Finding the

"Middle" of a Graph

and Why it Matters

Joseph Malkevitch

Mathematics and Computing Dept.

York College (CUNY)

Jamaica, New York 11451








What is the public's

perception of

mathematics and












Typical response:


Mathematics is a sea of problems involving symbols whose meaning I can not understand!




(x2)- 5 = ?


x/(x-3) + (x-4)/(x+7) = ?


X2(y - 4x) - y(3X2 + 3Y) = ?


 3x - 2(5x - 4) = 8 (Solve for x)



Typical response:


Mathematicians are aging males who wear beards and glasses and if they have any hair, it's a mess



Albert Einstein is the

greatest living




Neither the general public nor surprisingly large parts of the scientific community are aware of:

a. The dramatic growth in new mathematical tools.

b. The role of mathematics in the development of many recent technologies.


Medical imaging

Fax and wireless communication

Compact disc and CD-ROM technology

Fuel efficient transportation

Mapping the human genome

c. The emerging role of mathematics in areas outside of the sciences and engineering.




Political science

Applications: elections, legislative apportionment, weighted voting, political games


Applications: bankruptcy, fair allocation, cost sharing, equilibrium prices, efficiency, games


Applications: bargaining, optimization, security, fairness, new financial instruments


Applications: internal cost sharing, financial instruments, generational equity

Fine arts

Applications: fabric design, classification of patterns




Mathematics can be

viewed as a subject which

has concerns with various


in contrast with various




0. Arithmetic

1. Algebra

2. Geometry

3. Trigonometry

4. Calculus

5. Linear (Matrix) Algebra

6. Probability and Statistics

7. Graph Theory

8. Modern Algebra

9. Coding Theory

10. Knot Theory


(Many more!)



1. Optimization

2. Growth and Change

3. Information

4. Fairness and Equity

5. Risk

6. Shape and Space

7. Pattern and Symmetry

8. Order and Disorder

9. Reconstruction

10. Conflict and Cooperation

11. Unintuitive behavior



What is the best

location for a

 hospital in this community?



How would your analysis differ

if the problem involved locating a

firehouse a school, or a super-market?

What difference is there if a facility must

be at a vertex versus located somewhere

along an edge?





* Two points in space

* Insulin molecules in a gorilla and chimpanzee

* Two relatives

* Preference schedules of two voters

* Two images or texts

* Two code words


Distances between two points in space:

Euclidean distance

Taxicab distance

Max distance

Abstract properties of distance:

1. Distance is a non-negative real number.


2. The distance between two points is zero if and only if they are the same.


3. The distance between p and q is the same as the distance between q and p.


4. The triangle inequality: The distance from p to q plus the distance from q to r is at least as large as the distance from p to r.




Eccentricity of a vertex v:

Maximum distance any vertex can be from v

Central vertex:

Vertex with minimum value for its eccentricity




Subgraph induced by the central vertices of a graph

Jordan's Theorem:

The center of a tree is either a single vertex or a pair of vertices joined by an edge.




A tree has one central vertex if and only

if the diameter of the tree is twice its radius.

Numbers are eccentricities of the vertices.


Branch weight:

Numbers show the branch weights of the vertices.