Reinhart's extremal normalized distance Laplacian conjecture

Let GG be a graph on nn vertices, and let DL(G)\mathcal{D}^\mathcal{L}(G) be its normalized distance Laplacian matrix. Write KPKn1,n2,n3KPK_{n_1,n_2,n_3} for the graph obtained by connecting two cliques on n1n_1 and n3n_3 vertices through a path on n2n_2 vertices. Reinhart's extremal conjecture. The maximum spectral radius of DL\mathcal{D}^\mathcal{L} among graphs on nn vertices tends to 22 as nn\to\infty, and this maximum is achieved by KPKn1,n2,n3KPK_{n_1,n_2,n_3} for some n1+n2+n3=n+2n_1+n_2+n_3=n+2.

This conjecture concerns the asymptotic extremal behavior of the normalized distance Laplacian spectral radius and identifies a proposed family of extremal graphs. The supplied source does not indicate whether the conjecture has been resolved.

Sources & referencesView supporting material

Primary source

Jacob Johnston and Michael Tait, “Extremal values for the spectral radius of the normalized distance Laplacian”, arXiv:2302.11459 (2023).

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