networks

degree of vertex

number of edges connected to vertex

sum of degrees

in a graph refers to the total number of edges connected to a vertex.

vertex is even

the degree is even

vertex is odd

the degree is odd

isomorphic

graphs look differet but have the same info. same amount of vertices and edges, same connections between vertices

connected graphs

all vertices are directly or indirectly connected

bridges

an edge in a connected graph that if removed leaves the graph no longer connected

planar graph

can be drawn on a plane without any of its edges crossing. some do not look planar. graph 1 may not look planar but it can be isomorphic to graph 2 which is planar, so graph 1 is also planar

faces of a graph

regions of space which may be enclosed in the connected edges of a graph or the space surrounding it

eulers formula

states that for any connected planar graph, the formula v + f = e + 2 holds, where V is the number of vertices, E is the number of edges, and F is the number of faces.

walk

sequence of edges linking successive vertices that connect two different vertices in a graph

trail

a walk with no repeated edges

path

walk with no repeated edges or vertices

circuit

walk that starts and ends at the same vertex with no repeated edges

cycle

starts and ends at same vertex, no repeated edges and no repeated vertices (except start and end vertices)

eulerian trail

follows every edge of graph with no repeated edges. conditions are that the graph is connected and there are exactly 0 or 2 odd vertices

how can an eulerian trali start?

if there are 0 odd vertices, it can start at any vertex.

if there are 2 odd vertices, trali starts at one odd vertice and ends at the other

eulerian circuit

starts and finished at the same vertex. graph is connected and has vertices that all have an even degree

hamiltonian path

visits every vertex with no repeated vertices

hamiiltonian cycle

visits every vertex with no repeat except starts and ends a tthe same vertex

what does hamiltonian path and circuit not need to do

They do not need to visit every edge.