Abstract
Let G = (V,E) be a simple graph with n vertices, e edges and d1 be the highest degree. Further let λ i , i = 1,2,...,n be the non-increasing eigenvalues of the Laplacian matrix of the graph G. In this paper, we obtain the following result: For connected graph G, λ2 = λ3 = ... = λn-1 if and only if G is a complete graph or a star graph or a (d1,d1) complete bipartite graph.
Also we establish the following upper bound for the number of spanning trees of G on n, e and d1 only:
The equality holds if and only if G is a star graph or a complete graph. Earlier bounds by Grimmett [5], Grone and Merris [6], Nosal [11], and Kelmans [2] were sharp for complete graphs only. Also our bound depends on n, e and d1 only.
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This work was done while the author was doing postdoctoral research in LRI, Université Paris-XI, Orsay, France.
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Das, K.C. A Sharp Upper Bound for the Number of Spanning Trees of a Graph. Graphs and Combinatorics 23, 625–632 (2007). https://doi.org/10.1007/s00373-007-0758-4
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DOI: https://doi.org/10.1007/s00373-007-0758-4