GRAPH THEORY
Graph G VE with n vertices V and m edges E
U is a finite set of points V Eu v2 Un
E is a set of pairs of two distinct points
Nl n and I El M
example G 1,233 1,23 22,33 91,33
I
Twa veces 4 v are adjecent if there is an edge e 4 v te
We also that u and v are incident toe and e is incident
say
to u and v as well 4 u 4 v adjecent
Two edges that share a vertex are also adjecent.ae
e f adjecent
Degree of VEV edges incident tov
example
j d1 3 9121 1 9131 2 d 47 2
the sum of
all degrees 2 edges am
Average degree is therefore Â
A graph is regular if all vertices have the same degree
If all vertices have degree k the graph is k regular
example a regulargraphon 4 vertices
j
A k regular graph on n vertices has Ikn nsk
edges if kn is even
,A graph G Vie is complete if each pair ofpoints is adjecent
A complete graph on n points is denoted by kn
there is an edge between vertices
every pair of
example Ku
j
A graph G is bipartite if V can be split into 2 sets V1 V2
such that there are only edges from vertices in V1 to V2 and
vice versa If every vertex in V1
is connected to every vertexin V2
the graph is complete bipartite In that case kV UV2 has edge
set E E V1V22luieV1 VrtV2
A complete bipartite graph with IVil m and Ihlen is
km. ame
Öi
bipartitegraph
ii
graphnu
complete bipartite
Ï
not a bipartitegraph
A walk in a graph G v e is a sequence of vertices vo.vn Vk
such that Vii and Vi are connected for all El k
The length of a walk is dended k
b
are all distinctthe walk is a path
g p j Walk is e9 2,53,42,3
zit path is eg 2,5 3,4
, A graph is connected if there is a path between any two
of its vertices
example riff NÄÄÄ
connededgraph onedeagraph
A graph G V E is a subgraph of G vie if
U EV E
JE
is a subset of
Subgraph is a graph within a largergraph
example j
G E
Ili
G
component of G is the maximal connected subgraph G IV E
there are no edges we could add to this subgraph while preserving
connectiveness
example
Ï Ï II
subgraph G of G maxima connected
IIIIergmp nen µ component subgraph of G
Thm A graph is connected it exists of exactly 1 component
So any disconnected graph consists of at least a components
A Walk VoUn Vkl is a closed walk cycle if Vo vr
A cycle with all vertices distinct is a circuit
j
example walk is e9 2,53,423,27
circuit is eg 2,53,21
Graph G VE with n vertices V and m edges E
U is a finite set of points V Eu v2 Un
E is a set of pairs of two distinct points
Nl n and I El M
example G 1,233 1,23 22,33 91,33
I
Twa veces 4 v are adjecent if there is an edge e 4 v te
We also that u and v are incident toe and e is incident
say
to u and v as well 4 u 4 v adjecent
Two edges that share a vertex are also adjecent.ae
e f adjecent
Degree of VEV edges incident tov
example
j d1 3 9121 1 9131 2 d 47 2
the sum of
all degrees 2 edges am
Average degree is therefore Â
A graph is regular if all vertices have the same degree
If all vertices have degree k the graph is k regular
example a regulargraphon 4 vertices
j
A k regular graph on n vertices has Ikn nsk
edges if kn is even
,A graph G Vie is complete if each pair ofpoints is adjecent
A complete graph on n points is denoted by kn
there is an edge between vertices
every pair of
example Ku
j
A graph G is bipartite if V can be split into 2 sets V1 V2
such that there are only edges from vertices in V1 to V2 and
vice versa If every vertex in V1
is connected to every vertexin V2
the graph is complete bipartite In that case kV UV2 has edge
set E E V1V22luieV1 VrtV2
A complete bipartite graph with IVil m and Ihlen is
km. ame
Öi
bipartitegraph
ii
graphnu
complete bipartite
Ï
not a bipartitegraph
A walk in a graph G v e is a sequence of vertices vo.vn Vk
such that Vii and Vi are connected for all El k
The length of a walk is dended k
b
are all distinctthe walk is a path
g p j Walk is e9 2,53,42,3
zit path is eg 2,5 3,4
, A graph is connected if there is a path between any two
of its vertices
example riff NÄÄÄ
connededgraph onedeagraph
A graph G V E is a subgraph of G vie if
U EV E
JE
is a subset of
Subgraph is a graph within a largergraph
example j
G E
Ili
G
component of G is the maximal connected subgraph G IV E
there are no edges we could add to this subgraph while preserving
connectiveness
example
Ï Ï II
subgraph G of G maxima connected
IIIIergmp nen µ component subgraph of G
Thm A graph is connected it exists of exactly 1 component
So any disconnected graph consists of at least a components
A Walk VoUn Vkl is a closed walk cycle if Vo vr
A cycle with all vertices distinct is a circuit
j
example walk is e9 2,53,423,27
circuit is eg 2,53,21
