"Middle" of a Graph
and Why it Matters
Mathematics and Computing Dept.
York College (CUNY)
Jamaica, New York 11451
What is the public's
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)
Mathematicians are aging males who wear beards and glasses and if they have any hair, it's a mess
Albert Einstein is the
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.
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 scienceApplications: elections, legislative apportionment, weighted voting, political games
EconomicsApplications: bankruptcy, fair allocation, cost sharing, equilibrium prices, efficiency, games
BusinessApplications: bargaining, optimization, security, fairness, new financial instruments
AccountingApplications: internal cost sharing, financial instruments, generational equity
Fine artsApplications: fabric design, classification of patterns
Mathematics can be
viewed as a subject which
has concerns with various
in contrast with various
5. Linear (Matrix) Algebra
6. Probability and Statistics
7. Graph Theory
8. Modern Algebra
9. Coding Theory
10. Knot Theory
2. Growth and Change
4. Fairness and Equity
6. Shape and Space
7. Pattern and Symmetry
8. Order and Disorder
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:
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
Vertex with minimum value for its eccentricity
Subgraph induced by the central vertices of a graph
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.
Numbers show the branch weights of the vertices.