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.

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

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.

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

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

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

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.

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.

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

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

- 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:

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

So,

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

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

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,

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

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

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.

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.

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).

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

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

#2 #2