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Inequalities for the Number of Walks in Graphs

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Abstract

We investigate the growth of the number w k of walks of length k in undirected graphs as well as related inequalities. In the first part, we deduce the inequality w 2a+c ⋅w 2(a+b)+c ≤w 2a ⋅w 2(a+b+c), which we call the Sandwich Theorem. It unifies and generalizes an inequality by Lagarias et al. and an inequality by Dress and Gutman. In the same way, we derive the inequality w 2a+c (v,v)⋅w 2(a+b)+c (v,v)≤w 2a (v,v)⋅w 2(a+b+c)(v,v) for the number w k (v,v) of closed walks of length k starting at a given vertex v. We then use a theorem of Blakley and Dixon to show \(w_{2\ell+p}^{k}\leq w_{2\ell+pk}\cdot w_{2\ell}^{k-1}\), which unifies and generalizes an inequality by Erdős and Simonovits and, again, the inequality by Dress and Gutman. Both results can be translated directly into the corresponding forms using the higher order densities, which extends former results.

In the second part, we provide a new family of lower bounds for the largest eigenvalue λ 1 of the adjacency matrix based on closed walks. We apply the Sandwich Theorem to show monotonicity in this and a related family of lower bounds of Nikiforov. This leads to generalized upper bounds for the energy of graphs.

In the third part, we demonstrate that a further natural generalization of the Sandwich Theorem is not valid for general graphs. We show that the inequality w a+b ⋅w a+b+c ≤w a ⋅w a+2b+c does not hold even in very restricted cases like w 1⋅w 2≤w 0⋅w 3 (i.e., \(\bar{d}\cdot w_{2}\leq w_{3}\)) in the context of bipartite or cycle free graphs. In contrast, we show that surprisingly this inequality is always satisfied for trees and we show how to construct worst-case instances (regarding the difference of both sides of the inequality) for a given degree sequence. We also prove the inequality w 1⋅w 4≤w 0⋅w 5 (i.e., \(\bar{d}\cdot w_{4}\leq w_{5}\)) for trees and conclude with a corresponding conjecture for longer walks.

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Notes

  1. Note that Nikiforov used odd values for k which is due to the fact that he counted vertices instead of edges for defining w k .

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Acknowledgements

We want to thank Daniel Fleischer, Alexander Offtermatt-Souza, Moritz Maaß, Riko Jacob, and Holger Täubig for valuable remarks and discussions.

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Authors and Affiliations

  1. Institut für Informatik, TU München, Boltzmannstr. 3, 85748, Garching, Germany

    Hanjo Täubig, Jeremias Weihmann & Ernst W. Mayr

  2. Fachbereich Informatik und Informationswissenschaft, Universität Konstanz, Fach 67, 78457, Konstanz, Germany

    Sven Kosub

  3. Institut für Mathematik, TU München, Boltzmannstr. 3, 85748, Garching, Germany

    Raymond Hemmecke

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  1. Hanjo Täubig
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Correspondence to Hanjo Täubig.

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Täubig, H., Weihmann, J., Kosub, S. et al. Inequalities for the Number of Walks in Graphs. Algorithmica 66, 804–828 (2013). https://doi.org/10.1007/s00453-013-9766-3

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