1. The isoperimetric constant of a Generalized Paley graph
Tony Shaheen
Spencer Johnson
Gustavo Subuyuj
This work was funded by the CSU Alliance PUMP Program (Preparing
Undergraduates through Mentoring towards PhDs). In this talk, we extend
the work that was done by Kevin Cramer, Nicole Shabazi, Anthony Shaheen, and
Mike Krebs on the usual Paley graphs.

2. Outline of talk:
Definition of generalized Paley graphs
Isoperimetric (expansion) constant of a graph
Bounds on the isoperimetric constant
of a generalized Paley graph

3. What is a generalized Paley Graph?
Let p be an odd prime,
, d = gcd(p-1,m).
Suppose further that d divides (p-1) / 2.
Construct the set
The vertices of the Generalized Paley graph
are the elements of
Two vertices v and w are adjacent iff
Notes:
The condition that d divides (p-1)/2 ensures that the gamma set is symmetric.
2) Here we are generalizing a definition from A.N. Elsawy, Paley graphs and their
Generalizations, Master’s Thesis. Elsawy only considers odd m.

4. m = 2, p = 13, d = gcd(p-1,m) = 2, (p-1)/2 = 6
When m = 2 and p is 1 mod 4, we have the familiar Paley graph.
Example:
m = 2, p = 13, d = gcd(p-1,m) = 2, (p-1)/2 = 6

5. Example:
m = 6, p = 17, d = gcd(p-1,m) = 2, (p-1)/2 = 8

6. Some Properties of Generalized Paley Graphs
1) is (p-1)/d regular
2) is connected
3) is the complete graph iff d = 1
4) is the cycle graph iff d = (p-1)/2

7. Some properties continued....
5) Let and
. Suppose that both
divide (p-1)/2. Then
and
are isomorphic iff
6) Given a fixed odd prime p, a complete list of
non-isomorphic generalized Paley graphs of size p is
given by the graphs where m ranges over the
divisors of (p-1)/2.

8. Let F be a subset of
vertices of a graph.
The boundary of F,
denoted by
consists of the set
of edges from F to
its complement.
For example, in the
graph to the left, let
F denote the pink
vertices. Then the
boundary of F consists
of the green edges.

9. Let X be a graph with vertex set V
Let X be a graph with vertex set V. The isoperimetric constant (or expansion constant) of X, denoted h(X), is defined by

10. Therefore, for any subset F (of at most
Let X be a graph with vertex set V. The isoperimetric constant (or expansion constant) of X, denoted h(X), is defined by
Therefore, for any subset F (of at most
half the size of the vertex set V) we
have that

11. - Bound: is a symmetric set so it can be written as
where k = (p-1)/2d and
The isoperimetric constant of a generalized Paley graph
satisfies the following bound:

12. Example:
m = 6, p = 17, d = gcd(p-1,m) = 2, (p-1)/2 = 8

13. So,

14. Bound #2: , d = gcd(p-1,m). Let p be an odd prime,
Suppose further that d divides (p-1)/2 and that d > 1. Then:
where
is the number of triples of rational numbers
satisfying

15. Bound #2: , d = gcd(p-1,m). Let p be an odd prime,
Suppose further that d divides (p-1)/2 and that d > 1. Then:
where
is the number of triples of rational numbers
satisfying
Note: The lower bound is derived using an estimate on the eigenvalues of the graph (which
are essentially Gauss sums. The upper bound is derived by estimating the number of
solutions to the equation

16. Example: Standard Paley graphs Let m = 2 and
. Then d = gcd(p-1,2) = 2.
The integer
is the number of triples of rational numbers
satisfying the conditions:
There are no such solutions. Thus,

17. Example: Standard Paley graphs Let m = 2 and
. Then d = gcd(p-1,2) = 2.
The integer
is the number of triples of rational numbers
satisfying the conditions:
There are no such solutions. Thus, So
becomes

18. Example: Standard Paley graphs Let m = 2 and
. Then d = gcd(p-1,2) = 2.
The integer
is the number of triples of rational numbers
satisfying the conditions:
There are no such solutions. Thus, So
becomes

19. Example: Standard Paley graphs Let m = 2 and
. Then d = gcd(p-1,2) = 2.
The integer
is the number of triples of rational numbers
satisfying the conditions:
There are no such solutions. Thus, So
becomes
Note: One can actually get (p-1)/4 for the upper bound in this case.

20. Example: Let m = 3 and p be an odd prime.
We cannot have p equal to 2 modulo 3, for then d would
equal 1. So suppose that
Then d = gcd(p-1,m) = 3.

21. Example: Let m = 3 and p be an odd prime.
We cannot have p equal to 2 modulo 3, for then d would
equal 1. So suppose that
Then d = gcd(p-1,m) = 3.
The integer
is the number of triples of rational numbers
satisfying the conditions:
There are 2 such solutions: (1/3, 1/3, 1/3) and (2/3, 2/3, 2/3).

22. Example: Let m = 3 and p be an odd prime.
We cannot have p equal to 2 modulo 3, for then d would
equal 1. So suppose that
Then d = gcd(p-1,m) = 3.
The integer
is the number of triples of rational numbers
satisfying the conditions:
There are 2 such solutions: (1/3, 1/3, 1/3) and (2/3, 2/3, 2/3). So
becomes

23. Example: Let m = 3, p an odd prime with

24. #2 #2