Maximizing Spectral Radius and Number of Spanning Trees in Bipartite Graphs
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.
KeywordsSpectral radius Ferrers graph Spanning trees Bipartite graph Resistance distance Laplacian
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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