Graph theory

GRAPH THEORY
Some Important definitions
• Electrical network-A network is an interconnection of passive
elements(R,L,C) and active elements (voltage source, current source).
• Node: Terminal common to two or more elements is called a node.
• Branch: Line replacing the network element in a graph. Each branch
joins two distinct nodes.
• Graph: A collection of nodes and branches. It shows the geometrical
interconnection of the elements of a network.
- Directed: A graph whose branches are oriented.
- Undirected: A graph whose branches are not oriented.
• Rank of a Graph: It is (n-1), where n is the number of nodes or vertices
of the graph.
• Subgraph: It is a subset of the nodes and the branches of the graph.
The subset is proper if it does not contain all the branches and nodes of
the graph.

• Path: It is a particular (improper) subgraph consisting of an ordered
sequence of branches having the following properties: -
- At the terminating nodes, only one branch is incident
- At the remaining nodes, called the internal nodes, two branches are
incident.
• Tree: A tree is a connected subgraph of a connected graph containing
all the nodes of the graph but containing no loops, i.e., there is a unique
path between every pair of nodes.
- A tree is also defined as any set of branches in the original graph that is
just sufficient to connect all the nodes.
- There branches are (n-1)
• Twig: The branches of the tree are called twigs.
• Link or Chord: Those branches of the graph which are not in the tree.
• Co-tree: All the links of a tree together constitute complement of the
tree and is called co-tree, in which the number of branches are: b-(n-1)
where b is the number of branches of the graph.
Number of twigs: nt= n-1;
Number of links: nl= b-nt = b-n+1
Branches B=(7)
Nodes N=(5)
Twigs (t)=(4)
Links (l)=(3)
----------- Links: Branches of co-tree[2,5,6]
Twigs Branches of tree[1,3,4,7]

Properties of Trees
• Each tree has (n-1) branches.
• The rank of a tree is (n-1). This is also the rank of the graph to which the
tree belongs.
• Tree has all the nodes. It has no closed loops.
• The number of terminal nodes or end vertices of every tree are two.
• Every connected graph has atleast one tree.
• A connected graph’s subgraph is a tree if there exists only one path
between any pair of nodes in it.
Incidence Matrix [A]
• The incident matrix translates the graphical data of a network into
algebraic form. • For a graph with n nodes and b branches, the complete
incidence matrix AC is an n*b matrix whose elements are defined by:
aij = 1; if branch j leaves node i
-1; if branch j enters node i
0; if branch j is not incident with node i.

Properties of Complete Incidence Matrix
• The rank of a complete incidence matrix is (n-1).
• Determinant of the complete incidence matrix is always zero.
• The sum of the entries in any column is zero.
• If one complete row is removed from a complete incidence matrix, it
results into a reduced incidence matrix [Ar].
• The total number of possible trees of any graph= determinant of [A
ATranspose].
Drawing graph from Complete Incidence Matrix-
(a)Given Matrix-
(b) Numbering of rows and branches
(c) Mark 3 nodes n draw the graph-

Drawing graph from Reduced Incidence Matrix-
a) Given reduced incidence matrix-
b) Make complete incidence matrix-
c) Make the graph with 4 nodes-
Question. explain Tie-Set Matrix with example?
Ans. A fundamental loop is a closed path of a given graph with only one
Link and rest of them as twigs. The number of fundamental loops for any
given graph = b - (n - 1) = number of Links These fundamental loop

currents are called Tie set currents and the orientation of the tie set
currents governed by the link.
Example-
Here bde forms a tree
b,d,e → Twigs
a,c,f → Links
Number of fundamental loops = 6 – 4 + 1 = 3
Fundamental loop 1 is a,b,e with b and e as twigs and a as Link.
i1 is Tie set current and the direction as same as link ‘a’
Similarly, loop2 → b,c,d → i2
loop3 → a,e,f → i3
Tie set matrix-
It gives the relation between tie set currents and branch currents. The
rows of a matrix represent the tie – set currents. The columns of a matrix
represent branches of the graph.
The order of the tie set matrix is (b – n + 1) × b.
The elements of tie set matrix

If jth branch currents is incident at ith tie set current at oriented
in same direction.
= –1, if jth branch current is incident at ith tie set current at oriented in
opposite direction.
= 0, If jth branch current is not incident with ith tie set current.
Example-
For the network graph construct the tie set matrix and write the equations
by considering branches a, b, c as the tree branches.
Solution: The tree from the given graph is
a, b, c → twigs
d, e → Links
Fundamental loop 1 → a d c → i1
Fundamental loop 2 → a b c e → i2
Tie set matrix is

Let ja, jb, jc, jd, je are branch currents,
then ja = i1 + i2
jb = i2
jc = –i1 – i2
jd = i1
je = i2
Let va, vb, vc, vd, ve are branch voltages, then
va – vc + vd = 0
va + vb – vc + ve = 0
Question.Explain fundamentals of cut sets and cut set matrix with
example?
Ans.Fundamental cut set is a cut through a given graph which divides into
two parts but in its path of cutting it should encounter only one twig.
The path of cut set forms a voltage line, it is called as cut set voltage. The
orientation of this cut–set voltage is given by the twig governing it.
The number if cut set for any graph = n – 1 = number of twigs.

here b, d, e forms as tree
b, d, e → twigs
a,c, f → Links Number of fundamental cut sets = 4 – 1 = 3
Fundamental cut set 1 is a, b, c with b as twig and a, c, as links
e1 is cut set voltage and the direction is same as twig ‘b’.
Similarly cut set 2 → c, d, f → e2
cut set 3 → a, e, f → e3
Cut Set Matrix-
It gives the relation between cut set voltages and branch voltages The rows
of a matrix represent the cut set voltages. The columns of a matrix
represent the branches of the graph.
The order of the cut set matrix is (n – 1) × b.
The elements of a cut set matrix,
If Jth branch is incident to ith cut set and oriented in same
direction.
= –1, If Jth branch is incident to ith cut set oriented in opposite direction
= 0, If Jth branch is not incident with ith cut set voltage.

Example-
For the network graph below construct the cut set matrix and write the
equilibrium equations by considering branches a, b, c as tree branches.
Solution: The tree from the given graph is,
a, b, c → twigs
d, e → Links
Fundamental cut set 1→ c d e → e1
Fundamental cut set 2 → b e → e2
Fundamental cut set 3 → a d e → e3
Cut set matrix is-

Let ja, jb, jc, jd, je be the branch currents, then
jc + jd + je = 0
jb – je = 0
ja – jd – je = 0
let va, vb, vc, vd, ve be the branch voltages, then
va = e3,
vd = e1 − e3
vb = e2,
ve = e1 − e2 − e3
vc = e1,

Graph theory

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Graph theory