Example Data Table
| Case |
Vertices |
Edges |
Components |
Density |
Cycle rank |
Average degree |
| Road network sample |
8 |
10 |
1 |
35.714% |
3 |
2.5 |
| Small forest sample |
9 |
7 |
2 |
19.444% |
0 |
1.556 |
| Dense study sample |
6 |
14 |
1 |
93.333% |
9 |
4.667 |
Formula Used
Undirected maximum edges: Emax = n(n - 1) / 2.
Directed maximum edges: Emax = n(n - 1), when loops are not counted.
Density: D = m / Emax. The page also shows D as a percentage.
Average degree: k = 2m / n for undirected graphs. For directed mode, k = m / n.
Cycle rank: r = m - n + c. Here c is the number of components.
Minimum edges for components: Emin = n - c.
Average path length: L = total path length / counted path pairs.
Global transitivity estimate: T = 3 × triangles / connected triples.
How To Use This Calculator
Choose the graph type first. Select simple graph when loops and repeated edges are not used.
Enter vertices, edges, and components. Add triangles and path totals when those values are known.
Enter a degree sequence as comma, space, or line separated values. It improves degree checks.
Press the calculate button. The result appears above the form and below the header section.
Use the CSV or PDF buttons to save the calculated report for study or documentation.
Understanding Graph Features
A graph is a set of vertices joined by edges. It can model routes, networks, tasks, games, circuits, and relationships. A graph features calculator turns raw counts into useful measures. It helps you see structure before drawing every connection.
Why The Main Measures Matter
Vertex count shows the number of objects. Edge count shows the number of links. Density compares existing edges with all possible edges. A sparse graph has few links. A dense graph has many links. Components show how many separate pieces exist. One component means the graph is connected.
Degrees And Balance
Degree values describe how many edges touch each vertex. In an undirected graph, the degree sum equals twice the edge count. This is the handshaking rule. Odd degree vertices are important. They help check Euler trails and circuits. A degree sequence also reveals hubs. Hubs are vertices with high degree.
Cycles, Trees, And Connectivity
The cycle rank estimates independent cycles in an undirected graph. It uses edges, vertices, and components. When an undirected graph has n vertices, c components, and n minus c edges, it behaves like a forest. If it also has one component, it is a tree. Extra edges usually create cycles.
Planar And Bipartite Clues
Planar checks are only quick tests. They do not prove every case. A simple planar graph usually has at most three n minus six edges when n is at least three. A simple bipartite planar graph usually has at most two n minus four edges. These limits flag impossible layouts early.
Using Results Wisely
Calculator results should guide review, not replace proof. Graph rules depend on assumptions. Loops, parallel edges, direction, and weights can change meaning. Always match the input to your model. Use the example table for practice. Then export the report for notes, lessons, or project documentation.
Best Input Practice
Start with a clear graph type. Use directed mode only when edge direction matters. Keep loops and repeated edges out of simple graph checks. Enter components after counting isolated vertices. Add degree data when you need stronger feedback. Add path totals when average path length is known. Small checks prevent misleading conclusions later. Review units and labels before sharing any final report.
FAQs
What is graph density?
Graph density compares actual edges with the maximum possible edges. A value near zero means the graph is sparse. A value near one means the graph is dense.
What does cycle rank mean?
Cycle rank estimates the number of independent cycles in an undirected graph. It is calculated from edges, vertices, and components.
Can this calculator test directed graphs?
Yes. Directed mode changes the maximum edge count and average degree formula. Some checks, like Euler parity and planar limits, are mainly for undirected graphs.
What is a component?
A component is a separate connected part of a graph. If a graph has one component, every vertex can be reached from the others.
Why enter a degree sequence?
A degree sequence lets the calculator check degree sum, odd vertices, graphical possibility, and Euler rules. It gives stronger results than edge count alone.
Does the planar check prove a graph is planar?
No. It is only a quick edge limit check. Passing the limit does not prove planarity. Failing it can show the graph cannot be simple planar.
What is average path length?
Average path length is the total shortest path length divided by the number of counted vertex pairs. Enter both values when you know them.
Can I export the result?
Yes. After calculation, use the CSV button for spreadsheet data. Use the PDF button for a simple printable report.