# How many nonisomorphic subgraphs does K3 have?

(20 pts) How many nonisomorphic subgraphs does K3 have? Please justify your answer. You must write out your reasoning.

The number of nonisomorphic subgraphs of K3 is 5.

## Understanding Nonisomorphic Subgraphs of K3

**Step 1:**To determine the number of nonisomorphic subgraphs of K3, we need to consider all possible combinations of vertices and edges within the graph. The first step is to understand the structure of the complete graph K3.

**Step 2:**Let's analyze the subgraphs of K3 systematically. We start with the empty graph (0 vertices and 0 edges), which is always a subgraph of any graph. Next, we consider subgraphs with 1 vertex, which can be any of the three vertices of K3. Since each vertex is distinct, we have three possibilities. Moving on to subgraphs with 2 vertices, we can form subgraphs by selecting two out of the three vertices and connecting them with the edge between them. There are three ways to choose two vertices, resulting in three subgraphs. Finally, for subgraphs with 3 vertices, there is only one possibility, which is the complete graph K3 itself.

**Step 3:**Combining all the possibilities, we have a total of 3 subgraphs with 1 vertex, 3 subgraphs with 2 vertices, and 1 subgraph with 3 vertices. Initially, this totals to 7 subgraphs. However, we must consider isomorphisms, which are subgraphs that are essentially the same despite different vertex labels and edge connections. Upon closer inspection, we realize that the three subgraphs with 2 vertices are all isomorphic to each other since they have the same structure but with different vertex labels. Therefore, we subtract 2 from the total count of subgraphs, resulting in 5 nonisomorphic subgraphs of K3. In conclusion, despite the initial count of 7 subgraphs, careful consideration of isomorphisms reveals that K3 actually has 5 nonisomorphic subgraphs. This understanding is crucial in graph theory and combinatorics to analyze and differentiate between different subgraphs within a larger graph.