Editing Glossary of graph theory (section)

==E==
{{glossary}}

{{term|E|''E''}}
{{defn|{{math|''E''(''G'')}} is the edge set of {{mvar|G}}; see {{gli|edge set}}.}}

{{term|ear}}
{{defn|An ear of a graph is a path whose endpoints may coincide but in which otherwise there are no repetitions of vertices or edges.}}

{{term|ear decomposition}}
{{defn|An [[ear decomposition]] is a partition of the edges of a graph into a sequence of ears, each of whose endpoints (after the first one) belong to a previous ear and each of whose interior points do not belong to any previous ear. An open ear is a simple path (an ear without repeated vertices), and an open ear decomposition is an ear decomposition in which each ear after the first is open; a graph has an open ear decomposition if and only if it is biconnected. An ear is odd if it has an odd number of edges, and an odd ear decomposition is an ear decomposition in which each ear is odd; a graph has an odd ear decomposition if and only if it is factor-critical.}}

{{term|eccentricity}}
{{defn|The eccentricity of a vertex is the farthest distance from it to any other vertex.}}

{{term|edge}}
{{defn|An edge is (together with vertices) one of the two basic units out of which graphs are constructed. Each edge has two (or in hypergraphs, more) vertices to which it is attached, called its endpoints. Edges may be directed or undirected; undirected edges are also called lines and directed edges are also called arcs or arrows. In an undirected [[simple graph]], an edge may be represented as the set of its vertices, and in a directed simple graph it may be represented as an ordered pair of its vertices. An edge that connects vertices {{mvar|x}} and {{mvar|y}} is sometimes written {{mvar|xy}}.}}

{{term|edge cut}}
{{defn|A set of {{gli|edge}}s whose removal {{gli|disconnected|disconnects}} the {{gli|graph}}. A one-edge cut is called a {{gli|bridge}}, {{gli|isthmus}}, or {{gli|cut edge}}.}}

{{term|edge set}}
{{defn|The set of edges of a given graph {{mvar|G}}, sometimes denoted by {{math|''E''(''G'')}}.}}

{{term|edgeless graph}}
{{defn|The [[edgeless graph]] or totally disconnected graph on a given set of vertices is the graph that has no edges. It is sometimes called the empty graph, but this term can also refer to a graph with no vertices.}}

{{term|embedding}}
{{defn|A [[graph embedding]] is a topological representation of a graph as a subset of a topological space with each vertex represented as a point, each edge represented as a curve having the endpoints of the edge as endpoints of the curve, and no other intersections between vertices or edges. A [[planar graph]] is a graph that has such an embedding onto the Euclidean plane, and a [[toroidal graph]] is a graph that has such an embedding onto a torus. The [[Genus (mathematics)|genus]] of a graph is the minimum possible genus of a two-dimensional [[manifold]] onto which it can be embedded.}}

{{term|empty graph}}
{{defn|no=1|An [[edgeless graph]] on a nonempty set of vertices.}}
{{defn|no=2|The [[order-zero graph]], a graph with no vertices and no edges.}}

{{term|end}}
{{defn|An [[End (graph theory)|end]] of an infinite graph is an equivalence class of rays, where two rays are equivalent if there is a third ray that includes infinitely many vertices from both of them.}}

{{term|endpoint}}
{{defn|One of the two vertices joined by a given edge, or one of the first or last vertex of a walk, trail or path. The first endpoint of a given directed edge is called the ''tail'' and the second endpoint is called the ''head''.}}

{{term|enumeration}}
{{defn|[[Graph enumeration]] is the problem of counting the graphs in a given class of graphs, as a function of their order. More generally, enumeration problems can refer either to problems of counting a certain class of combinatorial objects (such as cliques, independent sets, colorings, or spanning trees), or of algorithmically listing all such objects.}}

{{term|Eulerian}}
{{defn|An [[Eulerian path]] is a walk that uses every edge of a graph exactly once. An Eulerian circuit (also called an Eulerian cycle or an Euler tour) is a closed walk that uses every edge exactly once. An Eulerian graph is a graph that has an Eulerian circuit. For an undirected graph, this means that the graph is connected and every vertex has even degree. For a directed graph, this means that the graph is strongly connected and every vertex has in-degree equal to the out-degree. In some cases, the connectivity requirement is loosened, and a graph meeting only the degree requirements is called Eulerian.}}

{{term|even}}
{{defn|Divisible by two; for instance, an even cycle is a cycle whose length is even.}}

{{term|expander}}
{{defn|An [[expander graph]] is a graph whose edge expansion, vertex expansion, or spectral expansion is bounded away from zero.}}

{{term|expansion}}
{{defn|no=1|The edge expansion, isoperimetric number, or [[Cheeger constant (graph theory)|Cheeger constant]] of a graph {{mvar|G}} is the minimum ratio, over subsets {{mvar|S}} of at most half of the vertices of {{mvar|G}}, of the number of edges leaving {{mvar|S}} to the number of vertices in {{mvar|S}}.}}
{{defn|no=2|The vertex expansion, vertex isoperimetric number, or magnification of a graph {{mvar|G}} is the minimum ratio, over subsets {{mvar|S}} of at most half of the vertices of {{mvar|G}}, of the number of vertices outside but adjacent to {{mvar|S}} to the number of vertices in {{mvar|S}}.}}
{{defn|no=3|The unique neighbor expansion of a graph {{mvar|G}} is the minimum ratio, over subsets of at most half of the vertices of {{mvar|G}}, of the number of vertices outside {{mvar|S}} but adjacent to a unique vertex in {{mvar|S}} to the number of vertices in {{mvar|S}}.}}
{{defn|no=4|The spectral expansion of a {{mvar|d}}-regular graph {{mvar|G}} is the [[spectral gap]] between the largest eigenvalue {{mvar|d}} of its adjacency matrix and the second-largest eigenvalue.}}
{{defn|no=5|A family of graphs has [[bounded expansion]] if all its {{mvar|r}}-shallow minors have a ratio of edges to vertices bounded by a function of {{mvar|r}}, and polynomial expansion if the function of {{mvar|r}} is a polynomial.}}

{{glossary end}}