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The Coulson Integral Formula

In: Graph Energy

Author

Listed:
  • Xueliang Li

    (Nankai University, Center for Combinatorics)

  • Yongtang Shi

    (Nankai University, Center for Combinatorics)

  • Ivan Gutman

    (University of Kragujevac, Faculty of Science)

Abstract

In the theory of graph energy, the so-called Coulson integral formula (3.1) plays an outstanding role. This formula was obtained by Charles Coulson as early as 1940 [73] and reads: 3.1 $$\mathcal{E}(G) = \frac{1} {\pi }\int\limits_{-\infty }^{+\infty }\left [n -\frac{\mathrm{i}x\,\phi ^{\prime}(G,\mathrm{i}x)} {\phi (G,\mathrm{i}x)} \right ]\mathrm{d}x = \frac{1} {\pi }\int\limits_{-\infty }^{+\infty }\left [n - x \frac{\mathrm{d}} {\mathrm{d}x}\ln \phi (G,\mathrm{i}x)\right ]\mathrm{d}x$$ where G is a graph, ϕ(G,x) is the characteristic polynomial of G, ϕ′(G,x)=(d∕dx)ϕ(G,x) its first derivative, and $$\mathrm{i} = \sqrt{-1}$$ .

Suggested Citation

  • Xueliang Li & Yongtang Shi & Ivan Gutman, 2012. "The Coulson Integral Formula," Springer Books, in: Graph Energy, edition 127, chapter 0, pages 19-23, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4614-4220-2_3
    DOI: 10.1007/978-1-4614-4220-2_3
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