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Zeitschrift für Naturforschung A Contents Vol. 68a (2013)

Z. Naturforsch. 68a, 531 – 538 (2013)

doi:10.5560/ZNA.2013-0031

On the Kirchhoff Index of Graphs

Kinkar C. Das

Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea

Received December 20, 2012 / revised March 15, 2013 / published online May 22, 2013

Reprint requests to: K. C. D.; E-mail: kinkardas2003@googlemail.com

Let G be a connected graph of order n with Laplacian eigenvalues μ₁ ≥ μ₂ ≥…≥ μ_n−1 > μ_n = 0. The Kirchhoff index of G is defined as Kf = Kf(G) = n ∑_k=1ⁿ⁻¹1/μ_k.

In this paper. we give lower and upper bounds on Kf of graphs in terms on n, number of edges, maximum degree, and number of spanning trees. Moreover, we present lower and upper bounds on the Nordhaus–Gaddum-type result for the Kirchhoff index.

Key words: Graph Spectrum; Laplacian Spectrum (of Graph); Kirchhoff Index; Nordhaus–Gaddum-Type.

Mathematics Subject Classification 2000: 05C50; 15A18

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