On intersecting families of independent sets in trees

A family of sets is intersecting if every pair its intersect. star a with some element (a center) in each sets. The classical result Erdős, Ko, and Rado (1961) states that r-subsets [n] r≤n/2 has size at most star. Let G be graph, α(G) independence number, μ(G) the smallest maximal independent set G. We say r-EKR if, among all maximum-sized families vertices G, there In 2005 Holroyd Talbot conjectured graph for 1≤r≤μ(G)/2. verified conjecture 2011 chordal graphs containing an isolated vertex. For without it difficult to determine center largest star, which often necessary prove EKR. tree leaf property occurs on one leaves. proved T when r≤4, 2017 Borg other authors gave examples trees not having r≥5. split vertex degree least 3. spider exactly Here we spiders have r≤α(T), characterize leaves maximum pendant adjacent leaf. show r≤α(T). also consider two provide partial results locations centers their stars.