# Independence Fractals of Graphs as Models in Architecture

Document Type : Special Issue: AIMC 51

Authors

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

2 Department of Mathematics, Yazd University, Yazd, Iran

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)=∑ ikxk, where ik denote the number of independent sets of cardinality k in G. The independence fractal of G is the set I(G)=limk→∞ Roots (I({Gk},x)-1),  where Gk=G[G[...]], 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 Gk to present some animated structures for building.

Keywords

#### References

[1] S. Alikhani and Y. H. Peng, Independence roots and independence fractals of certain graphs, J. Appl. Math. Comput. 36 (1 - 2) (2011) 89 - 100.
[2] M. F. Barnsley, Fractals Everywhere, Academic Press, Inc., Boston, MA, 1988.
[3] M. F. Blanco and M. Pisonero, An application of graphs in architecture, Available at https://www.mi.sanu.ac.rs/vismath/proceedings/blanco.htm.
[4] J. I. Brown, K. Dilcher and R. J. Nowakowski, Roots of independence polynomials of well covered graphs, J. Algebraic Combin. 11 (3) (2000) 197-210.
[5] J. I. Brown, C. A. Hickman and R. J. Nowakowski, The independence fractal of graph, J. Combin. Theory Ser. B 87 (2) (2003) 209 - 230.
[6] C. A. Hickman, Roots of Chromatic and Independence Polynomials, Thesis (Ph.D.) Dalhousie University (Canada), ProQuest LLC, Ann Arbor, MI, 2001. 103 pp.
[7] C. Hoede and X. L. Li, Clique polynomials and independent set polynomials of graphs, 13th British Combinatorial Conference (Guildford, 1991), Discrete Math. 125 (1-3) (1994) 219 - 228.