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| Paper IPM / M / 15372 |
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| Abstract: | |
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A starlike tree is a tree with exactly one vertex of degree greater than two. The spectral radius of a graph G, that is denoted by λ(G), is the largest eigenvalue of G.
Let k and n1,…,nk be some positive integers. Let T(n1,…,nk) be the tree T ( T is a path or a starlike tree) such that T has a vertex v so that T\v is the disjoint union of the paths Pn1−1,…,Pnk−1 where every neighbor of v in T has degree one or two.
Let P=(p1,…,pk) and
Q=(q1,…,qk), where p1 ≥ … ≥ pk ≥ 1 and
q1 ≥ … ≥ qk ≥ 1 are integer. We say P majorizes Q and let P\succeqM Q,
if for every j, 1 ≤ j ≤ k,
∑i=1jpi ≥ ∑i=1jqi, with equality if j=k.
In this paper we show that if P majorizes Q, that is (p1,…,pk)\succeqM(q1,…,qk), then λ(T(q1,…,qk)) ≥ λ(T(p1,…,pk)).
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