Spectral Graph Theory and Quantum Mechanics

This graph compares the Von Neumann entropy of various graph families using the eigenvalues of their Laplacian matrices. The complete graph exhibits the highest entropy, reflecting its high symmetry and connectivity. In contrast, the path, tree, and cycle graphs all display much lower and closely clustered entropy values, indicating simpler, more regular structures with fewer connections.

This graph is very similar to the previous one, comparing the Von Neumann entropy across different graph families, but this time using the eigenvalues of the adjacency matrix instead of the Laplacian. While the overall trends remain similar, the entropy values are generally lower. As a result, the complete graph no longer shows the same rapid, exponential increase, highlighting how the choice of matrix affects the eigenvalues and thus the entropy scale and interpretation.

This graph displays the ratio of entropy values computed from the Laplacian and adjacency matrices, showing how the two measures compare across graph types.

• Complete and complete bipartite graphs show a steady upward slope in the entropy ratio, reflecting a smooth and predictable difference as the graphs grow.
• Path, tree, and star graphs have more irregular structures, causing their entropy ratios to fluctuate with changes in size.
• Cycle and lattice graphs, while symmetric, can still show moderate fluctuations, especially at small sizes, due to their lower connectivity compared to complete graphs.
• This suggests that higher connectivity and stronger symmetry (like in complete graphs) lead to smoother entropy ratios.

This graph displays the Euler characteristic (χ) for each graph type, a topological invariant defined as the number of vertices minus the number of edges (V − E). It shows how this value changes as the graph size increases. We observe that:

• Complete and complete bipartite graphs have rapidly decreasing (negative) Euler characteristics due to their quickly growing number of edges.
• Paths, stars, and trees maintain an Euler characteristic of 1 for all sizes because their number of edges is always one less than their number of vertices.
• Cycles have an Euler characteristic of 0 since the number of edges equals the number of vertices.

This graph shows the ratio of Laplacian entropy between torus graphs and lattice graphs. We can see that toruses consistently have higher entropy. This is because they have more uniform and symmetrical connectivity.

Use this interactive tool to build graphs and see how their Von Neumann entropy changes. Here’s what to keep in mind as you experiment:

• Add Vertices and Edges: Click "Add Vertex" to place points, then connect them by clicking between vertices. Generally, adding more vertices and edges increases entropy.
• Disconnected Graphs: If the graph has separate disconnected parts, the total entropy equals the sum of the entropies of those parts.
• Empty Graph: A graph with vertices but no edges (an empty graph) has an entropy value of zero.
• Rewiring Edges: Changing which vertices are connected without adding/removing edges can alter the overall entropy. Try changing the connectivity and observe how the entropy responds.
• Compare Graph Families: Try drawing the different types of graphs we discussed earlier to explore how their shapes influence entropy.