Graph Theory Notes Part 1

Recently, I have started learning graph theory, here are a list of terminologies that are commonly used in graph theory,

  • A graph G=(V,E)G=(V,E) is a collection VV of vertices or nodes and EV×VE\subset V \times V of edges.

  • Two vertices are adjacent if there is an edge that has them as endpoints. An edge and a vertex are said to be incident if the vertex is an endpoint of the edge.

  • If UU is a subset of the vertices, then the induced subgraph G[U]G[U] is the graph obtained by deleting all vertices outside UU, keeping only edges with both endpoints in UU.

  • The complement G\overline {G} of GG is a graph with the same vertex set as GG and E(G)={eE(G)}E(\overline{G})=\{e \notin E(G)\}.

  • A graph is bipartite if the vertex set can be partitioned into two non-empty disjoint sets V1V2V_1 \cup V_2 such that edges only run between V1V_1 and V2V_2 or there are no edge that has both endpoints in the set.

  • A graph is complete bipartite if GG is bipartite and all possible edges between the two sets V1,V2V_1, V_2 are drawn. In the case where V1=m,V2=n,|V_1|=m, |V_2|=n, such a graph is denoted by Km,nK_{m,n}.

  • Let k2.k\geq 2. A graph GG is said to be kk-partite if the vertex set can be partitioned into kk pairwise disjoint sets V1,V2,,VkV_1, V_2, \cdots , V_k such that no edge has both endpoints in the same set.

  • A complete k-partite graph is defined similarly as a complete bipartite. In the case where Ai=ni|A_i|=n_i, such a graph is denoted by Kn1,n2,,nkK_{n_1,n_2,\cdots ,n_k}. (Note that a 2-partite graph is simply a bipartite graph.)

  • An edge whose endpoints are the same is called a loop.

  • A graph where there is more than one edge joining a pair of vertices is called a multigraph.

  • A graph without loops and is not a multigraph is said to be simple.

  • A graph is planar if it is possible to draw it in the plane without crossing any edges.

  • The chromatic number of a graph is the minimum number of colours needed to colour the vertices without having the same colour for any two adjacent vertices.

  • A clique or complete graph on nn vertices, denoted KnK_n, is the nn-vertex graph with all (n2)\binom n2 possible edges.

  • A path is a sequence of distinct, pairwise-adjacent vertices. (A graph itself can also be called a path.) The length of a path is defined to be the number of edges in the path.

  • A walk is a sequence of not-necessarily-distinct, pairwise-adjacent vertices.

  • A cycle is a path for which the first and last vertices are adjacent. (A graph itself can also be called a cycle.) The length of a cycle is defined to be the number of vertices (or edges) in the path.

  • The degree d(v)d(v) of a vertex vv is the number of edges that are incident to vv.

  • The distance between two vertices u,vu,v in a graph is defined to be the length of the shortest path joining u,vu,v. (In the case the graph is disconnected, this may not be well-defined.)

  • A graph is connected if for any pair of vertices, there exists a path joining the two vertices. Otherwise, a graph is disconnected.

  • A tree is a connected graph with no cycles.

  • A forest is a not-necessarily-connected graph with no cycles.

Note by ChengYiin Ong
4 weeks, 1 day ago

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