Self-diclique circulant digraphs

Frick, Marietjie, Llano, Bernardo, and Zuazua, Rita. "Self-diclique circulant digraphs." Mathematica Bohemica 140.3 (2015): 361-367. <http://eudml.org/doc/271656>.

@article{Frick2015,
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 $\lbrace (x,y)\colon x\in X,\ y\in Y\rbrace $ (when $X\cap Y\ne \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 $( K(X,Y),K(X^\{\prime \},Y^\{\prime \}))$ is an arc of $\overrightarrow\{k\}(D)$ if and only if $Y\cap X^\{\prime \}\ne \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.},
author = {Frick, Marietjie, Llano, Bernardo, Zuazua, Rita},
journal = {Mathematica Bohemica},
keywords = {circulant digraph; diclique; diclique operator; self-diclique digraph; graph dynamics},
language = {eng},
number = {3},
pages = {361-367},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Self-diclique circulant digraphs},
url = {http://eudml.org/doc/271656},
volume = {140},
year = {2015},
}

TY - JOUR
AU - Frick, Marietjie
AU - Llano, Bernardo
AU - Zuazua, Rita
TI - Self-diclique circulant digraphs
JO - Mathematica Bohemica
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 140
IS - 3
SP - 361
EP - 367
AB - 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 $\lbrace (x,y)\colon x\in X,\ y\in Y\rbrace $ (when $X\cap Y\ne \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 $( K(X,Y),K(X^{\prime },Y^{\prime }))$ is an arc of $\overrightarrow{k}(D)$ if and only if $Y\cap X^{\prime }\ne \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.
LA - eng
KW - circulant digraph; diclique; diclique operator; self-diclique digraph; graph dynamics
UR - http://eudml.org/doc/271656
ER -