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  2. Jan 2, 2017 · A non-trivial connected graph is any connected graph that isn't this graph. A non-trivial connected component is a connected component that isn't the trivial graph, which is another way of say that it isn't an isolated point.

    • Finite Graphs. A graph is said to be finite if it has a finite number of vertices and a finite number of edges. A finite graph is a graph with a finite number of vertices and edges.
    • Infinite Graph: A graph is said to be infinite if it has an infinite number of vertices as well as an infinite number of edges.
    • Trivial Graph: A graph is said to be trivial if a finite graph contains only one vertex and no edge. A trivial graph is a graph with only one vertex and no edges.
    • Simple Graph: A simple graph is a graph that does not contain more than one edge between the pair of vertices. A simple railway track connecting different cities is an example of a simple graph.
  3. The graph Gis non-trivial if it contains at least one edge, i.e., E 6= ;. non-trivial Equivalently, Gis non-trivial if Gis not an empty graph. The order of G, denoted by jGj, is the number of vertices of G, i.e., jGj= jVj. order, jGj

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  4. Feb 18, 2022 · If G G contains vertices v,v′ v, v ′ and edge e = {v,v′}, e = {v, v ′}, then v, e,v′, e, v v, e, v ′, e, v is a nontrivial cycle which is not proper. Example 16.2.1 16.2. 1: A forest of trees. The graph in Figure 16.2.1 16.2. 1 is acyclic. Each of its connected components is a tree.

  5. A graph G is bipartite if it is the trivial graph or if its vertex set can be partitioned into two independent, non-empty sets A and B. We refer to { A, B } as a bipartiton of V(G). Note: Some people require a bipartite graph to be non-trivial. Examples include any even cycle, any tree, and the graph below.

  6. www.math.kit.edu · iag6 · lehreGraph Theory - KIT

    Several examples of graphs and their corresponding pictures follow: V = [5], E= f12;13;24g V = fA;B;C;D;Eg, E= fAB;AC;AD;AE;CEg De nition 1.2 (Graph variants). A directed graph is a pair G= (V;A) where V is a nite set and A V2. The directed graph edges of a directed graph are also called arcs. arc

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  8. A non-trivial graph consists of one or more vertices (or nodes) connected by edges. Each edge connects exactly two vertices, although any given vertex need not be connected by an edge. The degree of a vertex is the number of edges connected to that vertex. In the graph below, vertex \( A \) is of degree 3, while vertices \( B \) and \( C \) are ...

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