# Graphs: Expressing Briefs Spatially

DOI: 10.4018/978-1-4666-4647-6.ch004
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## Abstract

The purpose of this chapter is to introduce the mathematical foundations of graphs and explain their applicability to giving an abstract spatial expression to the brief. As one knows from different kinds of diagrams, visual displays are quite useful for recognizing relationships and summarizing complex information. Graphs allow overview of aspects such as grouping and circulation in a brief. Moreover, they provide connections to designs produced on the basis of the brief, as design representations like floor plans can be directly compared to a requirements graph. The graphs discussed in this chapter consist of spatial entities, for example, spaces in a design or activities in a brief, and relationships between pairs of entities, such as direct access, proximity, or belongingness to the same group. Of the various programs available for developing and working with graphs, one considers in detail Microsoft Visio, a widely available diagramming application.
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## Things And Their Relationships

When you read the word ‘graph,’ especially in conjunction with computers, I guess that the first things that come to mind are pie, bar, line, and other kinds of charts and visual displays routinely produced with spreadsheets. Some may even think of the graphs used to plot the values of variables such as x and y in a mathematical function like f(x) = x3 - 9x on a Cartesian plane. In this chapter we are dealing with an entirely different type of graphs. It is unfortunate that the term ‘graph’ is used so widely but thankfully there can be no confusion between the different types. The graphs in this chapter are fairly simple but strictly structured diagrams consisting of vertices (or nodes) and edges (or lines) that link pairs of vertices (Figure 1).

Figure 1.

A graph with eight vertices and seven edges

Visual displays like these belong to graph theory, an area of mathematics that began in 1736 with Euler’s study of how one could not take a walk in Königsberg that crossed its seven bridges only once (Aldous & Wilson, 2000; Bondy & Murty, 2008; Harary, 1994; Ore, 1996). When it comes to expressing a structure in terms of its parts and their links, there are few representations that work as well as graphs. Even when we know nothing about graph theory we can still use similar diagrams to describe successfully abstract structures such as the organization of a state or company (organization chart), the genealogy of a family (family tree), the routes and stations of a public transport system (topological map), the components and connections of an electrical circuit (circuit diagram) or the structure of a molecule in chemistry (structural formula). Such diagrams are rather familiar and intuitively understandable as visualizations of things and their relationships.

Architects use graph-like abstractions too: Bubble and relationship diagrams are common ways of expressing the overall structure of a brief in terms of its activities and their required connections or proximity (Figure 2). The same diagrams are also used to describe a design in similarly abstract terms, often as a preliminary to a schematic design but also as a schematization of a developed design. Judging from their use as navigation aids instead of schematic floor plans, for example in wayfinding signage for the visually impaired (Whitehouse, 1999), graph-like representations of buildings appear to extend beyond architectural conventions and practices.

Figure 2.

Relationship diagram

The purpose of this chapter is to apply graph theory to such abstract representations and explore their utility in briefing:

• Using graphs we can transform a brief into a spatial structure that gives us a clearer picture of the organization and complexity of the brief.

• This spatial structure can be compared to a design that is abstracted into a similar graph.

• Graphs describing a brief or design allow us to visualize and analyse patterns of use based on spatial relationships, for example pedestrian circulation in a building.

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