The Coulson Integral Formula

Xueliang Li, Yongtang Shi and Ivan Gutman
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Xueliang Li: Nankai University, Center for Combinatorics
Yongtang Shi: Nankai University, Center for Combinatorics
Ivan Gutman: University of Kragujevac, Faculty of Science

Chapter Chapter 3 in Graph Energy, 2012, pp 19-23 from Springer

Abstract: 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 $$\mathrm{i} = \sqrt{-1}$$ .

Keywords: Coulson Integral Formula; Energy Graph; Characteristic Polynomial; Well-known Cauchy Formula; Descartes Coordinate System (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1007/978-1-4614-4220-2_3

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