Marietjie Frick, University of South Africa, P.O. Box 392, UNISA, 0003, Pretoria, South Africa, e-mail: marietjie.frick@gmail.com; Bernardo Llano, Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, Colonia Vicentina, 09340, Mexico City, Mexico, e-mail: llano@xanum.uam.mx; Rita Zuazua, Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad Universitaria, Av. Universidad 3000, Circuito Exterior S/N, Delegación Coyoacán, C.P. 04510, Mexico City, Mexico, e-mail: ritazuazua@ciencias.unam.mx
Abstract: We study a particular digraph dynamical system, the so called digraph diclique operator. Dicliques have frequently appeared in the literature the last years in connection with the construction and analysis of different types of networks, for instance biochemical, neural, ecological, sociological and computer networks among others. Let $D=(V,A)$ be a reflexive digraph (or network). Consider $X$ and $Y$ (not necessarily disjoint) nonempty subsets of vertices (or nodes) of $D$. A disimplex $K(X,Y)$ of $D$ is the subdigraph of $D$ with vertex set $X\cup Y$ and arc set $\{(x,y) x\in X, y\in Y\}$ (when $X\cap Y\neq\varnothing$, loops are not considered). A disimplex $K(X,Y)$ of $D$ is called a diclique of $D$ if $K(X,Y)$ is not a proper subdigraph of any other disimplex of $D$. The diclique digraph $\overrightarrow{k}(D)$ of a digraph $D$ is the digraph whose vertex set is the set of all dicliques of $D$ and $( K(X,Y),K(X',Y'))$ is an arc of $\overrightarrow{k}(D)$ if and only if $Y\cap X'\neq\varnothing$. We say that a digraph $D$ is self-diclique if $\overrightarrow{k}(D)$ is isomorphic to $D$. In this paper, we provide a characterization of the self-diclique circulant digraphs and an infinite family of non-circulant self-diclique digraphs.
Keywords: circulant digraph; diclique; diclique operator; self-diclique digraph; graph dynamics
Classification (MSC 2010): 05C20, 68R10
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