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Graph Theory
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Gravity
For Chapter 6 from: https://www.coconino.edu/open-source-textbooks#college-mathematics-for-everyday-life-by-inigo-jameson-kozak-lanzetta-and-sonier
Key Concepts:
Terms in this set (17)
vertex
dots on a graph (does not specifically require any edges to be connected)
edge
lines that connect one or more vertices
graph
a picture of vertices and edges
loop
an edge that starts and ends at the same vertex
multiple edges
two or more edges connecting the same vertices
connected graph
a graph such that there is a path going from any one vertex to all the other vertices
disconnected graph
a graph where at least one vertex does not have a path to other vertices in the graph
order of a network
the number of vertices in the entire network (or graph)
adjacent vertices
two vertices connected by an edge
degree of a vertex
the number of edges that meet at a vertex
path
a sequence of vertices with each vertex adjacent to the next one that starts and ends at different vertices and travels over any edge only once
circuit
a path that starts and ends at the same vertex
bridge
an edge such that if it were removed from a connected graph, the graph would become disconnected
network
a connection of vertices through edges
spanning subgraph
a graph that joins all of the vertices of a more complex graph, but does not create a circuit
tree
a graph that is connected and has no circuits
Properties of Trees
1. If a graph is a tree, there is one and only one path joining any two vertices. Conversely, if there is one and only one path joining any two vertices of a graph, the graph must be a tree.
2. In a tree, every edge is a bridge. Conversely, if every edge of a connected graph is a bridge, then the graph must be a tree.
3. A tree with N vertices must have N-1 edges.
4. A connected graph with N vertices and N-1 edges must be a tree.
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