graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

, the outer boundary of a subset of the vertices of a

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

is the set of vertices in that are adjacent to vertices in , but not in themselves. The inner boundary is the set of vertices in that have a neighbor outside . The edge boundary is the set of edges with one endpoint in the inner boundary and one endpoint in the outer boundary. These boundaries and their sizes are particularly relevant for isoperimetric problems in graphs, separator theorems,

minimum cut In graph theory, a minimum cut or min-cut of a graph is a cut (a partition of the vertices of a graph into two disjoint subsets) that is minimal in some metric. Variations of the minimum cut problem consider weighted graphs, directed graphs, ter ...

expander graph In graph theory, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion. Expander constructions have spawned research in pure and applied mathematics, with several appli ...

s, and

percolation theory In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type of phase transition, since at a critical fraction of addition the network of small, disconnected ...

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Graph theory objects {{graph-stub