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Peripherality in networks: theory and applications

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Abstract

We investigate several related measures of peripherality and centrality for vertices and edges in networks, including the Mostar index which was recently introduced as a measure of peripherality for both edges and networks. We refute a conjecture on the maximum possible Mostar index of bipartite graphs. We asymptotically answer another problem on the maximum difference between the Mostar index and the irregularity of trees. We also prove a number of extremal bounds and computational complexity results about the Mostar index, irregularity, and measures of peripherality and centrality. We discuss graphs where the Mostar index is not an accurate measure of peripherality. We construct a general family of graphs with the property that the Mostar index is strictly greater for edges that are closer to the center. We also investigate centrality and peripherality in two graphs which represent the SuperFast and MOZART-4 systems of atmospheric chemical reactions by computing various measures of peripherality and centrality for the vertices and edges in these graphs. For both of these graphs, we find that the Mostar index is closer to a measure of centrality than peripherality of the edges. We also introduce some new indices which perform well as measures of peripherality on the SuperFast and MOZART-4 graphs.

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Acknowledgements

JG was supported by the Woodward Fund for Applied Mathematics at San Jose State University, a gift from the estate of Mrs. Marie Woodward in memory of her son, Henry Tynham Woodward. He was an alumnus of the Mathematics Department at San Jose State University and worked with research groups at NASA Ames.

Funding

This study was funded by SJSU Woodward Fund for Applied Mathematics.

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Authors and Affiliations

  1. San Jose State University, San Jose, CA, USA

    Jesse Geneson & Shen-Fu Tsai

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  1. Jesse Geneson
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  2. Shen-Fu Tsai
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Correspondence to Jesse Geneson.

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Appendix 1: MOZART-4 tables

Appendix 1: MOZART-4 tables

See Tables 3 and 4.

Table 3 Rank of each chemical species in MOZART-4 with respect to various centrality measures
Full size table
Table 4 Rank of edges in MOZART-4 with respect to edge degree, inverse edge eccentricity, inverse edge peripherality, inverse edge sum peripherality, and inverse Mostar index
Full size table

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Geneson, J., Tsai, SF. Peripherality in networks: theory and applications. J Math Chem 60, 1021–1079 (2022). https://doi.org/10.1007/s10910-022-01345-8

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  • DOI: https://doi.org/10.1007/s10910-022-01345-8

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