Geodesic
On Trees as Star Complements in Regular Graphs
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 2, pp. 621-636

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Let G be a connected r-regular graph (r gt; 3) of order n with a tree of order t as a star complement for an eigenvalue µ ∉ −1, 0. It is shown that n ≤ 1/2 (r + 1)t − 2. Equality holds when G is the complement of the Clebsch graph (with µ = 1, r = 5, t = 6, n = 16).
Keywords: eigenvalue, regular graph, star complement, tree 
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Rowlinson, Peter. On Trees as Star Complements in Regular Graphs. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 2, pp. 621-636. http://geodesic.mathdoc.fr/item/DMGT_2020_40_2_a16/