Basic Concepts of Graph Theory Cut-set Incidence Matrix Circuit Matrix Cut-set Matrix

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1 Basic Concepts of Graph Theory Cut-set Incidence Matrix Circuit Matrix Cut-set Matrix

2 Definition of Graph Definition: In a connected graph G of n nodes (vertices),the subgraph T that satisfies the following properties is called a tree. T is connected T contains all the vertices of G T contains no circuit, T contains exactly n- number of edges. In every connected graph G there exists at least one tree.

3 Tree & Co-tree Let G have p separated parts G, G 2,..., G p, that is G=G G 2... G p, and let T i be a tree in G i, i=,2,...,p, then,t=t T 2... T p is called a forest of G. DEFINITION: The complement of a tree is called a co-treeand the complement of a forest is called a co-forest. The edges of a tree or a forest are called branchesand the edges of a co-tree or co-forest are called links (chords).

4 Tree & Co-tree Examples v v 4 e 2 v 3 e 4 e 6 e e 3 e 5 9 possible trees and corresponding co-trees: T = { e, e, e, e } T = { e, e, e, e } T = { e, e, e, e } {, } {, } {, } T = e e T = e e T = e e T = { e, e, e, e } T = { e, e, e, e } T = { e, e, e, e } {, } {, e } T = { e, e } T = e e T = e T v 2 v = { e, e, e, e } T = { e, e, e, e } { e, e } T { e, e} T = = T 3 6 = { e, e, e, e } T = { e, e } 9 2 6

5 Rank & Nullity DEFINITION: Let G be a graph and let b and l be respectively the number of branches and chords of G, then band lare called respectively the rank and the nullity of the graph. THEOREM: Let G have n nodes, eedges and pconnected parts, then its rank and nullity are given respectively by b = n - p and l = e n + p

6 Fundamental Circuit (f-circuit) DEFINITION: Let G be a connected graph and let T and T be tree and co-tree respectively, that is G=T T. Let a linke T and its unique tree path (a path which is formed by the branches of T) define a circuit. This circuit is called the fundamental circuit (f-circuit) of G. All such circuits defined by all the chords of T are called the fundamental circuits (f-circuits) of G. If G is not connected, then the f-circuits are defined with respect to a forest.

7 f-circuit Example Note that the number of f-circuits is given by the nullity of G and that, with respect to a chosen tree T of G, each f- circuit contains one and only link. Consider the following graph v v 4 e 2 v 3 e 4 e 5 e 6 v 5 e v 2 e 3 f-circuits: c f ={e 3,e,e 2 }, c f2 ={e 6,e 8,e 4,e 5 }, c f3 ={e 7,e 8,e 4,e 5 } e 8 e 7 v 6 Nullity of G l=e-n+p=8-6+=3

8 Cut-Set DEFINITION: The cut-set of a graph G is the subgraph G x of G consisting of the set of edges satisfying the following properties: The removal of G x from G reduces the rank of G exactly by one. No proper subgraph of G x has this property. If G is connected,then the first property in the above definition can be replaced by the following phrase. The removal of G x from G separates the given connected graph G into exactly two connected subgraphs.

9 Cut-set example Consider the following graph and the following set of edges v v 4 e 2 v 3 e 4 e 5 e 6 v 5 e e 3 e 8 v 6 e 7 v 2 G ={e,e 2 } G 2 ={e 4,e 6,e 7 } G 3 ={e 2,e 3,e 4,e 8 } G 4 ={e 2,e 3,e 6 } is also a cut-set Cut-set is not a cut-set, because the removal of G 3 from G results in three connected subgraphs is not a cut-set, because a subset of G 4 is cut-set

10 Fundamental cut-set (f-cutset) DEFINITION: Let G be a connected graph and let T be its tree. The branch e t T defines a unique cut-set (a cut-set which is formed by e t and the links of G). This cut-set is called the fundamental cut-set (fcutset) of G. All such cut-sets defined by all the branches of T are called the fundamental cut-sets (fcutsets) of G. If G is not connected then the f-cut sets are defined with respect to a forest. Note that the number of fundamental cut-sets is given by the rank of G and with respect to a chosen tree T of G, each fundamental cut-set contains one and only one branch.

11 f-cutsetexample Consider the following graph with T={e,e 2,e 4,e 5,e 8 } v v 4 e 2 v 3 e 4 e 5 e 6 v 5 e v 2 e 3 f-cutsets: e 8 v 6 e 7 x f ={e,e 3 } x f2 ={e 2,e 3 } x f3 ={e 4,e 6,e 7 } x f4 ={e 5,e 6,e 7 } x f5 ={e 8,e 6,e 7 }

12 Matrices of Directed Graphs v v 4 e 6 v 6 e 2 e 4 e 8 e v 3 e e v 8 v 2 e 3 e 5 v 5 e 7 v 7 e 9 The edge e which has a direction from node v to node v 2 simply indicates that any transmission from v to v 2 along e is assumed to be positive. Any transmission from v 2 to v along e is assumed to be negative.

13 Incidence Matrix a ij = DEFINITION: Let eand nrepresent respectively the number of edges and nodes of a graph G. The incidence matrix A = [ a ] a ij n e having nrows and ecolumnswith its elements are defined as if if if edge edge edge j j j incident to node incident to node not incident to i i node and oriented "outward" and oriented "inward" i

14 Incidence Matrix (2) v v 2 v 3 v 4 v 5 v 6 v 7 v 8 e e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e e Incident Matrix: Property: Any column of A contains exactly two nonzero entries of opposite sign node edge A=

15 Reduced Incidence Matrix DEFINITION: For a connected graph G, the matrix A, obtained by deleting any one of the rows of the incidence matrix A a is called the reduced incidence matrix. Note that since any column of A a contains exactly two nonzero entries of opposite sign, one can uniquely determine the incident matrix when the reduced incident matrix is given. Note also that the rank of A a is n-.

16 Circuit Matrix In a graph G, let k be the number of circuits and let an arbitrary circuit orientation be assigned to each one of these circuits. DEFINITION: The circuit matrix B = [ b ] ij k e for a graph G of eedges and kcircuits is defined as b ij = if if if edge edge edge j j j incident to circuit i with "same"orientation incident to circuit i with "opposite" orientation not incident to circuit i

17 Circuit Matrix Consider the following graph e 3 v v 2 v 3 v 4 e e 2 e 4 e 5 e 6 c 6 c5 c 4 c 3 c 2 c e e e e e e c c c c c c = B

18 f-circuit Matrix Let b i represent the row of B that corresponds to circuit c i. The circuits c i,...,c j are independent if the rows b i,... b j are independent. DEFINITION: The f-circuit matrix B f of a graph G with respect to some tree T is defined as the circuit matrix consisting of the fundamental circuits of G only whose orientations are chosen in the same direction as that of defining links. The fundamental circuit matrix B f of a graph G with respect to some tree T can always be written as B = f l e [ U B f 2 ]

19 f-circuit Matrix (2) Consider the following graphwith T = {e,e 3,e 5 } e 3 v v 2 v 3 v 4 e e 2 e 4 e 5 e 6 [ ] f f edge B U B = =

20 Incidence & Circuit Matrices THEOREM: If the column orderings of the circuit and incident matrices are identical then A a B T f = Also B f A a A B T a T = = BA T a =

21 Matrices of Oriented Graphs Consider the following graph e 3 v v 2 v 3 v 4 e e 2 e 4 e 5 e 6 c 6 c5 c 4 c 3 c 2 c ; = f edge B = A a

22 Cut-set Matrix In a graph G let x be the number of cut-sets having arbitrary orientations. Then, we have the following definition. DEFINITION: The cut-set matrix Q = [ q ] ij x e for a graph G of eedges and xcut-sets is defined as q ij = if if if edge edge edge j j j in cut -set i with in cut -set i with not in cut -set i e e j j, x, x i i "same"orientation "opposite" orientation

23 Cut-set Matrix (2) Consider the following graph and its seven possible cut-sets e 2 e v x v e e 4 5 x 4 x 6 x 7 e 3 Q= v 3 v e 6 2 x x 2 5 x 3

24 f-cut-set Matrix DEFINITION: The f-cutset matrix Q f of a graph G with respect to some tree T is defined as the cut-set matrix consisting of the fundamental cut-set of G only whose orientations are chosen in the same direction as that of defining branches. The fundamental cut-set matrix A f of a graph G with respect to some tree T can always be written as Q f b e Recall that b = n- = [ U Q b b f b ( eb) ]

25 f-cut-set Matrix (2) Consider the following graphwith T = {e 2,e 4,e 5 } v edge e 2 e v e e 4 5 e 3 Q f = v 3 v e 6 2

26 Cut-set & Circuit Matrices THEOREM: If the column orderings of the circuit and incident matrices are identical then QB T f = Also B f Q QB T T = = BQ T =

27 Cut-set & Circuit Matrices Consider the following graph e 3 v v 2 v 3 v e e 2 e 4 e 5 e 3 v v 2 v 3 v e e 2 e 4 e 5 e 6 = B = Q

28 FUNDAMENTAL POSTULATES Now, Let G be a connected graph having e edges and let T x = x ( ) ( ) ( ) t, x2 t, Lxe t and y T = y ( ) ( ) ( ) t, y2 t, L ye t be two vectors where x i and y i, i=,...,e, correspond to the across and through variables associated with the edge i respectively.

29 FUNDAMENTAL POSTULATES 2. POSTULATE Let B be the circuit matrix of the graph G having e edges then we can write the following algebraic equation for the across variables of G (e.g., edge voltage) Bx= KVL 3. POSTULATE Let Q be the cut-set matrix of the graph G having e edges then we can write the following algebraic equation for the through variables of G (e.g., edge current) Qy = KCL

30 Fundamental Circuit & Cut-set Equations Consider a graph G and a tree T in G. Let the vectorsv and i partitioned as Then ] T ; [ ] T link branch branch link v = [ v v i= i i B v f v link [ ] link = U B = Q i= [ U Q ] =B f 2 f 2 v v v branch branch i f branch =Q f f i i i link branch link = fundamental circuit equation fundamental cut-set equation

31 Series & Parallel Edges Definition: Two edges e i and e k are said to be connected in series if they have exactly one common vertexof degree two. Definition: Two edges e i and e k are said to be connected in parallel if they are incident at the same pair of vertices v i and v k. v v i e k e k e i e i v k

32 General Procedure. Draw a graph and then identify a tree. 2. Place all control-voltage branches for voltagecontrolled dependent sources in the tree, if possible. 3. Place all control-current branches for current-controlled dependent sources in the cotree, if possible. 4. Find incidence, f-circuit, or f-cutset matrix. 5. Replace voltage, current sources with short, open circuits, respectively. 6. Formulate the matrix equation. /5/25 32

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