Co-tree of a graph is

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Option B: Set of all the links in a given tree

Explanation & Concepts:
In Graph Theory applied to electrical networks, a tree is a connected subgraph containing all nodes but no loops. The co-tree is the complement of the tree, consisting of all the branches of the original graph that are not included in the chosen tree, known as links.

Working Principle: If a connected graph has $n$ nodes and $b$ branches, any spanning tree contains exactly $t = n - 1$ branches. The remaining $l = b - (n - 1)$ branches do not form part of the tree and are classified as links. The set of these $l$ links collectively forms the co-tree of the graph.

Key Formulas: $b = t + l$ — Total branches equal the sum of tree branches ($t$) and links ($l$); $l = b - n + 1$ — Number of links (co-tree branches) in a connected graph

Key Points: A tree must include all nodes ($n$) of the graph.; A tree contains exactly $n-1$ branches.; The co-tree is essential for forming fundamental loop equations in circuit analysis.; The number of links determines the number of independent loops in a circuit.

Advantages: Simplifies network analysis using Kirchhoff's laws; Provides a systematic way to define independent loops

Disadvantages: Choice of tree is not unique; different trees lead to different sets of links; Selection process becomes complex in highly dense graphs

Applications: Mesh analysis for electrical circuits; Determining the circuit topology for matrix-based network solutions

Additional Info: The number of links $l$ is also known as the nullity of the graph.. Option A is incorrect because it describes the tree itself, not the complement.. Option C is incorrect because nodes are vertices, while co-trees are composed of edges (branches).

Answer: B is correct — The co-tree is defined as the set of all branches of a graph that are not present in the selected spanning tree, which are termed as links.

Exam Tip: Remember that the number of links equals the number of fundamental loops, which is exactly the number of equations required for mesh analysis: $l = b - n + 1$.

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Option A: Set of all branch in a given tree

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Option C: Sets of all the node in given tree.

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Option D: None of above

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