Determining Conditions for the Edge Isoperimetric Inequality for Graphs in Z2

The edge isoperimetric inequality problem is presented along with a programming and graphical approach to analyzing properties of the shapes of vertex sets that fulfill the condition of minimal edge boundary for a given vertex set size. Background Graph Theory For those who are unfamiliar with graph theory, a brief overview should suffice. A graph G(V,E) is defined as a combination of a set of vertices V (represented as points) and a set of edges E (connectors between these points), specifically some multiset of the subsets of V. Figure 1 Figure 1 above is an example of a graph in which V = {A, B, C, D, E} and E = {(A,B),(B,C), (A,C), (A,D), (D,E)}. The Edge Isoperimetric Inequality Now most of us are familiar with the problem of finding the maximal area that can be circumscribed using a perimeter of fixed length. The solution can be proven using calculus to be that of a circle. Likewise, in a metric space, the solution of minimal volume is that of the Euclidean ball. (A metric space (X, d) is a set X, along with a function d:X*X  R+ that acts as a distance function.) All other areas circumscribed by this fixed length is then less than or equal to the area of this circle, hence the inequality. An equivalent problem is that of finding the minimal length for a given fixed area. Victoria Wang Mentor: Prof. Ellen Veomett SMC SRP ‘12 9/17/2012 This leads us into the edge isoperimetric inequality, which similarly, attempts to find a minimal edge boundary for a given vertex set of fixed size. The edge boundary of a given set is given as e(A) = {(x,y) є E:| A ⋂ {x,y}| =1}. In other words, all of the edges that connect from the given vertex set to the surrounding vertices on a grid that are “one away” are part of the edge boundary. This includes vertices that are along a diagonal or to the left, right, above, or below a vertex in your set. In my research we examined the properties of the shapes of a vertex set with minimal edges for a given number of vertices in Z2. For example, let’s us take seven as the size of the vertex set that we are attempting to find a minimal edge boundary for. In Figure 2 below, we are taking the vertex set in blue to be one example of the potential vertex set shapes of size 7. Figure 3 presents the same shape, with the edge boundary of the vertex set in red.