Graph Energy Calculator

sum of |eigenvalues|

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About Graph Energy Calculator

A graph energy calculator computing E(G) = Σ|λᵢ|, the sum of absolute values of adjacency matrix eigenvalues. Introduced by Gutman (1978) for molecular orbital theory. E(K_n) = 2(n-1). Hyperenergetic: E(G) > E(K_n). Integral energy: all eigenvalues integer. Client-side.

Graph Energy Calculator Features

  • E(G) value
  • Eigenvalues
  • Hyperenergetic
  • McClelland
  • Common graphs
Graph energy E(G) = Σ|λᵢ|: sum of absolute adjacency eigenvalues. Chemical origin: total π-electron energy in Hückel theory. E(K_n) = 2(n-1). McClelland bounds: √(2mn) ≤ E ≤ √(n(m + n·det^(2/n))). Extensive research in mathematical chemistry.

How to Use

Select graph:

  • E: Energy
  • λ: Eigenvalues
  • Bounds: McClelland

Hyperenergetic Graphs

G is hyperenergetic if E(G) > E(K_n) = 2(n-1). Complete graph is NOT always the maximum energy graph! Many hyperenergetic graphs exist: line graphs of complete graphs, Kneser graphs. Surprising phenomenon.

Chemical Applications

Hückel molecular orbital theory: E(G) ≈ total π-electron energy. Stability prediction: higher energy → more stable conjugated molecule. Applies to benzenoid hydrocarbons, fullerenes, carbon nanotubes.

Step-by-Step Instructions

  1. 1Select graph.
  2. 2Compute eigenvalues.
  3. 3Sum absolute values.
  4. 4Check hyperenergetic.
  5. 5Apply McClelland.

Graph Energy Calculator — Frequently Asked Questions

Why sum absolute eigenvalues?+

In molecular orbital theory, each eigenvalue λ corresponds to an energy level. Electrons fill levels symmetrically around 0. Total energy sums |λ| weighted by occupation. For graph theory: E captures spectral magnitude.

What's McClelland's bound?+

E(G) ≤ √(n·(m + √(n·(n-1)·(S² - m²/n)))). Upper and lower bounds involving n (vertices), m (edges), S² (sum of squared eigenvalues). Useful for extremal graph energy problems.

What graphs maximize energy?+

Open problem! For n vertices: NOT always K_n. The maximum energy graph is unknown for most n. Koolen-Moulton: E ≤ (n/2)(1+√n) for all graphs. Constructing extremal examples is an active research area.

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