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Maximizing Spectral Radius and Number of Spanning Trees in Bipartite Graphs

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Mathematical Programming and Game Theory

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Abstract

The problems of maximizing the spectral radius and the number of spanning trees in a class of bipartite graphs with certain degree constraints are considered. In both the problems, the optimal graph is conjectured to be a Ferrers graph. Known results towards the resolution of the conjectures are described. We give yet another proof of a formula due to Ehrenborg and van Willigenburg for the number of spanning trees in a Ferrers graph. The main tool is a result which gives several necessary and sufficient conditions under which the removal of an edge in a graph does not affect the resistance distance between the end vertices of another edge.

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Acknowledgements

I sincerely thank Ranveer Singh for a careful reading of the manuscript. Support from the JC Bose Fellowship, Department of Science and Technology, Government of India, is gratefully acknowledged.

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Authors and Affiliations

  1. Indian Statistical Institute, 7, S. J. S Sansanwal Marg, New Delhi, 110016, India

    Ravindra B. Bapat

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  1. Ravindra B. Bapat
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Correspondence to Ravindra B. Bapat .

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Editors and Affiliations

  1. Indian Statistical Institute, New Delhi, India

    S.K. Neogy

  2. Indian Statistical Institute, New Delhi, India

    Ravindra B. Bapat

  3. Indian Statistical Institute, New Delhi, India

    Dipti Dubey

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Bapat, R.B. (2018). Maximizing Spectral Radius and Number of Spanning Trees in Bipartite Graphs. In: Neogy, S., Bapat, R., Dubey, D. (eds) Mathematical Programming and Game Theory. Indian Statistical Institute Series. Springer, Singapore. https://doi.org/10.1007/978-981-13-3059-9_2

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