1. Graph Representations and their Applications (formal description)
Dr. Offer Shai
Tel-Aviv University, Israel

2. Suggested approach employs discrete mathematical models called combinatorial representations that are based on graph theory, matroid theory and discrete linear programming. In this lecture I will introduce only the graph representations.
Graph representation gk includes the knowledge of graph theory and is augmented with additional mathematical properties.
Graph representations are classified to different types – G1,G2…Gn according to the different sets of augmented mathematical properties (gk  Gl). Gl gk

3. The essence of our approach is to transform engineering system si belonging to engineering domain Dj (si Dj) into graph representation gk, so that the topology and the physical laws of si correspond to the topology and the mathematical properties of gk.
gk = T(si)
topology(gk)=T(topology(sj))
properties(gk)=T(laws(sj)) Gl Dj T gk si

4. Solving Engineering Problems
This transformation is used to substitute the physical problems related with engineering system si with mathematical problem related with gk
T(problem(si)) = problem(gk)
solution(problem(si))=T(solution(problem(gk)) Gl Dj
T-1 T
solution(problem(gk))
problem(gk)
solution(problem(si))
problem(si)

5. Solving Engineering Problems
This transformation is used to substitute the physical problems related with engineering system si with mathematical problem related with gk
T(problem(si)) = problem(gk)
solution(problem(si))=T(solution(problem(gk)) Gl
Domain of trusses T
truss1
graph1
problem(g1)
problem(truss1)

6. Transforming knowledge to other engineering fields.
It was found that the same type of graph representations, say Gl can be associated with more than one engineering domain, say Da and Db
Then for each engineering system siDa we can construct a system si’ DB so that
T(si)=gi  Gl =T(s’i) Db Gl Da T’ T
s’i gi si

7. Transforming knowledge to other engineering fields.
Thus one can devise a new piece of knowledge ka in Da, such as device, theorem or method, by transforming already existing knowledge kb from Db first to Gj and then to Da.
kg=T’(kb); ka=T-1(kg) ka=T-1(T’(kb)) 2 3 6 5 4 1 C D B A Db Gl Da T’
T-1 kb kg ka

8. Duality between engineering systems
Since graph representations are mathematical entities, mathematical relations can be established between them, such as duality relations:
(giG1)=Tdual(gj G2) G1 G2
Tdual gj gi

9. Duality between engineering systems
Duality between graph representations yield the duality relations between the represented engineering systems. G1 G2 Db Da D T’ gj gi sj T si

10. Transforming Knowledge
The relations between engineering fields open channels for transforming engineering knowledge.

11. Transforming Knowledge
A truss comprises knowledge including its physical properties, such as forces, statical laws, theorems and more…. sj Db

12. Transforming Knowledge
All the knowledge of the truss is transformed into the mathematical knowledge of the graph representation gj , including graph theory laws, flows and more … sj Db G2 gj T’

13. Transforming Knowledge
All the knowledge of the graph representation is transformed into the knowledge in the dual graph representation gi , including graph theory laws, potentials, potential differences and more… G2 gj G1 gi D

14. Transforming Knowledge
From the knowledge of the dual graph representation we can construct a new engineering system – mechanism, whose knowledge includes physical properties, such as velocities, kinematical laws, theorems and more … Da si G1 gi T

15. Transforming Knowledge