let $r$ be a commutative ring with identity. let $g(r)$ denote the maximal graph associated to $r$, i.e., $g(r)$ is a graph with vertices as the elements of $r$, where two distinct vertices $a$ and $b$ are adjacent if and only if there is a maximal ideal of $r$ containing both. let $gamma(r)$ denote the restriction of $g(r)$ to non-unit elements of $r$. in this paper we study the various graphi...

We present a general method for evaluating and visualizing evolutionary dynamics of self-replicators using a graph-based representation for genealogy. Through a transformation from the space of species and mutations to the space of nodes and links, evolutionary dynamics are understood as a flow in graph space. Mapping functions are introduced to translate graph nodes to points in an n-dimension...

Recent results have shown that the Glauber dynamics for graph colourings has optimal mixing time when (i) the graph is triangle-free and D-regular and the number of colours k is a small constant fraction smaller than 2D, or (ii) the graph has maximum degree D and k = 2D. We extend both these results to prove that the Glauber dynamics has optimal mixing time when the graph has maximum degree D a...

Quantum walks obey unitary dynamics: they form closed quantum systems. The system becomes open if the walk suffers from imperfections represented as missing links on the underlying basic graph structure, described by dynamical percolation. Openness of the system's dynamics creates decoherence, leading to strong mixing. We present a method to analytically solve the asymptotic dynamics of coined,...

In the field of biological regulation, models extracted from experimental works are usually complex networks comprising intertwined feedback circuits. R. Thomas and coworkers introduced a qualitative description of such regulatory networks, then they used the concept of circuit-characteristic states to identify all steady states and functional circuits. These characteristic states play an essen...

Using the intrinsic relationship between the external equitable partition (EEP) and the spectral properties of the graph Laplacian, we characterize convergence and observability properties of consensus dynamics on networks. In particular, we establish the relationship between the original consensus dynamics and the associated consensus of the quotient graph under varied initial conditions, and ...

This paper studies the problem of fault detection and isolation (FDI) for multi-agent systems (MAS) via complex Laplacian subject to actuator faults. A planar formation of point agents in the plane using simple and linear interaction rules related to complex Laplacian is achieved. The communication network is a directed, and yet connected graph with a fixed topology. The loss of symmetry in the...