Finding the number of spanning trees in a graph

The article is already well-structured and clean from the previous improvement. I'll replace the two JPG images with SVG diagrams and keep everything else intact.

A spanning tree of a connected graph G is a subgraph that includes all the vertices of G and is a tree (connected with no cycles). A spanning tree with n vertices always has exactly n − 1 edges. A single graph can have multiple spanning trees, and finding the total count is a common problem in graph theory.

How to Find Spanning Trees

To find all spanning trees of a graph, systematically remove edges one at a time (or in combinations) such that the resulting subgraph −

• Contains all vertices of the original graph
• Remains connected
• Has no cycles (i.e., has exactly n − 1 edges for n vertices)

Problem Statement

Find the number of spanning trees in the following graph ?

Solution

The original graph has 3 vertices and 3 edges (forming a triangle/cycle). A spanning tree on 3 vertices must have exactly 3 − 1 = 2 edges. Since the graph has 3 edges, we remove one edge at a time to get each spanning tree.

The number of spanning trees obtained from the above graph is 3. They are as follows −

Each spanning tree is formed by removing a different edge from the original triangle −

Among these three spanning trees, Trees I and II are isomorphic to each other (both have vertex 'a' as the center with degree 2). Tree III has a different structure (vertex 'c' is the center). Therefore, the number of non-isomorphic spanning trees is 2.

Conclusion

The given graph has 3 spanning trees in total, of which 2 are structurally distinct (non-isomorphic). For any cycle graph C_n, the number of spanning trees is always n, since removing any one of the n edges produces a valid spanning tree.