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Ordering Starlike Trees by the Totality of Their Spectral Moments

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Abstract

The k-th spectral moment Mk(G) of the adjacency matrix of a graph G represents the number of closed walks of length k in G. We study here the partial order ≼ of graphs, defined by G ≼ H if Mk(G) ≤ Mk(H) for all k ≥ 0, and are interested in the question when is ≼ a linear order within a specified set of graphs? Our main result is that ≼ is a linear order on each set of starlike trees with constant number of vertices. Recall that a connected graph G is a starlike tree if it has a vertex u such that the components of G − u are paths, called the branches of G. It turns out that the ≼ ordering of starlike trees with constant number of vertices coincides with the shortlex order of sorted sequence of their branch lengths.

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Acknowledgments

The author is indebted to Stephan Wagner, Francesco Belardo and Milan Pokorny for reading and discussing the initial draft of the article, and to reviewers for helping to improve the presentation.

Funding

The author was supported by the Serbian Ministry of Education, Science and Technological Development via the Mathematical Institute of SASA and by the Serbian Academy of Sciences and Arts through the project F-159.

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Authors and Affiliations

  1. Mathematical Institute, Serbian Academy of Sciences and Arts, Kneza Mihaila 36, 11000, Belgrade, Serbia

    Dragan Stevanović

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  1. Dragan Stevanović
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Correspondence to Dragan Stevanović.

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Code availability (software application or custom code)

Octave files for generation of various types of starlike trees and counting their walks are available from the author upon request.

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Stevanović, D. Ordering Starlike Trees by the Totality of Their Spectral Moments. Order 39, 77–94 (2022). https://doi.org/10.1007/s11083-021-09566-3

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