j)$ if $ineq j$ and $c_{ii}=sum_{j=1}^{n}(1+D-d_G(v_i, v_j))$. The complementary transmission $CT_G(v)$ of a vertex $v$ is defined as $CT_G(v)=sum_{u in ...