\n
## Diagram: Catalog of Small Graphs (H₁–H₂₈)
### Overview
The image displays a 7-row by 4-column grid containing 28 distinct, simple undirected graphs. Each graph is composed of nodes (solid black dots), edges (straight lines connecting nodes), and loops (circles attached to a single node). Every graph is labeled below its diagram with the notation "H" followed by a subscript number, sequentially from H₁ to H₂₈. The grid is framed by thin black lines, separating each graph into its own cell.
### Components/Axes
* **Grid Structure:** 7 rows x 4 columns.
* **Graph Elements:**
* **Nodes:** Represented as solid black circles (●).
* **Edges:** Represented as straight black lines connecting two nodes.
* **Loops:** Represented as open circles (○) attached to a single node.
* **Labels:** Each graph has a unique identifier placed directly below it, formatted as `H` with a numerical subscript (e.g., H₁, H₂, ..., H₂₈).
* **Spatial Layout:** The graphs are arranged in a strict grid. The legend (the labeling system) is consistently placed at the bottom-center of each cell.
### Detailed Analysis
The graphs are presented in order of increasing complexity. Below is a breakdown by row:
**Row 1 (H₁-H₄):**
* **H₁:** A single isolated node. (0 edges, 0 loops)
* **H₂:** A single node with one loop attached. (0 edges, 1 loop)
* **H₃:** Two isolated nodes, not connected. (0 edges, 0 loops)
* **H₄:** Two nodes. One node is isolated, the other has a loop. (0 edges, 1 loop)
**Row 2 (H₅-H₈):**
* **H₅:** Two nodes, each with a loop. No edge between them. (0 edges, 2 loops)
* **H₆:** Two nodes connected by a single edge. (1 edge, 0 loops)
* **H₇:** Two nodes connected by an edge. One node has a loop. (1 edge, 1 loop)
* **H₈:** Two nodes connected by an edge. Both nodes have loops. (1 edge, 2 loops)
**Row 3 (H₉-H₁₂):**
* **H₉:** Three isolated nodes. (0 edges, 0 loops)
* **H₁₀:** Three nodes. Two are isolated, one has a loop. (0 edges, 1 loop)
* **H₁₁:** Three nodes. One is isolated, two each have a loop. (0 edges, 2 loops)
* **H₁₂:** Three nodes, each with a loop. No edges. (0 edges, 3 loops)
**Row 4 (H₁₃-H₁₆):**
* **H₁₃:** Three nodes. Two are connected by an edge, one is isolated. (1 edge, 0 loops)
* **H₁₄:** Three nodes. Two are connected by an edge, the isolated node has a loop. (1 edge, 1 loop)
* **H₁₅:** Three nodes. Two are connected by an edge, and one of those connected nodes has a loop. The third node is isolated. (1 edge, 1 loop)
* **H₁₆:** Three nodes. Two are connected by an edge, and both of those nodes have loops. The third node is isolated. (1 edge, 2 loops)
**Row 5 (H₁₇-H₂₀):**
* **H₁₇:** Three nodes. Two are connected by an edge, and both have loops. The third node is isolated and also has a loop. (1 edge, 3 loops)
* **H₁₈:** Three nodes. Two are connected by an edge, and both have loops. The third node is isolated and has a loop. (Identical structure to H₁₇, visually similar placement). (1 edge, 3 loops)
* **H₁₉:** Three nodes forming a path: Node A connected to Node B, which is connected to Node C. (2 edges, 0 loops)
* **H₂₀:** Three nodes forming a path (A-B-C). Node A has a loop. (2 edges, 1 loop)
**Row 6 (H₂₁-H₂₄):**
* **H₂₁:** Three nodes forming a path (A-B-C). Node B (the middle node) has a loop. (2 edges, 1 loop)
* **H₂₂:** Three nodes forming a path (A-B-C). Node C has a loop. (2 edges, 1 loop)
* **H₂₃:** Three nodes forming a path (A-B-C). Nodes A and B have loops. (2 edges, 2 loops)
* **H₂₄:** Three nodes forming a path (A-B-C). Nodes A and C have loops. (2 edges, 2 loops)
**Row 7 (H₂₅-H₂₈):**
* **H₂₅:** Three nodes all connected to each other, forming a triangle (a complete graph K₃). (3 edges, 0 loops)
* **H₂₆:** A triangle (K₃). One node has a loop. (3 edges, 1 loop)
* **H₂₇:** A triangle (K₃). Two nodes have loops. (3 edges, 2 loops)
* **H₂₈:** A triangle (K₃). All three nodes have loops. (3 edges, 3 loops)
### Key Observations
1. **Progressive Complexity:** The catalog shows a clear progression from the simplest possible graph (a single node, H₁) to a more complex, fully connected graph with loops on all vertices (H₂₈).
2. **Systematic Enumeration:** The graphs appear to be a systematic enumeration of all possible distinct (non-isomorphic) simple graphs on up to 3 vertices, including the consideration of loops. The sequence explores all combinations of connectivity and loop placement.
3. **Disconnected vs. Connected:** The first four rows contain many disconnected graphs. From H₁₉ onward, all graphs are connected.
4. **Loop Placement:** Loops are treated as a feature that can be added independently to any node in any graph configuration.
### Interpretation
This image serves as a **visual reference catalog for small graph structures** in graph theory. It is likely used for educational purposes to illustrate fundamental concepts such as:
* **Graph Components:** Nodes, edges, and loops.
* **Connectivity:** The difference between connected and disconnected graphs.
* **Graph Isomorphism:** The distinct graphs (H₁-H₂₈) represent non-isomorphic classes—meaning no two graphs in the set can be transformed into each other simply by relabeling the vertices.
* **Enumeration:** It demonstrates the process of listing all possible graphs for a small number of vertices (n=1, 2, and 3), including the allowance of loops.
The methodical layout and labeling suggest this figure is from a textbook, lecture note, or reference material designed to provide a concrete, visual foundation for abstract graph-theoretic concepts. The progression allows a learner to see how complexity builds from the most basic element.
