Graph theory[1]
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This document discusses graphs and networks. It defines key terms related to graphs like nodes, branches, trees, and loops. It explains properties of trees, including that a tree with n nodes will have n-1 branches and no closed paths. It also discusses incidence matrices, cut-set matrices, and loop matrices which relate to representing graphs and networks mathematically through matrices. Formulas are provided for relationships between the number of trees, branches, and nodes in a graph.
![Graph theory[1]](https://image.slidesharecdn.com/graphtheory1-151213070323/85/Graph-theory-1-7-320.jpg)
![ Relation between twig and link L=B-N+1
If the graph as N number of nodes , tree will have N-1
branches.
Number of trees of a graph T = determinant [A].[A]^t
Orthogonal relationship [A].[B]^t =0 or [B][A]^t = 0
[Vb] = [Q]^t .[Vt]
[Vb] = [A]^t[Vn]](https://image.slidesharecdn.com/graphtheory1-151213070323/85/Graph-theory-1-13-320.jpg)
![Graph theory[1]](https://image.slidesharecdn.com/graphtheory1-151213070323/85/Graph-theory-1-14-320.jpg)
Graph theory[1]
- 1. BY DEEP NANDI BRANCH:- EE(2nd year) ROLL NO.- 31
- 2. When all the elements (e.g. R, L,C) in a network are replaced by lines with circles or dots at both ends the configuration is called graph of the network.
- 3. BRANCH NODE TREE TREE LINK TREE BRANCH LOOP MATRIX
- 4. : A branch is a line segment representing one network element or a combination of element connected between two points. : A node point is defined as an end point of a line segment and exist at the junction between two branches. : It is an inter connected open set of branches which include all the nodes of given graph.
- 5. 1 2 3
- 6. PROPERTIES OF TREE It consist of all the nodes of the graph . If the graph has n number of nodes the tree will n-1 number of branches There will be no closed path in the tree . There can be many possible different for a given graph depending on the number of nodes and branches.
- 8. INCIDENCE MATRIX Row -> node , column -> branch Algebraic sum of column entries of an incidence matrix is zero. Determinant of incidence matrix of a closed loop is zero . 1 2 3 1 2 4 3 5
- 9. Row -> cut-set , column -> branch Draw the tree of the graph . Take the direction of cut-set as the twig direction . The direction along the branch of cut-set is taken as positive one and if the direction is opposite it is taken as negative one .
- 11. R0w -> loop current , column -> branches Draw the tree from the given graph Take the direction of loop as a link direction. The branch having the same direction as a loop direction is taken as 1 and if opposite direction it is taken -1.
- 12. 1 2 3 4 1 2 3 4 5I
- 13. Relation between twig and link L=B-N+1 If the graph as N number of nodes , tree will have N-1 branches. Number of trees of a graph T = determinant [A].[A]^t Orthogonal relationship [A].[B]^t =0 or [B][A]^t = 0 [Vb] = [Q]^t .[Vt] [Vb] = [A]^t[Vn]