Independence Fractals of Graphs as Models in Architecture

Document Type: Special Issue: International Conference on Architecture and Mathematics

Authors

1 Faculty of Art and Architecture, Islamic Azad University, Yazd Branch, Yazd, Iran

2 Department of Mathematics, Yazd University, Yazd, Iran

10.22052/mir.2019.169780.1112

Abstract

Architectural science requires interdisciplinary science interconnection in order to improve this science. Graph theory and geometrical fractal are two examples of branches of mathematics which have applications in architecture and design. In architecture,
the vertices are the rooms and the edges are the direct connections between each two rooms.
The independence polynomial of a graph $G$ is the polynomial
$I(G,x)=\sum i_kx^k$, where $i_k$ denote the number of independent sets
of cardinality $k$ in $G$. The independence fractal of $G$ is the set ${\cal I}(G)=\lim _{k\rightarrow \infty} Roots (I({G^{k}},x)-1),$ where $G^{k}=G[G[\cdots]]$, and $G[H]$ is the
lexicographic product for two graphs $G$ and $H$. In this paper, we consider graphical presentation of a ground plane as a graph $G$ and use
the sequences of limit roots of independence polynomials of $G^k$ to present some animated structures for building.

Keywords