Graph Theory
Two vertices v ≠ w are said to be adjacent or neighbors if they are joined by edge vw. Two edges e ≠ f are said to be adjacent is they have an end in common.
Adjacency
The adjacency matrix A of a graph G is the |V(G)|x|V(G)| matrix whose elements a(u,v) count the number of edges that join vertices u and v, with loops counted as two edges.
Adjacency matrix
A graph is bipartite if its vertex set can be divided into disjoint sets U and V such that every edge has an end in U and an end in V, i.e. U and V are both independent sets. We may denote such a graph with bipartition U and V as G(U, V, E), or as G(U, V). A bipartite graph is complete if it is both simple and every vertex in U is adjacent to every vertex in V, and is denoted Kn,m where n = |U| and m = |V|.
Bipartite graph
The distance between two vertices in a graph is the number of edges in a shortest path connecting them. If they are not connected, their distance is defined to be infinite.
Distance
The circumference of a graph is the length of a longest cycle in the graph. The girth of an acyclic graph is defined to be infinity.
Circumference
A graph is said to be complete if all vertices are pairwise adjacent. The unique complete graph on n vertices is denoted by Kⁿ.
Complete graph
Two vertices u and v of an undirected graph G are said to be connected if G contains a path from u to v. G is considered connected if its vertices are pairwise connected. Equivalently, G is connected if for partition of its vertices into two nonempty sets U and V, there is an edge with one end in U and and one in V. A graph which is not connected is disconnected.
Connectivity
A cycle on three or more vertices is a simple graph whose vertices can be arranged in a cyclic sequence that such two vertices are adjacent if they are consecutive in the sequence and nonadjacent otherwise. The length of a cycle is the number of its edges. A cycle of length k is a k-cycle, and is denoted C_k.
Cycle
A graph is an ordered pair G = (V, E) consisting of a set of vertices or nodes V and a set of edges or lines E, which are 2-element subsets of V.
Definition of a graph
The degree of a vertex v, denoted by deg(v), is the number of edges incident to v, with loops counted twice. For a simple graph, deg(v) is the number of adjacent vertices. δ(G), d(G) and Δ(G) denote the minimum, average and maximum degrees of the vertices of G. A graph of order n and size m has average degree 2m/n by the handshaking theorem.
Degree
The degree sequence of a graph G is the sequence of degrees of each vertex of G in non-increasing order.
Degree sequence
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Deletion
The diameter of a graph is the maximum eccentricity of any vertex.
Diameter
There are many tests one can use to show two graphs are not isomorphic. A graph invariant is a property which does not change between graphs that are isomorphic. An isomorphism preserves: - order and size - adjacency - vertex degrees - many other things
Disproving isomorphism
The eccentricity of a vertex v is the maximum distance between v and all other vertices.
Eccentricity
A simple graph is edge-transitive if for any edge uv and xy, there is an automorphism α such that α(u)α(v) = xy.
Edge-transitive graph
A graph is finite if both its edge set and vertex set are finite. Otherwise it is infinite. An infinite graph whose vertices have finite degree is locally finite.
Finiteness
A graph which contains no cycle, i.e. an acyclic graph, is called a forest. A connected forest is called a tree. A vertex of a tree with degree one is called a leaf.
Forests and trees and leaves
The girth of a graph is the length of a shortest cycle in the graph. The girth of an acyclic graph is defined to be infinity.
Girth
An automorphism of a graph G = (V, E) is a permutation µ: V→V of its vertices such that for all x,y ∈ V, xy ∈ E ↔ µ(x)µ(y) ∈ E, that is, µ is an isomorphism from G to itself.
Graph automorphism
Let G be a simple graph. The complement H (or denoted by G with a line on top) of G is the simple graph whose vertex set is V and whose edges are the pairs of nonadjacent vertices of G. Two graphs are isomorphic if and only if their complements are isomorphic. A graph isomorphic to its complement is self-complementary, and must have order n ≡ 0,1 (mod 4).
Graph complement
Let G = (V, E) and G' = (V', E') be two graphs. We call G and G' isomorphic, and write G ≅ G', if there exists a bijection φ: V → V' with xy ∈ E ↔ φ(x)φ(y) ∈ E' for all x,y ∈ V. Such a map φ is called an isomorphism, and if G = G', it is an automorphism. φ preserves vertex adjacency and degree. An isomorphism induces a bijection between edge sets.
Graph isomorphism
∑deg(v) = 2*|E(G)|, for v in V(G).
Handshaking theorem
A vertex v is incident with an edge e if v ∈ e. The vertices incident with an edge are its ends or end points, and an edge is said to join its ends.
Incidence and end point
The incidence matrix M of a graph G is the |V(G)|x|E(G)| matrix whose elements m(v,e) count the number of times vertex v and edge e are incident.
Incidence matrix
Two nonadjacent vertices are independent. An independent set, or stable set, is a set of pairwise independent vertices.
Independence
For some non-empty subset S of V(G), we say the subgraph induced by S is the subgraph of G whose vertex set is S and whose edges are precisely those which join two vertices of S, and is denoted by G[S]. Such a subgraph is unique.
Induced subgraph
A graph that contains parallel edges.
Multigraph
The order of a graph is the cardinality of its vertex set, i.e. |V(G)|.
Order of a graph
A path is a simple graph whose vertices can be arranged in a linear sequence that such two vertices are adjacent if they are consecutive in the sequence and nonadjacent otherwise. The length of a path is the number of edges. A path of length k is called a k-path, and is denoted P_k.
Path
In general, proving that two graphs are isomorphic is tedious. One must either construct an isomorphism or prove one exists. Don't be tempted to think two graphs that share common features are necessarily isomorphic (eg 2 k-regular graphs with the same order).
Proving isomorphism
A pseudograph is a graph that contains a loop and possibly parallel edges.
Pseudograph
The radius of a graph is the minimum eccentricity of any vertex.
Radius
A graph is regular if every vertex has the same degree. A regular graph with vertices of degree k is called a k-regular graph.
Regular graph
A simple graph is a graph with no loops or parallel edges.
Simple graph
The size of a graph is the cardinality of its edge set, i.e. |E(G)|
Size of a graph
A subgraph G of H is a spanning subgraph if V(H) = V(G). A graph of size m has 2^m spanning subgraphs.
Spanning subgraph
If G and H are two graphs with V(G) ⊆ V(H) and E(G) ⊆ E(H), we say G is a subgraph of H and write G ⊆ H.
Subgraph
A graph is trivial if it has order 0 or 1.
Trivial graph
An edge with two identical ends is a loop, while one with distinct ends is a link. Two edges with the same pair of edges are parallel.
Types of edges
For a graph G = (A, B), its vertex is denoted A = V(G), and its edge set by B = E(G).
Vertex and edge set
Two vertices u and v are called similar is there is an automorphism µ such that µ(u) = v. A graph in which every vertex is similar is called vertex-transitive. A graph with no similar vertices is asymmetric.
Vertex-transitive graph
A walk is an alternating sequence of vertices and edges, beginning and ending with a vertex, where each vertex is incident to both the edge that precedes it and the edge that follows it in the sequence, and where the vertices that precede and follow an edge are the end vertices of that edge. A walk is closed if its first and last vertices are the same, and open if they are different. A trail is walk whose edges are distinct. A path is a walk with no repeated vertices and therefore no repeated edges. Any walk from vertex u to v contains a path from u to v.
Walk, trail
The n-dimensional cube, denoted Q_k, has {0,1}-words of length k as vertices which are adjacent if and only if they differ at exactly one position. Q_k is k-regular, with |V(Q_k)| = 2^k and |E(Q_k)| = k*2^(k-1).
n-dimensional cube