so then...

Preface- the symbol є means

*is an element of*

A

__graph__is a pair of sets (V,E) such that E={{u,v}: u,v є V}

An element of V is a

__vertex__(V is the "groundset"), an element of E is an edge.

[Vertices are connected by edges, thereby making a graph, fairly simple.]

If e={u,v} є E(G) then: (1.) e

__joins__u and v, (2.) e is

__incident__with u (or v), and (3.) u is

__adjacent__to v.

The

__degree of a vertex__, denoted as deg(v), is the number of vertices that are adjacent to v. An isolated vertex has degree zero.

The

__neighborhood of a vertex__is the set of vertices adjacent to it. This is denoted as N(v). Please note that deg(v)=|N(v)|, that is, the degree of some vertex v is equal to the cardinality of v's neighborhood.

The graph of G' is a

__subgraph__of G if V(G') is a subset of V(G) and E(G') is a subest of E(G). However, if V(G') and E(G') do not form an independent graph, then G' is not a subgraph of G.

A

__walk__in G, from u to v, is a sequence of adjacent vertices starting with u and ending with v.

A

__trail__in G is a walk that repeats no edge.

A

__path__in G is a walk that repeats no vertex.

A

__closed walk__is a walk with at least one edge, and that starts and ends at the same vertex.

A

__circuit__is a closed walk that repeats no edge. (closed trail)

A

__cycle__is a closed walk that repeats no vertex (except the first and last, since they're the same). (closed path)

A graph is

__connected__if there is a walk between any two vertices.

A

__component__of a graph is a maximal connected subgraph.

That's our lesson in graph theory today, and if you actually read that, then you are a bigger loser than I am.

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